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# Systems of Equations and Inequalities
## Solving Systems with Gaussian Elimination
### Learning Objectives
1. Use row operations on a matrix (IA 4.5.2)
2. Solve systems of equations using matrices (IA 4.5.3)
### Objective 1: Use row operations on a matrix (IA 4.5.2)
In the last section, we learned how to write the augmented matrix for a system of equations.
Once a system of equations is in its augmented matrix form, we will solve by elimination by performing operations on the rows that will lead us to the solution. Our goal will be to get 1 on the diagonal of the matrix and all entries below the diagonal must be zeros.
### Row Operations
In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.
1. Interchange any two rows.
2. Multiply a row by any real number except 0.
3. Add a nonzero multiple of one row to another row.
These actions are called row operations and will help us use the matrix to solve a system of equations.
### Practice Makes Perfect
### Objective 2: Solve systems of equations using matrices (IA 4.5.3)
To solve a system of equations using matrices, we transform the augmented matrix into a matrix in row-echelon form using row operations. For a consistent and independent system of equations, the augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.
Once we get the augmented matrix into row-echelon form, we can write the equivalent system of equations and solve for at least one variable. We then substitute this value in another equation to continue to solve for the other variables.
### Practice Makes Perfect
Carl Friedrich Gauss lived during the late 18th century and early 19th century, but he is still considered one of the most prolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.
We first encountered Gaussian elimination in Systems of Linear Equations: Two Variables. In this section, we will revisit this technique for solving systems, this time using matrices.
### Writing the Augmented Matrix of a System of Equations
A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
For example, consider the following
system of equations.
We can write this system as an augmented matrix:
We can also write a matrix containing just the coefficients. This is called the coefficient matrix.
A three-by-three system of equations such as
has a coefficient matrix
and is represented by the augmented matrix
Notice that the matrix is written so that the variables line up in their own columns: x-terms go in the first column, y-terms in the second column, and z-terms in the third column. It is very important that each equation is written in standard form
so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.
### Writing a System of Equations from an Augmented Matrix
We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.
### Performing Row Operations on a Matrix
Now that we can write systems of equations in augmented matrix form, we will examine the various row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows.
Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown.
We use row operations corresponding to equation operations to obtain a new matrix that is row-equivalent in a simpler form. Here are the guidelines to obtaining row-echelon form.
1. In any nonzero row, the first nonzero number is a 1. It is called a leading 1.
2. Any all-zero rows are placed at the bottom on the matrix.
3. Any leading 1 is below and to the right of a previous leading 1.
4. Any column containing a leading 1 has zeros in all other positions in the column.
To solve a system of equations we can perform the following row operations to convert the coefficient matrix to row-echelon form and do back-substitution to find the solution.
1. Interchange rows. (Notation:
)
2. Multiply a row by a constant. (Notation:
)
3. Add the product of a row multiplied by a constant to another row. (Notation:
Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in row-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method that uses row operations to obtain a 1 as the first entry so that row 1 can be used to convert the remaining rows.
### Solving a System of Linear Equations Using Matrices
We have seen how to write a system of equations with an augmented matrix, and then how to use row operations and back-substitution to obtain row-echelon form. Now, we will take row-echelon form a step farther to solve a 3 by 3 system of linear equations. The general idea is to eliminate all but one variable using row operations and then back-substitute to solve for the other variables.
### Key Concepts
1. An augmented matrix is one that contains the coefficients and constants of a system of equations. See .
2. A matrix augmented with the constant column can be represented as the original system of equations. See .
3. Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.
4. We can use Gaussian elimination to solve a system of equations. See , , and .
5. Row operations are performed on matrices to obtain row-echelon form. See .
6. To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions. See and .
7. A calculator can be used to solve systems of equations using matrices. See .
8. Many real-world problems can be solved using augmented matrices. See and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, write the augmented matrix for the linear system.
For the following exercises, write the linear system from the augmented matrix.
For the following exercises, solve the system by Gaussian elimination.
### Extensions
For the following exercises, use Gaussian elimination to solve the system.
### Real-World Applications
For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. |
# Systems of Equations and Inequalities
## Solving Systems with Inverses
### Learning Objectives
1. Evaluate the determinant of a 2×2 matrix (IA 4.6.1)
2. Evaluate the determinant of a 3x3 matrix (IA 4.6.2)
### Objective 1: Evaluate the determinant of a 2×2 matrix (IA 4.6.1)
If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant.
### Practice Makes Perfect
Find the determinant of the 2x2 matrices.
### Objective 2: Evaluate the determinant of a 3×3 matrix (IA 4.6.2)
To evaluate the determinant of a 3×3 matrix, we must be able to evaluate the minor of an entry in the determinant.
The minor of an entry is the 2×2 determinant found by eliminating the row and column in the 3×3 determinant that contains the entry.
For example, to find the minor of entry a1, we eliminate the row and column which contain it. So, we eliminate the first row and first column. Then we write the 2×2 determinant that remains.
To find the minor of entry b2, we eliminate the row and column that contain it. So, we eliminate the second row and second column. Then we write the 2×2 determinant that remains.
### Strategy for evaluating the determinant of a 3x3 matrix
To evaluate a 3×3 determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier.
When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.
### Expanding by minors along the first row to evaluate a 3x3 determinant.
To evaluate a 3×3 determinant by expanding by minors along the first row, we use the following pattern:
NOTE: We can evaluate the determinant of a matrix by expanding minors along any row or column. When a row or a column has a zero entry, expanding by that row or column results in less calculations.
### Practice Makes Perfect
Soriya plans to invest $10,500 into two different bonds to spread out her risk. The first bond has an annual return of 10%, and the second bond has an annual return of 6%. In order to receive an 8.5% return from the two bonds, how much should Soriya invest in each bond? What is the best method to solve this problem?
There are several ways we can solve this problem. As we have seen in previous sections, systems of equations and matrices are useful in solving real-world problems involving finance. After studying this section, we will have the tools to solve the bond problem using the inverse of a matrix.
### Finding the Inverse of a Matrix
We know that the multiplicative inverse of a real number
is
and
For example,
and
The multiplicative inverse of a matrix is similar in concept, except that the product of matrix
and its inverse
equals the identity matrix. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. We identify identity matrices by
where
represents the dimension of the matrix. Observe the following equations.
The identity matrix acts as a 1 in matrix algebra. For example,
A matrix that has a multiplicative inverse has the properties
A matrix that has a multiplicative inverse is called an invertible matrix. Only a square matrix may have a multiplicative inverse, as the reversibility,
is a requirement. Not all square matrices have an inverse, but if
is invertible, then
is unique. We will look at two methods for finding the inverse of a
matrix and a third method that can be used on both
and
matrices.
### Finding the Multiplicative Inverse Using Matrix Multiplication
We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication.
### Finding the Multiplicative Inverse by Augmenting with the Identity
Another way to find the multiplicative inverse is by augmenting with the identity. When matrix
is transformed into
the augmented matrix
transforms into
For example, given
augment
with the identity
Perform row operations with the goal of turning
into the identity.
1. Switch row 1 and row 2.
2. Multiply row 2 by
and add to row 1.
3. Multiply row 1 by
and add to row 2.
4. Add row 2 to row 1.
5. Multiply row 2 by
The matrix we have found is
### Finding the Multiplicative Inverse of 2×2 Matrices Using a Formula
When we need to find the multiplicative inverse of a
matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.
If
is a
matrix, such as
the multiplicative inverse of
is given by the formula
where
If
then
has no inverse.
### Finding the Multiplicative Inverse of 3×3 Matrices
Unfortunately, we do not have a formula similar to the one for a
matrix to find the inverse of a
matrix. Instead, we will augment the original matrix with the identity matrix and use row operations to obtain the inverse.
Given a
matrix
augment
with the identity matrix
To begin, we write the augmented matrix with the identity on the right and
on the left. Performing elementary row operations so that the identity matrix appears on the left, we will obtain the inverse matrix on the right. We will find the inverse of this matrix in the next example.
### Solving a System of Linear Equations Using the Inverse of a Matrix
Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices:
is the matrix representing the variables of the system, and
is the matrix representing the constants. Using matrix multiplication, we may define a system of equations with the same number of equations as variables as
To solve a system of linear equations using an inverse matrix, let
be the coefficient matrix, let
be the variable matrix, and let
be the constant matrix. Thus, we want to solve a system
For example, look at the following system of equations.
From this system, the coefficient matrix is
The variable matrix is
And the constant matrix is
Then
looks like
Recall the discussion earlier in this section regarding multiplying a real number by its inverse,
To solve a single linear equation
for
we would simply multiply both sides of the equation by the multiplicative inverse (reciprocal) of
Thus,
The only difference between a solving a linear equation and a system of equations written in matrix form is that finding the inverse of a matrix is more complicated, and matrix multiplication is a longer process. However, the goal is the same—to isolate the variable.
We will investigate this idea in detail, but it is helpful to begin with a
system and then move on to a
system.
### Key Equations
### Key Concepts
1. An identity matrix has the property
See .
2. An invertible matrix has the property
See .
3. Use matrix multiplication and the identity to find the inverse of a
matrix. See .
4. The multiplicative inverse can be found using a formula. See .
5. Another method of finding the inverse is by augmenting with the identity. See .
6. We can augment a
matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse. See .
7. Write the system of equations as
and multiply both sides by the inverse of
See and .
8. We can also use a calculator to solve a system of equations with matrix inverses. See .
### Section Exercises
### Verbal
### Algebraic
In the following exercises, show that matrix
is the inverse of matrix
For the following exercises, find the multiplicative inverse of each matrix, if it exists.
For the following exercises, solve the system using the inverse of a
matrix.
For the following exercises, solve a system using the inverse of a
matrix.
### Technology
For the following exercises, use a calculator to solve the system of equations with matrix inverses.
### Extensions
For the following exercises, find the inverse of the given matrix.
### Real-World Applications
For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. |
# Systems of Equations and Inequalities
## Solving Systems with Cramer's Rule
### Learning Objectives
1. Use Cramer’s Rule to solve systems of equations (IA 4.6.3)
### Objective 1: Use Cramer’s Rule to solve systems of equations (IA 4.6.3)
Cramer’s Rule uses determinants to solve systems of equations.
### Practice Makes Perfect
### Practice Makes Perfect
We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.
### Evaluating the Determinant of a 2×2 Matrix
A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an invertible matrix and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.
### Using Cramer’s Rule to Solve a System of Two Equations in Two Variables
We will now introduce a final method for solving systems of equations that uses determinants. Known as Cramer’s Rule, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750 in Introduction à l'Analyse des lignes Courbes algébriques. Cramer’s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.
Cramer’s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.
To understand Cramer’s Rule, let’s look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables.
We eliminate one variable using row operations and solve for the other. Say that we wish to solve for
If equation (2) is multiplied by the opposite of the coefficient of
in equation (1), equation (1) is multiplied by the coefficient of
in equation (2), and we add the two equations, the variable
will be eliminated.
Now, solve for
Similarly, to solve for
we will eliminate
Solving for
gives
Notice that the denominator for both
and
is the determinant of the coefficient matrix.
We can use these formulas to solve for
and
but Cramer’s Rule also introduces new notation:
1.
determinant of the coefficient matrix
2.
determinant of the numerator in the solution of
3.
determinant of the numerator in the solution of
The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express
and
as a quotient of two determinants.
### Evaluating the Determinant of a 3 × 3 Matrix
Finding the determinant of a 2×2 matrix is straightforward, but finding the determinant of a 3×3 matrix is more complicated. One method is to augment the 3×3 matrix with a repetition of the first two columns, giving a 3×5 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.
Find the determinant of the 3×3 matrix.
1. Augment
with the first two columns.
2. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.
3. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.
The algebra is as follows:
### Using Cramer’s Rule to Solve a System of Three Equations in Three Variables
Now that we can find the determinant of a 3 × 3 matrix, we can apply Cramer’s Rule to solve a system of three equations in three variables. Cramer’s Rule is straightforward, following a pattern consistent with Cramer’s Rule for 2 × 2 matrices. As the order of the matrix increases to 3 × 3, however, there are many more calculations required.
When we calculate the determinant to be zero, Cramer’s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.
Consider a 3 × 3 system of equations.
where
If we are writing the determinant
we replace the
column with the constant column. If we are writing the determinant
we replace the
column with the constant column. If we are writing the determinant
we replace the
column with the constant column. Always check the answer.
### Understanding Properties of Determinants
There are many properties of determinants. Listed here are some properties that may be helpful in calculating the determinant of a matrix.
### Key Concepts
1. The determinant for
is
See .
2. Cramer’s Rule replaces a variable column with the constant column. Solutions are
See .
3. To find the determinant of a 3×3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). See .
4. To solve a system of three equations in three variables using Cramer’s Rule, replace a variable column with the constant column for each desired solution:
See .
5. Cramer’s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. See and .
6. Certain properties of determinants are useful for solving problems. For example:
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the determinant.
For the following exercises, solve the system of linear equations using Cramer’s Rule.
For the following exercises, solve the system of linear equations using Cramer’s Rule.
### Technology
For the following exercises, use the determinant function on a graphing utility.
### Real-World Applications
For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.
For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.
For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown in .
### Review Exercises
### Systems of Linear Equations: Two Variables
For the following exercises, determine whether the ordered pair is a solution to the system of equations.
For the following exercises, use substitution to solve the system of equations.
For the following exercises, use addition to solve the system of equations.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.
### Systems of Linear Equations: Three Variables
For the following exercises, solve the system of three equations using substitution or addition.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.
### Systems of Nonlinear Equations and Inequalities: Two Variables
For the following exercises, solve the system of nonlinear equations.
For the following exercises, graph the inequality.
For the following exercises, graph the system of inequalities.
### Partial Fractions
For the following exercises, decompose into partial fractions.
### Matrices and Matrix Operations
For the following exercises, perform the requested operations on the given matrices.
### Solving Systems with Gaussian Elimination
For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.
For the following exercises, write the augmented matrix from the system of linear equations.
For the following exercises, solve the system of linear equations using Gaussian elimination.
### Solving Systems with Inverses
For the following exercises, find the inverse of the matrix.
For the following exercises, find the solutions by computing the inverse of the matrix.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.
### Solving Systems with Cramer's Rule
For the following exercises, find the determinant.
For the following exercises, use Cramer’s Rule to solve the linear systems of equations.
### Practice Test
Is the following ordered pair a solution to the system of equations?
For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.
For the following exercises, graph the following inequalities.
For the following exercises, write the partial fraction decomposition.
For the following exercises, perform the given matrix operations.
For the following exercises, use Gaussian elimination to solve the systems of equations.
For the following exercises, use the inverse of a matrix to solve the systems of equations.
For the following exercises, use Cramer’s Rule to solve the systems of equations.
For the following exercises, solve using a system of linear equations. |
# Analytic Geometry
## Introduction to Analytic Geometry
The Greek mathematician Menaechmus (c. 380–c. 320 BCE) is generally credited with discovering the shapes formed by the intersection of a plane and a right circular cone. Depending on how he tilted the plane when it intersected the cone, he formed different shapes at the intersection–beautiful shapes with near-perfect symmetry.
It was also said that Aristotle may have had an intuitive understanding of these shapes, as he observed the orbit of the planet to be circular. He presumed that the planets moved in circular orbits around Earth, and for nearly 2000 years this was the commonly held belief.
It was not until the Renaissance movement that Johannes Kepler noticed that the orbits of the planet were not circular in nature. His published law of planetary motion in the 1600s changed our view of the solar system forever. He claimed that the sun was at one end of the orbits, and the planets revolved around the sun in an oval-shaped path.
Other objects in the solar system (and perhaps other systems) follow a similar elliptical path, including the spectacular rings of Saturn. Using this understanding as a basis, 19th century mathematicians like James Clerk Maxwell and Sofya Kovalevskaya showed that despite their appearance through the telescopes of the day (and even in current telescopes), the rings are not solid and continuous, but are rather composed of small particles. Even after the Voyager and Cassini missions have provided close-up and detailed data regarding the ring structures, full understanding of their construction relies heavily on mathematical analysis. Of particular interest are the influences of Saturn's moons and moonlets, and the ways they both disrupt and preserve the ring structure.
In this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected by a plane. We will begin by studying each of three figures created in this manner. We will develop defining equations for each figure and then learn how to use these equations to solve a variety of problems. |
# Analytic Geometry
## The Ellipse
### Learning Objectives
1. Complete the square of a binomial expression. (IA 9.2.1)
2. Graph a circle. (IA 11.1.4)
### Objective 1: Complete the square of a binomial expression. (IA 9.2.1)
But what happens if we have to solve an equation where the trinomial is not a perfect square?
For example, ? For these types of equations, we can use a process called completing the square.
Recall .
We can use the Binomial Squares Pattern to make a perfect square.
### Practice Makes Perfect
Determine what number would have to be added to the given terms to create a perfect square trinomial. Then rewrite as a binomial squared.
### Objective 2: Graph a circle. (IA 11.1.4)
A circle is all points in a plane that are a fixed distance from a given point in the plane. The given point is called the center, (h, k) and the fixed distance is called the radius, r, of the circle.
The standard or graphing form of the equation of a circle with center, (h, k) and radius, r, is .
The general form of the equation of a circle is . If we are given an equation in general form, we can change it to standard, also called the graphing form, by completing the squares in both x and y. Then we can graph the circle using its center and radius.
### Practice Makes Perfect
Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in , is such a room.Architect of the Capitol. http://www.aoc.gov. Accessed April 15, 2014. It is an semi-circular room called a whispering chamber because the shape makes it possible for sound to travel along the walls and dome. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.
### Writing Equations of Ellipses in Standard Form
A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in .
Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points
in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See .
Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See .
In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the x- and y-axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.
To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.
### Deriving the Equation of an Ellipse Centered at the Origin
To derive the equation of an ellipse centered at the origin, we begin with the foci
and
The ellipse is the set of all points
such that the sum of the distances from
to the foci is constant, as shown in .
If
is a vertex of the ellipse, the distance from
to
is
The distance from
to
is
. The sum of the distances from the foci to the vertex is
If
is a point on the ellipse, then we can define the following variables:
By the definition of an ellipse,
is constant for any point
on the ellipse. We know that the sum of these distances is
for the vertex
It follows that
for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.
Thus, the standard equation of an ellipse is
This equation defines an ellipse centered at the origin. If
the ellipse is stretched further in the horizontal direction, and if
the ellipse is stretched further in the vertical direction.
### Writing Equations of Ellipses Centered at the Origin in Standard Form
Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we’ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.
The key features of the ellipse are its center, vertices, co-vertices, foci, and lengths and positions of the major and minor axes. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.
### Writing Equations of Ellipses Not Centered at the Origin
Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translated
units horizontally and
units vertically, the center of the ellipse will be
This translation results in the standard form of the equation we saw previously, with
replaced by
and y replaced by
### Graphing Ellipses Centered at the Origin
Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form
for horizontal ellipses and
for vertical ellipses.
### Graphing Ellipses Not Centered at the Origin
When an ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point,
we use the standard forms
for horizontal ellipses and
for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.
### Solving Applied Problems Involving Ellipses
Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See . In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper. When these chambers are placed in unexpected places, such as the ones inside Bush International Airport in Houston and Grand Central Terminal in New York City, they can induce surprised reactions among travelers.
### Key Equations
### Key Concepts
1. An ellipse is the set of all points
in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
2. When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See and .
3. When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See and .
4. When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See and .
5. Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.
For the following exercises, find the foci for the given ellipses.
### Graphical
For the following exercises, graph the given ellipses, noting center, vertices, and foci.
For the following exercises, use the given information about the graph of each ellipse to determine its equation.
For the following exercises, given the graph of the ellipse, determine its equation.
### Extensions
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula
### Real-World Applications
|
# Analytic Geometry
## The Hyperbola
### Learning Objectives
1. Use the Distance Formula. (IA 11.1.1)
2. Graph a hyperbola with center at (0,0). (IA 11.4.1)
### Objective 1: Use the Distance Formula. (IA 11.1.1)
### Practice Makes Perfect
Use the Distance Formula.
### Objective 2: Graph a hyperbola with center at (0,0). (IA 11.4.1)
A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola.
The line through the foci is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola.
Notice that, unlike the equation of an ellipse, the denominator of is not always and the denominator of is not always .
Notice that when the term is positive, the transverse axis is on the x-axis. When the term is positive, the transverse axis is on the y-axis.
### Practice Makes Perfect
Graph a hyperbola with center at (0,0).
What do paths of comets, supersonic booms, ancient Grecian pillars, and natural draft cooling towers have in common? They can all be modeled by the same type of conic. For instance, when something moves faster than the speed of sound, a shock wave in the form of a cone is created. A portion of a conic is formed when the wave intersects the ground, resulting in a sonic boom. See .
Most people are familiar with the sonic boom created by supersonic aircraft, but humans were breaking the sound barrier long before the first supersonic flight. The crack of a whip occurs because the tip is exceeding the speed of sound. The bullets shot from many firearms also break the sound barrier, although the bang of the gun usually supersedes the sound of the sonic boom.
### Locating the Vertices and Foci of a Hyperbola
In analytic geometry, a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other. See .
Like the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all points
in a plane such that the difference of the distances between
and the foci is a positive constant.
Notice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the difference of two distances, whereas the ellipse is defined in terms of the sum of two distances.
As with the ellipse, every hyperbola has two axes of symmetry. The transverse axis is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The conjugate axis is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two asymptotes that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle. See .
In this section, we will limit our discussion to hyperbolas that are positioned vertically or horizontally in the coordinate plane; the axes will either lie on or be parallel to the x- and y-axes. We will consider two cases: those that are centered at the origin, and those that are centered at a point other than the origin.
### Deriving the Equation of a Hyperbola Centered at the Origin
Let
and
be the foci of a hyperbola centered at the origin. The hyperbola is the set of all points
such that the difference of the distances from
to the foci is constant. See .
If
is a vertex of the hyperbola, the distance from
to
is
The distance from
to
is
The difference of the distances from the foci to the vertex is
If
is a point on the hyperbola, we can define the following variables:
By definition of a hyperbola,
is constant for any point
on the hyperbola. We know that the difference of these distances is
for the vertex
It follows that
for any point on the hyperbola. As with the derivation of the equation of an ellipse, we will begin by applying the distance formula. The rest of the derivation is algebraic. Compare this derivation with the one from the previous section for ellipses.
This equation defines a hyperbola centered at the origin with vertices
and co-vertices
### Writing Equations of Hyperbolas in Standard Form
Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin.
### Hyperbolas Centered at the Origin
Reviewing the standard forms given for hyperbolas centered at
we see that the vertices, co-vertices, and foci are related by the equation
Note that this equation can also be rewritten as
This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices.
### Hyperbolas Not Centered at the Origin
Like the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translated
units horizontally and
units vertically, the center of the hyperbola will be
This translation results in the standard form of the equation we saw previously, with
replaced by
and
replaced by
Like hyperbolas centered at the origin, hyperbolas centered at a point
have vertices, co-vertices, and foci that are related by the equation
We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given.
### Graphing Hyperbolas Centered at the Origin
When we have an equation in standard form for a hyperbola centered at the origin, we can interpret its parts to identify the key features of its graph: the center, vertices, co-vertices, asymptotes, foci, and lengths and positions of the transverse and conjugate axes. To graph hyperbolas centered at the origin, we use the standard form
for horizontal hyperbolas and the standard form
for vertical hyperbolas.
### Graphing Hyperbolas Not Centered at the Origin
Graphing hyperbolas centered at a point
other than the origin is similar to graphing ellipses centered at a point other than the origin. We use the standard forms
for horizontal hyperbolas, and
for vertical hyperbolas. From these standard form equations we can easily calculate and plot key features of the graph: the coordinates of its center, vertices, co-vertices, and foci; the equations of its asymptotes; and the positions of the transverse and conjugate axes.
### Solving Applied Problems Involving Hyperbolas
As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy, physics, engineering, and architecture. The design efficiency of hyperbolic cooling towers is particularly interesting. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to generate power efficiently. Because of their hyperbolic form, these structures are able to withstand extreme winds while requiring less material than any other forms of their size and strength. See . For example, a 500-foot tower can be made of a reinforced concrete shell only 6 or 8 inches wide!
The first hyperbolic towers were designed in 1914 and were 35 meters high. Today, the tallest cooling towers are in France, standing a remarkable 170 meters tall. In we will use the design layout of a cooling tower to find a hyperbolic equation that models its sides.
### Key Equations
### Key Concepts
1. A hyperbola is the set of all points
in a plane such that the difference of the distances between
and the foci is a positive constant.
2. The standard form of a hyperbola can be used to locate its vertices and foci. See .
3. When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. See and .
4. When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. See and .
5. Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.
For the following exercises, find the equations of the asymptotes for each hyperbola.
### Graphical
For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.
For the following exercises, given information about the graph of the hyperbola, find its equation.
For the following exercises, given the graph of the hyperbola, find its equation.
### Extensions
For the following exercises, express the equation for the hyperbola as two functions, with
as a function of
Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.
### Real-World Applications
For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.
For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information. |
# Analytic Geometry
## The Parabola
### Learning Objectives
1. Graph vertical parabolas. (IA 11.2.1)
2. Graph horizontal parabolas. (IA 11.2.2)
### Objective 1: Graph vertical parabolas. (IA 11.2.1)
A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here.
### Practice Makes Perfect
Graph vertical parabolas.
### Objective 2: Graph horizontal parabolas. (IA 11.2.2)
Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the x and y in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.
### Practice Makes Perfect
Katherine Johnson is the pioneering NASA mathematician who was integral to the successful and safe flight and return of many human missions as well as satellites. Prior to the work featured in the movie Hidden Figures, she had already made major contributions to the space program. She provided trajectory analysis for the Mercury mission, in which Alan Shepard became the first American to reach space, and she and engineer Ted Sopinski authored a monumental paper regarding placing an object in a precise orbital position and having it return safely to Earth. Many of the orbits she determined were made up of parabolas, and her ability to combine different types of math enabled an unprecedented level of precision. Johnson said, "You tell me when you want it and where you want it to land, and I'll do it backwards and tell you when to take off."
Johnson's work on parabolic orbits and other complex mathematics resulted in successful orbits, Moon landings, and the development of the Space Shuttle program. Applications of parabolas are also critical to other areas of science. Parabolic mirrors (or reflectors) are able to capture energy and focus it to a single point. The advantages of this property are evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes, microphones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little maintenance. In this section we will explore the parabola and its uses, including low-cost, energy-efficient solar designs.
### Graphing Parabolas with Vertices at the Origin
In The Ellipse, we saw that an ellipse is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a parabola. See .
Like the ellipse and hyperbola, the parabola can also be defined by a set of points in the coordinate plane. A parabola is the set of all points
in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
In Quadratic Functions, we learned about a parabola’s vertex and axis of symmetry. Now we extend the discussion to include other key features of the parabola. See . Notice that the axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus.
The line segment that passes through the focus and is parallel to the directrix is called the latus rectum. The endpoints of the latus rectum lie on the curve. By definition, the distance
from the focus to any point
on the parabola is equal to the distance from
to the directrix.
To work with parabolas in the coordinate plane, we consider two cases: those with a vertex at the origin and those with a vertex at a point other than the origin. We begin with the former.
Let
be a point on the parabola with vertex
focus
and directrix
as shown in . The distance
from point
to point
on the directrix is the difference of the y-values:
The distance from the focus
to the point
is also equal to
and can be expressed using the distance formula.
Set the two expressions for
equal to each other and solve for
to derive the equation of the parabola. We do this because the distance from
to
equals the distance from
to
We then square both sides of the equation, expand the squared terms, and simplify by combining like terms.
The equations of parabolas with vertex
are
when the x-axis is the axis of symmetry and
when the y-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.
The key features of a parabola are its vertex, axis of symmetry, focus, directrix, and latus rectum. See . When given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the parabola.
A line is said to be tangent to a curve if it intersects the curve at exactly one point. If we sketch lines tangent to the parabola at the endpoints of the latus rectum, these lines intersect on the axis of symmetry, as shown in .
### Writing Equations of Parabolas in Standard Form
In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.
### Graphing Parabolas with Vertices Not at the Origin
Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated
units horizontally and
units vertically, the vertex will be
This translation results in the standard form of the equation we saw previously with
replaced by
and
replaced by
To graph parabolas with a vertex
other than the origin, we use the standard form
for parabolas that have an axis of symmetry parallel to the x-axis, and
for parabolas that have an axis of symmetry parallel to the y-axis. These standard forms are given below, along with their general graphs and key features.
### Solving Applied Problems Involving Parabolas
As we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the Olympic torch, have a very unique reflecting property. When rays of light parallel to the parabola’s axis of symmetry are directed toward any surface of the mirror, the light is reflected directly to the focus. See . This is why the Olympic torch is ignited when it is held at the focus of the parabolic mirror.
Parabolic mirrors have the ability to focus the sun’s energy to a single point, raising the temperature hundreds of degrees in a matter of seconds. Thus, parabolic mirrors are featured in many low-cost, energy efficient solar products, such as solar cookers, solar heaters, and even travel-sized fire starters.
### Key Equations
### Key Concepts
1. A parabola is the set of all points
in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
2. The standard form of a parabola with vertex
and the x-axis as its axis of symmetry can be used to graph the parabola. If
the parabola opens right. If
the parabola opens left. See .
3. The standard form of a parabola with vertex
and the y-axis as its axis of symmetry can be used to graph the parabola. If
the parabola opens up. If
the parabola opens down. See .
4. When given the focus and directrix of a parabola, we can write its equation in standard form. See .
5. The standard form of a parabola with vertex
and axis of symmetry parallel to the x-axis can be used to graph the parabola. If
the parabola opens right. If
the parabola opens left. See .
6. The standard form of a parabola with vertex
and axis of symmetry parallel to the y-axis can be used to graph the parabola. If
the parabola opens up. If
the parabola opens down. See .
7. Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.
For the following exercises, rewrite the given equation in standard form, and then determine the vertex
focus
and directrix
of the parabola.
### Graphical
For the following exercises, graph the parabola, labeling the focus and the directrix.
For the following exercises, find the equation of the parabola given information about its graph.
For the following exercises, determine the equation for the parabola from its graph.
### Extensions
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.
### Real-World Applications
|
# Analytic Geometry
## Rotation of Axes
### Learning Objectives
1. Using rotation of axes formulas.
2. Identify conic sections by their equations. (IA 11.4.3)
### Objective 1: Using rotation of axes formulas.
If a point
on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle
from the positive
x
-axis, then the coordinates of the point with respect to the new axes are
The following rotations of axes formulas define the relationship between (x,y) and (x’,y’):
### Practice Makes Perfect
Using rotation of axes formulas:
### Objective 2: Identify conic sections by their equations. (IA 11.4.3)
We can identify a conic from its equations by looking at the signs and coefficients of the variables that are squared.
### Practice Makes Perfect
Identify conic sections by their equations.
As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See .
Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in . A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.
### Identifying Nondegenerate Conics in General Form
In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.
where
and
are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.
You may notice that the general form equation has an
term that we have not seen in any of the standard form equations. As we will discuss later, the
term rotates the conic whenever
is not equal to zero.
### Finding a New Representation of the Given Equation after Rotating through a Given Angle
Until now, we have looked at equations of conic sections without an
term, which aligns the graphs with the x- and y-axes. When we add an
term, we are rotating the conic about the origin. If the x- and y-axes are rotated through an angle, say
then every point on the plane may be thought of as having two representations:
on the Cartesian plane with the original x-axis and y-axis, and
on the new plane defined by the new, rotated axes, called the x'-axis and y'-axis. See .
We will find the relationships between
and
on the Cartesian plane with
and
on the new rotated plane. See .
The original coordinate x- and y-axes have unit vectors
and
The rotated coordinate axes have unit vectors
and
The angle
is known as the angle of rotation. See . We may write the new unit vectors in terms of the original ones.
Consider a vector in the new coordinate plane. It may be represented in terms of its coordinate axes.
Because
we have representations of
and
in terms of the new coordinate system.
### Writing Equations of Rotated Conics in Standard Form
Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to transform the equation of a conic given in the form
into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in the
and
coordinate system without the
term, by rotating the axes by a measure of
that satisfies
We have learned already that any conic may be represented by the second degree equation
where
and
are not all zero. However, if
then we have an
term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle
where
### Identifying Conics without Rotating Axes
Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is
If we apply the rotation formulas to this equation we get the form
It may be shown that
The expression does not vary after rotation, so we call the expression invariant. The discriminant,
is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.
### Key Equations
### Key Concepts
1. Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
2. A nondegenerate conic section has the general form
where
and
are not all zero. The values of
and
determine the type of conic. See .
3. Equations of conic sections with an
term have been rotated about the origin. See .
4. The general form can be transformed into an equation in the
and
coordinate system without the
term. See and .
5. An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine which conic section is represented based on the given equation.
For the following exercises, find a new representation of the given equation after rotating through the given angle.
For the following exercises, determine the angle
that will eliminate the
term and write the corresponding equation without the
term.
### Graphical
For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.
For the following exercises, graph the equation relative to the
system in which the equation has no
term.
For the following exercises, determine the angle of rotation in order to eliminate the
term. Then graph the new set of axes.
For the following exercises, determine the value of
based on the given equation. |
# Analytic Geometry
## Conic Sections in Polar Coordinates
Most of us are familiar with orbital motion, such as the motion of a planet around the sun or an electron around an atomic nucleus. Within the planetary system, orbits of planets, asteroids, and comets around a larger celestial body are often elliptical. Comets, however, may take on a parabolic or hyperbolic orbit instead. And, in reality, the characteristics of the planets’ orbits may vary over time. Each orbit is tied to the location of the celestial body being orbited and the distance and direction of the planet or other object from that body. As a result, we tend to use polar coordinates to represent these orbits.
In an elliptical orbit, the periapsis is the point at which the two objects are closest, and the apoapsis is the point at which they are farthest apart. Generally, the velocity of the orbiting body tends to increase as it approaches the periapsis and decrease as it approaches the apoapsis. Some objects reach an escape velocity, which results in an infinite orbit. These bodies exhibit either a parabolic or a hyperbolic orbit about a body; the orbiting body breaks free of the celestial body’s gravitational pull and fires off into space. Each of these orbits can be modeled by a conic section in the polar coordinate system.
### Identifying a Conic in Polar Form
Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph. Consider the parabola
shown in .
In The Parabola, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line). In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus
at the pole, and a line, the directrix, which is perpendicular to the polar axis.
If
is a fixed point, the focus, and
is a fixed line, the directrix, then we can let
be a fixed positive number, called the eccentricity, which we can define as the ratio of the distances from a point on the graph to the focus and the point on the graph to the directrix. Then the set of all points
such that
is a conic. In other words, we can define a conic as the set of all points
with the property that the ratio of the distance from
to
to the distance from
to
is equal to the constant
For a conic with eccentricity
1. if
the conic is an ellipse
2. if
the conic is a parabola
3. if
the conic is an hyperbola
With this definition, we may now define a conic in terms of the directrix,
the eccentricity
and the angle
Thus, each conic may be written as a polar equation, an equation written in terms of
and
### Graphing the Polar Equations of Conics
When graphing in Cartesian coordinates, each conic section has a unique equation. This is not the case when graphing in polar coordinates. We must use the eccentricity of a conic section to determine which type of curve to graph, and then determine its specific characteristics. The first step is to rewrite the conic in standard form as we have done in the previous example. In other words, we need to rewrite the equation so that the denominator begins with 1. This enables us to determine
and, therefore, the shape of the curve. The next step is to substitute values for
and solve for
to plot a few key points. Setting
equal to
and
provides the vertices so we can create a rough sketch of the graph.
### Defining Conics in Terms of a Focus and a Directrix
So far we have been using polar equations of conics to describe and graph the curve. Now we will work in reverse; we will use information about the origin, eccentricity, and directrix to determine the polar equation.
### Key Concepts
1. Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus
at the pole, and a line, the directrix, which is perpendicular to the polar axis.
2. A conic is the set of all points
where eccentricity
is a positive real number. Each conic may be written in terms of its polar equation. See .
3. The polar equations of conics can be graphed. See , , and .
4. Conics can be defined in terms of a focus, a directrix, and eccentricity. See and .
5. We can use the identities
and
to convert the equation for a conic from polar to rectangular form. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.
For the following exercises, convert the polar equation of a conic section to a rectangular equation.
For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.
### Extensions
Recall from Rotation of Axes that equations of conics with an
term have rotated graphs. For the following exercises, express each equation in polar form with
as a function of
### Chapter Review Exercises
### The Ellipse
For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.
For the following exercises, graph the ellipse, noting center, vertices, and foci.
For the following exercises, use the given information to find the equation for the ellipse.
### The Hyperbola
For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.
For the following exercises, graph the hyperbola, labeling vertices and foci.
For the following exercises, find the equation of the hyperbola.
### The Parabola
For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.
For the following exercises, graph the parabola, labeling vertex, focus, and directrix.
For the following exercises, write the equation of the parabola using the given information.
### Rotation of Axes
For the following exercises, determine which of the conic sections is represented.
For the following exercises, determine the angle
that will eliminate the
term, and write the corresponding equation without the
term.
For the following exercises, graph the equation relative to the
system in which the equation has no
term.
### Conic Sections in Polar Coordinates
For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.
For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.
For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.
### Practice Test
For the following exercises, write the equation in standard form and state the center, vertices, and foci.
For the following exercises, sketch the graph, identifying the center, vertices, and foci.
For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.
For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.
For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.
For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.
For the following exercises, determine which conic section is represented by the given equation, and then determine the angle
that will eliminate the
term.
For the following exercises, rewrite in the
system without the
term, and graph the rotated graph.
For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.
For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci. |
# Sequences, Probability, and Counting Theory
## Introduction to Sequences, Probability and Counting Theory
A lottery winner has some big decisions to make regarding what to do with the winnings. Buy a new home? A luxury convertible? A cruise around the world?
The likelihood of winning the lottery is slim, but we all love to fantasize about what we could buy with the winnings. One of the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series of regular payments, called an annuity, over an extended period of time.
This decision is often based on many factors, such as tax implications, interest rates, and investment strategies. There are also personal reasons to consider when making the choice, and one can make many arguments for either decision. However, most lottery winners opt for the lump sum.
In this chapter, we will explore the mathematics behind situations such as these. We will take an in-depth look at annuities. We will also look at the branch of mathematics that would allow us to calculate the number of ways to choose lottery numbers and the probability of winning. |
# Sequences, Probability, and Counting Theory
## Sequences and Their Notations
### Learning Objectives
1. Write the first few terms of a sequence (IA 12.1.1)
2. Find a formula for the general term (nth term) of a sequence (IA 12.1.2)
### Objective 1: Write the first few terms of a sequence (IA 12.1.1).
A patient takes a 30 mg antibiotic capsule. At the end of that hour, the amount of antibiotic remaining in her body is only 90% of the amount in the beginning of that hour. The 30mg dose is taken at time t = 1 hour. How much of this dose remains at the end of 1 hour? 2hours? 3 hours? 4 hours?
This ordered list of numbers 27, 24.3, 21.87, 19.68, … is a sequence. Each number in the list is a term.
A sequence is a function whose domain is the counting numbers. A sequence may have an infinite number of terms or a finite number of terms. Our sequence has three dots (ellipsis) at the end which indicates the list never ends. If the domain is the set of all counting numbers, then the sequence is an infinite sequence.
Often when working with sequences we do not want to write out all the terms. We want a more compact way to show how each term is defined. When we worked with functions, we wrote and we said the expression 2x was the rule that defined values in the range.
While a sequence is a function, we do not use the usual function notation. Instead of writing the function as , we would write it as . The is the , the term in the nth position where n is a value in the domain. The formula for writing the nth term of the sequence is called the general term or formula of the sequence.
General sequence terms are denoted as follows:
### Practice Makes Perfect
Write the first few terms of a sequence.
### Objective 2: Find a formula for the general term (nth term) of a sequence (IA 12.1.2)
Sometimes we have a few terms of a sequence and it would be helpful to know the general term or . To find the general term, we look for patterns in the terms. Often the patterns involve multiples or powers. We also look for a pattern in the signs of the terms.
### Practice Makes Perfect
A video game company launches an exciting new advertising campaign. They predict the number of online visits to their website, or hits, will double each day. The model they are using shows 2 hits the first day, 4 hits the second day, 8 hits the third day, and so on. See .
If their model continues, how many hits will there be at the end of the month? To answer this question, we’ll first need to know how to determine a list of numbers written in a specific order. In this section, we will explore these kinds of ordered lists.
### Writing the Terms of a Sequence Defined by an Explicit Formula
One way to describe an ordered list of numbers is as a sequence. A sequence is a function whose domain is a subset of the counting numbers. The sequence established by the number of hits on the website is
The ellipsis (…) indicates that the sequence continues indefinitely. Each number in the sequence is called a term. The first five terms of this sequence are 2, 4, 8, 16, and 32.
Listing all of the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at the end of the month would require listing out as many as 31 terms. A more efficient way to determine a specific term is by writing a formula to define the sequence.
One type of formula is an explicit formula, which defines the terms of a sequence using their position in the sequence. Explicit formulas are helpful if we want to find a specific term of a sequence without finding all of the previous terms. We can use the formula to find the nth term of the sequence, where
is any positive number. In our example, each number in the sequence is double the previous number, so we can use powers of 2 to write a formula for the
term.
The first term of the sequence is
the second term is
the third term is
and so on. The
term of the sequence can be found by raising 2 to the
power. An explicit formula for a sequence is named by a lower case letter
with the subscript
The explicit formula for this sequence is
Now that we have a formula for the
term of the sequence, we can answer the question posed at the beginning of this section. We were asked to find the number of hits at the end of the month, which we will take to be 31 days. To find the number of hits on the last day of the month, we need to find the 31st term of the sequence. We will substitute 31 for
in the formula.
If the doubling trend continues, the company will get
hits on the last day of the month. That is over 2.1 billion hits! The huge number is probably a little unrealistic because it does not take consumer interest and competition into account. It does, however, give the company a starting point from which to consider business decisions.
Another way to represent the sequence is by using a table. The first five terms of the sequence and the
term of the sequence are shown in .
Graphing provides a visual representation of the sequence as a set of distinct points. We can see from the graph in that the number of hits is rising at an exponential rate. This particular sequence forms an exponential function.
Lastly, we can write this particular sequence as
A sequence that continues indefinitely is called an infinite sequence. The domain of an infinite sequence is the set of counting numbers. If we consider only the first 10 terms of the sequence, we could write
This sequence is called a finite sequence because it does not continue indefinitely.
### Investigating Alternating Sequences
Sometimes sequences have terms that are alternate. In fact, the terms may actually alternate in sign. The steps to finding terms of the sequence are the same as if the signs did not alternate. However, the resulting terms will not show increase or decrease as
increases. Let’s take a look at the following sequence.
Notice the first term is greater than the second term, the second term is less than the third term, and the third term is greater than the fourth term. This trend continues forever. Do not rearrange the terms in numerical order to interpret the sequence.
### Investigating Piecewise Explicit Formulas
We’ve learned that sequences are functions whose domain is over the positive integers. This is true for other types of functions, including some piecewise functions. Recall that a piecewise function is a function defined by multiple subsections. A different formula might represent each individual subsection.
### Finding an Explicit Formula
Thus far, we have been given the explicit formula and asked to find a number of terms of the sequence. Sometimes, the explicit formula for the
term of a sequence is not given. Instead, we are given several terms from the sequence. When this happens, we can work in reverse to find an explicit formula from the first few terms of a sequence. The key to finding an explicit formula is to look for a pattern in the terms. Keep in mind that the pattern may involve alternating terms, formulas for numerators, formulas for denominators, exponents, or bases.
### Writing the Terms of a Sequence Defined by a Recursive Formula
Sequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,…. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.
Each term of the Fibonacci sequence depends on the terms that come before it. The Fibonacci sequence cannot easily be written using an explicit formula. Instead, we describe the sequence using a recursive formula, a formula that defines the terms of a sequence using previous terms.
A recursive formula always has two parts: the value of an initial term (or terms), and an equation defining
in terms of preceding terms. For example, suppose we know the following:
We can find the subsequent terms of the sequence using the first term.
So the first four terms of the sequence are
.
The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms.
To find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We were told previously that the eighth and ninth terms are 21 and 34, so
### Using Factorial Notation
The formulas for some sequences include products of consecutive positive integers. , written as
is the product of the positive integers from 1 to
For example,
An example of formula containing a factorial is
The sixth term of the sequence can be found by substituting 6 for
The factorial of any whole number
is
We can therefore also think of
as
### Key Equations
### Key Concepts
1. A sequence is a list of numbers, called terms, written in a specific order.
2. Explicit formulas define each term of a sequence using the position of the term. See , , and .
3. An explicit formula for the
term of a sequence can be written by analyzing the pattern of several terms. See .
4. Recursive formulas define each term of a sequence using previous terms.
5. Recursive formulas must state the initial term, or terms, of a sequence.
6. A set of terms can be written by using a recursive formula. See and .
7. A factorial is a mathematical operation that can be defined recursively.
8. The factorial of
is the product of all integers from 1 to
See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, write the first four terms of the sequence.
For the following exercises, write the first eight terms of the piecewise sequence.
For the following exercises, write an explicit formula for each sequence.
For the following exercises, write the first five terms of the sequence.
For the following exercises, write the first eight terms of the sequence.
For the following exercises, write a recursive formula for each sequence.
For the following exercises, evaluate the factorial.
For the following exercises, write the first four terms of the sequence.
### Graphical
For the following exercises, graph the first five terms of the indicated sequence
For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.
For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.
### Technology
Follow these steps to evaluate a sequence defined recursively using a graphing calculator:
For the following exercises, use the steps above to find the indicated term or terms for the sequence.
Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.
Using a TI-83, do the following.
For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.
### Extensions
|
# Sequences, Probability, and Counting Theory
## Arithmetic Sequences
### Learning Objectives
1. Determine if a sequence is arithmetic (IA 12.2.1)
2. Find the general term (nth term) of an arithmetic sequence (IA 12.2.2)
### Objective 1: Determine if a sequence is arithmetic (IA 12.2.1)
An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.
The difference between consecutive terms, d, and is called the common difference, for n greater than or equal to two.
### Practice Makes Perfect
Determine if each sequence is arithmetic. If so, indicate the common difference.
### Practice Makes Perfect
### Objective 2: Find the general term (nth term) of an arithmetic sequence (IA 12.2.2)
In the last section, we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.
Let’s write the first few terms of a sequence where the first term is and the common difference is d. We will then look for a pattern.
As we look for a pattern we see that each term starts with .
The first term adds 0d to the , the second term adds 1d, the third term adds 2d, the fourth term adds 3d, and the fifth term adds 4d. The number of ds that were added to is one less than the number of the term. We then have the formula for the general term of an arithmetic sequence.
### Practice Makes Perfect
Find the general term (nth term) of an arithmetic sequence.
Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.
As an example, consider a woman who starts a small contracting business. She purchases a new truck for $25,000. After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years. The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years. In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck’s value.
### Finding Common Differences
The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is –3,400.
The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any term of the sequence, and add 3 to find the subsequent term.
### Writing Terms of Arithmetic Sequences
Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of
and
into formula below.
### Using Recursive Formulas for Arithmetic Sequences
Some arithmetic sequences are defined in terms of the previous term using a recursive formula. The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.
### Using Explicit Formulas for Arithmetic Sequences
We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.
To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.
The common difference is
, so the sequence represents a linear function with a slope of
. To find the
-intercept, we subtract
from
. You can also find the
-intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in .
Recall the slope-intercept form of a line is
When dealing with sequences, we use
in place of
and
in place of
If we know the slope and vertical intercept of the function, we can substitute them for
and
in the slope-intercept form of a line. Substituting
for the slope and
for the vertical intercept, we get the following equation:
We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is
, which simplifies to
### Finding the Number of Terms in a Finite Arithmetic Sequence
Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.
### Solving Application Problems with Arithmetic Sequences
In many application problems, it often makes sense to use an initial term of
instead of
In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:
### Key Equations
### Key Concepts
1. An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
2. The constant between two consecutive terms is called the common difference.
3. The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See .
4. The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See and .
5. A recursive formula for an arithmetic sequence with common difference
is given by
See .
6. As with any recursive formula, the initial term of the sequence must be given.
7. An explicit formula for an arithmetic sequence with common difference
is given by
See .
8. An explicit formula can be used to find the number of terms in a sequence. See .
9. In application problems, we sometimes alter the explicit formula slightly to
See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the common difference for the arithmetic sequence provided.
For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.
For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.
For the following exercises, write the first five terms of the arithmetic series given two terms.
For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.
For the following exercises, find the first term given two terms from an arithmetic sequence.
For the following exercises, find the specified term given two terms from an arithmetic sequence.
For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence.
For the following exercises, write a recursive formula for each arithmetic sequence.
For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.
For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.
For the following exercises, write an explicit formula for each arithmetic sequence.
For the following exercises, find the number of terms in the given finite arithmetic sequence.
### Graphical
For the following exercises, determine whether the graph shown represents an arithmetic sequence.
For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence.
### Technology
For the following exercises, follow the steps to work with the arithmetic sequence
using a graphing calculator:
For the following exercises, follow the steps given above to work with the arithmetic sequence
using a graphing calculator.
### Extensions
|
# Sequences, Probability, and Counting Theory
## Geometric Sequences
### Learning Objectives
1. Determine if a sequence is geometric (IA 12.3.1).
2. Find the general term (nth term) of a geometric sequence (IA 12.3.2).
### Objective 1: Determine if a sequence is geometric (IA 12.3.1)
A sequence is called a geometric sequence if the ratio between consecutive terms is always the same.
The ratio between consecutive terms in a geometric sequence is r, the common ratio, where n is greater than or equal to two.
### Practice Makes Perfect
Determine if each sequence is geometric. If so, indicate the common ratio.
### Practice Makes Perfect
### Objective 2: Find the general term (nth term) of a geometric sequence (IA 12.3.2)
Let’s find the formula for the general term of a geometric sequence.
Let’s write the first few terms of the sequence where the first term is and the common ratio is . We will then look for a pattern.
### Practice Makes Perfect
Find the general term (nth term) of a geometric sequence.
Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.
### Finding Common Ratios
The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.
### Writing Terms of Geometric Sequences
Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is
and the common ratio is
we can find subsequent terms by multiplying
to get
then multiplying the result
to get
and so on.
The first four terms are
### Using Recursive Formulas for Geometric Sequences
A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.
### Using Explicit Formulas for Geometric Sequences
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.
Let’s take a look at the sequence
This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is
The graph of the sequence is shown in .
### Solving Application Problems with Geometric Sequences
In real-world scenarios involving geometric sequences, we may need to use an initial term of
instead of
In these problems, we can alter the explicit formula slightly by using the following formula:
### Key Equations
### Key Concepts
1. A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
2. The constant ratio between two consecutive terms is called the common ratio.
3. The common ratio can be found by dividing any term in the sequence by the previous term. See .
4. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See and .
5. A recursive formula for a geometric sequence with common ratio
is given by
for
.
6. As with any recursive formula, the initial term of the sequence must be given. See .
7. An explicit formula for a geometric sequence with common ratio
is given by
See .
8. In application problems, we sometimes alter the explicit formula slightly to
See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the common ratio for the geometric sequence.
For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.
For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.
For the following exercises, write the first five terms of the geometric sequence, given any two terms.
For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.
For the following exercises, find the specified term for the geometric sequence, given the first four terms.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write a recursive formula for each geometric sequence.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write an explicit formula for each geometric sequence.
For the following exercises, find the specified term for the geometric sequence given.
For the following exercises, find the number of terms in the given finite geometric sequence.
### Graphical
For the following exercises, determine whether the graph shown represents a geometric sequence.
For the following exercises, use the information provided to graph the first five terms of the geometric sequence.
### Extensions
|
# Sequences, Probability, and Counting Theory
## Series and Their Notations
### Learning Objectives
1. Use summation notation to write a sum. (IA 12.1.5)
2. Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)
### Objective 1: Use summation notation to write a sum. (IA 12.1.5)
A series is the sum of the terms of a sequence. For example, 1 + 6 + 11+ 16 + 21 + 26 + 31 is the sum of the first seven terms arithmetic sequence with general term,
We write a series by using the summation notation. In order to write that summation, we will need to find the general term of our sequence and the summation will look like:
For the series, 1 + 6 + 11 + 16 + 21 + 26 + 31 + .... the summation notation is
### Practice Makes Perfect
Use summation notation to write the sum.
### Objective 2: Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)
### Practice Makes Perfect
A parent decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.
### Using Summation Notation
To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.
The of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation
represents the partial sum.
Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma,
to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series.
If we interpret the given notation, we see that it asks us to find the sum of the terms in the series
for
through
We can begin by substituting the terms for
and listing out the terms of this series.
We can find the sum of the series by adding the terms:
### Using the Formula for Arithmetic Series
Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference,
The sum of the terms of an arithmetic sequence is called an arithmetic series. We can write the sum of the first
terms of an arithmetic series as:
We can also reverse the order of the terms and write the sum as
If we add these two expressions for the sum of the first
terms of an arithmetic series, we can derive a formula for the sum of the first
terms of any arithmetic series.
Because there are
terms in the series, we can simplify this sum to
We divide by 2 to find the formula for the sum of the first
terms of an arithmetic series.
Use the formula to find the sum of each arithmetic series.
### Using the Formula for Geometric Series
Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio,
We can write the sum of the first
terms of a geometric series as
Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first
terms of a geometric series. We will begin by multiplying both sides of the equation by
Next, we subtract this equation from the original equation.
Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for
divide both sides by
Use the formula to find the indicated partial sum of each geometric series.
### Using the Formula for the Sum of an Infinite Geometric Series
Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first
terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is
This series can also be written in summation notation as
where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.
### Determining Whether the Sum of an Infinite Geometric Series is Defined
If the terms of an infinite geometric sequence approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:
The common ratio
As
gets very large, the values of
get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with
approach 0; the sum of a geometric series is defined when
Determine whether the sum of the infinite series is defined.
### Finding Sums of Infinite Series
When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first
terms of a geometric series.
We will examine an infinite series with
What happens to
as
increases?
The value of
decreases rapidly. What happens for greater values of
As
gets very large,
gets very small. We say that, as
increases without bound,
approaches 0. As
approaches 0,
approaches 1. When this happens, the numerator approaches
This give us a formula for the sum of an infinite geometric series.
Find the sum, if it exists.
### Solving Annuity Problems
At the beginning of the section, we looked at a problem in which a parent invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the parent invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.
We can find the value of the annuity right after the last deposit by using a geometric series with
and
After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.
We can find the value of the annuity after
deposits using the formula for the sum of the first
terms of a geometric series. In 6 years, there are 72 months, so
We can substitute
into the formula, and simplify to find the value of the annuity after 6 years.
After the last deposit, the parent will have a total of $4,320.44 in the account. Notice, the parent made 72 payments of $50 each for a total of
This means that because of the annuity, the parent earned $720.44 interest in their college fund.
### Key Equations
### Key Concepts
1. The sum of the terms in a sequence is called a series.
2. A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See .
3. The sum of the terms in an arithmetic sequence is called an arithmetic series.
4. The sum of the first
terms of an arithmetic series can be found using a formula. See and .
5. The sum of the terms in a geometric sequence is called a geometric series.
6. The sum of the first
terms of a geometric series can be found using a formula. See and .
7. The sum of an infinite series exists if the series is geometric with
8. If the sum of an infinite series exists, it can be found using a formula. See , , and .
9. An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, express each description of a sum using summation notation.
For the following exercises, express each arithmetic sum using summation notation.
For the following exercises, use the formula for the sum of the first
terms of each arithmetic sequence.
For the following exercises, express each geometric sum using summation notation.
For the following exercises, use the formula for the sum of the first
terms of each geometric sequence, and then state the indicated sum.
For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.
### Graphical
For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.
For the following exercises, use the geometric series
### Numeric
For the following exercises, find the indicated sum.
For the following exercises, use the formula for the sum of the first terms of an arithmetic series to find the sum.
For the following exercises, use the formula for the sum of the first
terms of a geometric series to find the partial sum.
For the following exercises, find the sum of the infinite geometric series.
For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.
### Extensions
### Real-World Applications
|
# Sequences, Probability, and Counting Theory
## Counting Principles
### Learning Objectives
1. Solve counting problems using the addition principle.
2. Solve counting problems using the multiplication principle.
### Objective 1: Solve counting problems using the addition principle.
In probability theory, an outcome is a possible result of an experiment or trial.
In probability theory, an event is a set of outcomes of an experiment.
Disjoint events cannot happen at the same time. In other words, they are mutually exclusive.
The addition principle applies when we are making only one selection.
### Practice Makes Perfect
Solve counting problems using the addition principle.
### Objective 2: Solve counting problems using the multiplication principle.
The Multiplication Principle applies when we are making more than one selection.
### Practice Makes Perfect
Solve counting problems using the multiplication principle.
A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!
We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.
### Using the Addition Principle
The company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. The Addition Principle tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. By the Addition Principle, there are 8 total options, as we can see in .
### Using the Multiplication Principle
The Multiplication Principle applies when we are making more than one selection. Suppose we are choosing an appetizer, an entrée, and a dessert. If there are 2 appetizer options, 3 entrée options, and 2 dessert options on a fixed-price dinner menu, there are a total of 12 possible choices of one each as shown in the tree diagram in .
The possible choices are:
1. soup, chicken, cake
2. soup, chicken, pudding
3. soup, fish, cake
4. soup, fish, pudding
5. soup, steak, cake
6. soup, steak, pudding
7. salad, chicken, cake
8. salad, chicken, pudding
9. salad, fish, cake
10. salad, fish, pudding
11. salad, steak, cake
12. salad, steak, pudding
We can also find the total number of possible dinners by multiplying.
We could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.
### Finding the Number of Permutations of n Distinct Objects
The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation.
### Finding the Number of Permutations of n Distinct Objects Using the Multiplication Principle
To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.
There are four options for the first place, so we write a 4 on the first line.
After the first place has been filled, there are three options for the second place so we write a 3 on the second line.
After the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally, we find the product.
There are 24 possible permutations of the paintings.
A family of five is having portraits taken. Use the Multiplication Principle to find the following.
### Finding the Number of Permutations of n Distinct Objects Using a Formula
For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of
objects and we want to choose
objects from the set in order, we write
Another way to write this is
a notation commonly seen on computers and calculators. To calculate
we begin by finding
the number of ways to line up all
objects. We then divide by
to cancel out the
items that we do not wish to line up.
Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is
Using factorials, we get the same result.
There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.
Note that the formula stills works if we are choosing all
objects and placing them in order. In that case we would be dividing by
or
which we said earlier is equal to 1. So the number of permutations of
objects taken
at a time is
or just
A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.
### Find the Number of Combinations Using the Formula
So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with combinations. A selection of
objects from a set of
objects where the order does not matter can be written as
Just as with permutations,
can also be written as
In this case, the general formula is as follows.
An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are
ways to order 3 paintings. There are
or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are not selecting 1 painting. There are 4 paintings we could choose not to select, so there are 4 ways to select 3 of the 4 paintings.
### Finding the Number of Subsets of a Set
We have looked only at combination problems in which we chose exactly
objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can be ordered. How many different pizzas are possible?
To answer this question, we need to consider pizzas with any number of toppings. There is
way to order a pizza with no toppings. There are
ways to order a pizza with exactly one topping. If we continue this process, we get
There are 32 possible pizzas. This result is equal to
We are presented with a sequence of choices. For each of the
objects we have two choices: include it in the subset or not. So for the whole subset we have made
choices, each with two options. So there are a total of
possible resulting subsets, all the way from the empty subset, which we obtain when we say “no” each time, to the original set itself, which we obtain when we say “yes” each time.
### Finding the Number of Permutations of n Non-Distinct Objects
We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the
permutations we counted are duplicates. The general formula for this situation is as follows.
In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are
ways to order the stars and
ways to order the moon.
There are 3,326,400 ways to order the sheet of stickers.
### Key Equations
### Key Concepts
1. If one event can occur in
ways and a second event with no common outcomes can occur in
ways, then the first or second event can occur in
ways. See .
2. If one event can occur in
ways and a second event can occur in
ways after the first event has occurred, then the two events can occur in
ways. See .
3. A permutation is an ordering of
objects.
4. If we have a set of
objects and we want to choose
objects from the set in order, we write
5. Permutation problems can be solved using the Multiplication Principle or the formula for
See and .
6. A selection of objects where the order does not matter is a combination.
7. Given
distinct objects, the number of ways to select
objects from the set is
and can be found using a formula. See .
8. A set containing
distinct objects has
subsets. See .
9. For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See .
### Section Exercises
### Verbal
For the following exercises, assume that there are
ways an event
can happen,
ways an event
can happen, and that
are non-overlapping.
Answer the following questions.
### Numeric
For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.
For the following exercises, compute the value of the expression.
For the following exercises, find the number of subsets in each given set.
For the following exercises, find the distinct number of arrangements.
### Extensions
### Real-World Applications
|
# Sequences, Probability, and Counting Theory
## Binomial Theorem
### Learning Objectives
1. Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)
### Objective 1: Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)
Pascal’s triangle helps us find the coefficients of the terms in the expansion of a binomial.
To find the coefficients of the terms, we write our expansion again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients. The array to the right is called Pascal’s Triangle.
Notice that in each expansion the powers of a in each term decrease from n to 0, and the powers of b increase from 0 to n.
Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.
To find the coefficients of the expansion of the binomial , go to the row that has the value n as a second entry.
### Practice Makes Perfect
Use Pascal’s Triangle to expand a binomial.
A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find
without multiplying the binomial by itself
times.
### Identifying Binomial Coefficients
In Counting Principles, we studied combinations. In the shortcut to finding
we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation
instead of
but it can be calculated in the same way. So
The combination
is called a binomial coefficient. An example of a binomial coefficient is
### Using the Binomial Theorem
When we expand
by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand
we might multiply
by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.
First, let’s examine the exponents. With each successive term, the exponent for
decreases and the exponent for
increases. The sum of the two exponents is
for each term.
Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:
These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.
Another way to see the coefficients is to examine the expansion of a binomial in general form,
to successive powers 1, 2, 3, and 4.
Can you guess the next expansion for the binomial
See , which illustrates the following:
1. There are
terms in the expansion of
2. The degree (or sum of the exponents) for each term is
3. The powers on
begin with
and decrease to 0.
4. The powers on
begin with 0 and increase to
5. The coefficients are symmetric.
To determine the expansion on
we see
thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of
the pattern is as follows:
1. Introduce
and then for each successive term reduce the exponent on
by 1 until
is reached.
2. Introduce
and then increase the exponent on
by 1 until
is reached.
The next expansion would be
But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as Pascal's Triangle, shown in . Pascal didn't invent the triangle. The underlying principles had been developed and written about for over 1500 years, first by the Indian mathematician (and poet) Pingala in the second century BCE. Others throughout Asia and Europe worked with the concepts throughout, and the triangle was first published in its graphical form by Omar Khayyam, an Iranian mathematician and astronomer, for whom the triangle is named in Iran. French mathematician Blaise Pascal repopularized it when he republished it and used it to solve a number of probability problems.
To generate Pascal’s Triangle, we start by writing a 1. In the row below, row 2, we write two 1’s. In the 3rd row, flank the ends of the rows with 1’s, and add
to find the middle number, 2. In the
row, flank the ends of the row with 1’s. Each element in the triangle is the sum of the two elements immediately above it.
To see the connection between Pascal’s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.
### Using the Binomial Theorem to Find a Single Term
Expanding a binomial with a high exponent such as
can be a lengthy process.
Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.
Note the pattern of coefficients in the expansion of
The second term is
The third term is
We can generalize this result.
### Key Equations
### Key Concepts
1.
is called a binomial coefficient and is equal to
See .
2. The Binomial Theorem allows us to expand binomials without multiplying. See .
3. We can find a given term of a binomial expansion without fully expanding the binomial. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, evaluate the binomial coefficient.
For the following exercises, use the Binomial Theorem to expand each binomial.
For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.
For the following exercises, find the indicated term of each binomial without fully expanding the binomial.
### Graphical
For the following exercises, use the Binomial Theorem to expand the binomial
Then find and graph each indicated sum on one set of axes.
### Extensions
|
# Sequences, Probability, and Counting Theory
## Probability
### Learning Objectives
1. Introduction to Sample Spaces and Computing Basic Probabilities.
### Objective 1: Introduction to Sample Spaces and Computing Basic Probabilities.
Many events in life are inherently uncertain: will it snow tomorrow? Am I going to get an ‘A’ in this course? None of these questions can be answered with certainty, however, we might say that some are unlikely, and others are more likely.
The probability of an event is a description of how likely it is that an event will happen. A probability is a number between 0 and 1 (that is, between 0% and 100%), where probabilities closer to 100% are very likely to occur, and probabilities closer to 0% are very unlikely to occur. A probability of 0% means the event is impossible, and a probability of 100% means the event will certainly occur.
A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. It is defined by its sample space, events within the sample space, and probabilities associated with each event.
The sample space S for a probability model is the set of all possible outcomes. For example, the sample space for rolling a dice is the set 1,2,3,4,5,6.This notation is referred to as roster notation.
An event A is a subset of the sample space S. For example, the event “Rolling an even number” is the subset 2,4,6.
To calculate the probability of an event, we divide the number of possible outcomes of the event by the number of possible outcomes of the sample space.
It is important to note that in order to use this formula, all outcomes must be equally likely to happen.
For example, the probability of rolling an even number with a standard dice is:
### Practice Makes Perfect
Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in . They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions.
### Constructing Probability Models
Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is An event is any subset of a sample space.
The likelihood of an event is known as probability. The probability of an event
is a number that always satisfies
where 0 indicates an impossible event and 1 indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like .
The sum of the probabilities listed in a probability model must equal 1, or 100%.
### Computing Probabilities of Equally Likely Outcomes
Let
be a sample space for an experiment. When investigating probability, an event is any subset of
When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in
Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in
so the probability of the event is
### Computing the Probability of the Union of Two Events
We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events
is the event that occurs if either or both events occur.
Suppose the spinner in is spun. We want to find the probability of spinning orange or spinning a
There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is
There are a total of 6 sections, and 2 of them have a
So the probability of spinning a
is
If we added these two probabilities, we would be counting the sector that is both orange and a
twice. To find the probability of spinning an orange or a
we need to subtract the probability that the sector is both orange and has a
The probability of spinning orange or a
is
### Computing the Probability of Mutually Exclusive Events
Suppose the spinner in is spun again, but this time we are interested in the probability of spinning an orange or a
There are no sectors that are both orange and contain a
so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is
Notice that with mutually exclusive events, the intersection of
and
is the empty set. The probability of spinning an orange is
and the probability of spinning a
is
We can find the probability of spinning an orange or a
simply by adding the two probabilities.
The probability of spinning an orange or a
is
### Using the Complement Rule to Compute Probabilities
We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event
denoted
is the set of outcomes in the sample space that are not in
For example, suppose we are interested in the probability that a horse will lose a race. If event
is the horse winning the race, then the complement of event
is the horse losing the race.
To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.
The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is
the probability of the horse losing the race is simply
### Computing Probability Using Counting Theory
Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.
Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are
ways to select 2 phones that are not defective. There are 8 phones, so there are
ways to select 2 phones. The probability of selecting 2 phones that are not defective is:
### Key Equations
### Key Concepts
1. Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
2. The probabilities in a probability model must sum to 1. See .
3. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. See .
4. To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See .
5. To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See .
6. The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See .
7. In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See .
### Section Exercises
### Verbal
### Numeric
For the following exercises, use the spinner shown in to find the probabilities indicated.
For the following exercises, two coins are tossed.
For the following exercises, four coins are tossed.
For the following exercises, one card is drawn from a standard deck of
cards. Find the probability of drawing the following:
For the following exercises, two dice are rolled, and the results are summed.
For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:
For the following exercises, use this scenario: a bag of M&Ms contains
blue,
brown,
orange,
yellow,
red, and
green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.
### Extensions
Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting
numbers from the numbers
to
After the player makes his selections,
winning numbers are randomly selected from numbers
to
A win occurs if the player has correctly selected
or
of the
winning numbers. (Round all answers to the nearest hundredth of a percent.)
### Real-World Applications
Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).United States Census Bureau. http://www.census.gov
### Chapter Review Exercises
### Sequences and Their Notation
### Arithmetic Sequences
### Geometric Sequences
### Series and Their Notation
### Counting Principles
### Binomial Theorem
### Probability
For the following exercises, assume two die are rolled.
For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)
### Practice Test
For the following exercises, use the spinner in . |
# Functions
## Introduction to Functions
Toward the end of the twentieth century, the values of stocks of internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. The graph above tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1,100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000.
Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties. |
# Functions
## Functions and Function Notation
A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.
### Determining Whether a Relation Represents a Function
A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.
The domain is
The range is
Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter
Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter
A function
is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain,
is paired with exactly one element in the range,
Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as
Notice that each element in the domain,
is not paired with exactly one element in the range,
For example, the term “odd” corresponds to three values from the range,
and the term “even” corresponds to two values from the range,
This violates the definition of a function, so this relation is not a function.
compares relations that are functions and not functions.
### Using Function Notation
Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.
To represent “height is a function of age,” we start by identifying the descriptive variables
for height and
for age. The letters
and
are often used to represent functions just as we use
and
to represent numbers and
and
to represent sets.
Remember, we can use any letter to name the function; the notation
shows us that
depends on
The value
must be put into the function
to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.
We can also give an algebraic expression as the input to a function. For example
means “first add a and b, and the result is the input for the function f.” The operations must be performed in this order to obtain the correct result.
### Representing Functions Using Tables
A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.
lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function
where
identifies months by an integer rather than by name.
defines a function
Remember, this notation tells us that
is the name of the function that takes the input
and gives the output
displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.
### Finding Input and Output Values of a Function
When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.
When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.
### Evaluation of Functions in Algebraic Forms
When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function
can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.
### Evaluating Functions Expressed in Formulas
Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation
expresses a functional relationship between
and
We can rewrite it to decide if
is a function of
### Evaluating a Function Given in Tabular Form
As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.
The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See .http://www.kgbanswers.com/how-long-is-a-dogs-memory-span/4221590. Accessed 3/24/2014.
At times, evaluating a function in table form may be more useful than using equations. Here let us call the function
The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function
at the input value of “goldfish.” We would write
Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function
seems ideally suited to this function, more so than writing it in paragraph or function form.
### Finding Function Values from a Graph
Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).
### Determining Whether a Function is One-to-One
Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.
However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in .
This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.
To visualize this concept, let’s look again at the two simple functions sketched in (a) and (b). The function in part (a) shows a relationship that is not a one-to-one function because inputs
and
both give output
The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.
### Using the Vertical Line Test
As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.
The most common graphs name the input value
and the output value
and we say
is a function of
or
when the function is named
The graph of the function is the set of all points
in the plane that satisfies the equation
If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in tell us that
and
However, the set of all points
satisfying
is a curve. The curve shown includes
and
because the curve passes through those points.
The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See .
### Using the Horizontal Line Test
Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.
### Identifying Basic Toolkit Functions
In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use
as the input variable and
as the output variable.
We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in .
### Key Equations
### Key Concepts
1. A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. See and .
2. Function notation is a shorthand method for relating the input to the output in the form
See and .
3. In tabular form, a function can be represented by rows or columns that relate to input and output values. See .
4. To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See and .
5. To solve for a specific function value, we determine the input values that yield the specific output value. See .
6. An algebraic form of a function can be written from an equation. See and .
7. Input and output values of a function can be identified from a table. See .
8. Relating input values to output values on a graph is another way to evaluate a function. See .
9. A function is one-to-one if each output value corresponds to only one input value. See .
10. A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See .
11. The graph of a one-to-one function passes the horizontal line test. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the relation represents a function.
For the following exercises, determine whether the relation represents
as a function of
For the following exercises, evaluate
at the indicated values
### Graphical
For the following exercises, use the vertical line test to determine which graphs show relations that are functions.
For the following exercises, determine if the given graph is a one-to-one function.
### Numeric
For the following exercises, determine whether the relation represents a function.
For the following exercises, determine if the relation represented in table form represents
as a function of
For the following exercises, use the function
represented in .
For the following exercises, evaluate the function
at the values
and
For the following exercises, evaluate the expressions, given functions
and
1.
2.
3.
### Technology
For the following exercises, graph
on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
on the given domain. Determine the corresponding range. Show each graph.
### Real-World Applications
|
# Functions
## Domain and Range
Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller/horror entries from the early 2000s—I am Legend, Hannibal, The Ring, The Grudge, and The Conjuring. shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.
### Finding the Domain of a Function Defined by an Equation
In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.
We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See .
We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write
We will discuss interval notation in greater detail later.
Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.
Before we begin, let us review the conventions of interval notation:
1. The smallest term from the interval is written first.
2. The largest term in the interval is written second, following a comma.
3. Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
4. Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.
See for a summary of interval notation.
### Using Notations to Specify Domain and Range
In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example,
describes the behavior of
in set-builder notation. The braces
are read as “the set of,” and the vertical bar | is read as “such that,” so we would read
as “the set of x-values such that 10 is less than or equal to
and
is less than 30.”
compares inequality notation, set-builder notation, and interval notation.
To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol,
to combine two unconnected intervals. For example, the union of the sets
and
is the set
It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is
### Finding Domain and Range from Graphs
Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See .
We can observe that the graph extends horizontally from
to the right without bound, so the domain is
The vertical extent of the graph is all range values
and below, so the range is
Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.
### Finding Domains and Ranges of the Toolkit Functions
We will now return to our set of toolkit functions to determine the domain and range of each.
### Graphing Piecewise-Defined Functions
Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function
With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.
If we input 0, or a positive value, the output is the same as the input.
If we input a negative value, the output is the opposite of the input.
Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.
We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income
would be
if
and
if
### Key Concepts
1. The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
2. The domain of a function can be determined by listing the input values of a set of ordered pairs. See .
3. The domain of a function can also be determined by identifying the input values of a function written as an equation. See , , and .
4. Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See .
5. For many functions, the domain and range can be determined from a graph. See and .
6. An understanding of toolkit functions can be used to find the domain and range of related functions. See , , and .
7. A piecewise function is described by more than one formula. See and .
8. A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the domain of each function using interval notation.
### Graphical
For the following exercises, write the domain and range of each function using interval notation.
For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.
### Numeric
For the following exercises, given each function
evaluate
and
For the following exercises, given each function
evaluate
and
For the following exercises, write the domain for the piecewise function in interval notation.
### Technology
### Extension
### Real-World Applications
|
# Functions
## Rates of Change and Behavior of Graphs
Gasoline costs have experienced some wild fluctuations over the last several decades. http://www.eia.gov/totalenergy/data/annual/showtext.cfm?t=ptb0524. Accessed 3/5/2014. lists the average cost, in dollars, of a gallon of gasoline for the years 2005–2012. The cost of gasoline can be considered as a function of year.
If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed per year. In this section, we will investigate changes such as these.
### Finding the Average Rate of Change of a Function
The price change per year is a rate of change because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the average rate of change over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.
The Greek letter
(delta) signifies the change in a quantity; we read the ratio as “delta-y over delta-x” or “the change in
divided by the change in
” Occasionally we write
instead of
which still represents the change in the function’s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.
In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was
On average, the price of gas increased by about 19.6¢ each year.
Other examples of rates of change include:
1. A population of rats increasing by 40 rats per week
2. A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)
3. A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)
4. The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage
5. The amount of money in a college account decreasing by $4,000 per quarter
### Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant
As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. shows examples of increasing and decreasing intervals on a function.
While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is the location of a local maximum. The function value at that point is the local maximum. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is the location of a local minimum. The function value at that point is the local minimum. The plural form is “local minima.” Together, local maxima and minima are called local extrema, or local extreme values, of the function. (The singular form is “extremum.”) Often, the term local is replaced by the term relative. In this text, we will use the term local.
Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.
For the function whose graph is shown in , the local maximum is 16, and it occurs at
The local minimum is
and it occurs at
To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. illustrates these ideas for a local maximum.
These observations lead us to a formal definition of local extrema.
### Analyzing the Toolkit Functions for Increasing or Decreasing Intervals
We will now return to our toolkit functions and discuss their graphical behavior in , , and .
### Use A Graph to Locate the Absolute Maximum and Absolute Minimum
There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The
coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively.
To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See .
Not every function has an absolute maximum or minimum value. The toolkit function
is one such function.
### Key Equations
### Key Concepts
1. A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. See .
2. Identifying points that mark the interval on a graph can be used to find the average rate of change. See .
3. Comparing pairs of input and output values in a table can also be used to find the average rate of change. See .
4. An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. See and .
5. The average rate of change can sometimes be determined as an expression. See .
6. A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See .
7. A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
8. A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
9. Minima and maxima are also called extrema.
10. We can find local extrema from a graph. See and .
11. The highest and lowest points on a graph indicate the maxima and minima. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the average rate of change of each function on the interval specified for real numbers
or
### Graphical
For the following exercises, consider the graph of
shown in .
For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.
For the following exercises, consider the graph shown in .
For the following exercises, consider the graph in .
### Numeric
For the following exercises, find the average rate of change of each function on the interval specified.
### Technology
For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.
### Extension
### Real-World Applications
|
# Functions
## Composition of Functions
Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.
Using descriptive variables, we can notate these two functions. The function
gives the cost
of heating a house for a given average daily temperature in
degrees Celsius. The function
gives the average daily temperature on day
of the year. For any given day,
means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature
For example, we could evaluate
to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the cost function at that temperature. We would write
By combining these two relationships into one function, we have performed function composition, which is the focus of this section.
### Combining Functions Using Algebraic Operations
Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.
Suppose we need to add two columns of numbers that represent a husband and wife’s separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that year’s incomes and then collecting all the data in a new column. If
is the wife’s income and
is the husband’s income in year
and we want
to represent the total income, then we can define a new function.
If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write
Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.
For two functions
and
with real number outputs, we define new functions
and
by the relations
### Create a Function by Composition of Functions
Performing algebraic operations on functions combines them into a new function, but we can also create functions by composing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output. The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a composite function. We represent this combination by the following notation:
We read the left-hand side as
composed with
at
and the right-hand side as
of
of
The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol
is called the composition operator. We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important not to confuse function composition with multiplication because, as we learned above, in most cases
It is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the equation above, the function
takes the input
first and yields an output
Then the function
takes
as an input and yields an output
In general,
and
are different functions. In other words, in many cases
for all
We will also see that sometimes two functions can be composed only in one specific order.
For example, if
and
then
but
These expressions are not equal for all values of
so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value
Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.
### Evaluating Composite Functions
Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner function’s output as the input for the outer function.
### Evaluating Composite Functions Using Tables
When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.
### Evaluating Composite Functions Using Graphs
When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the
and
axes of the graphs.
### Evaluating Composite Functions Using Formulas
When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.
While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition
To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like
we substitute the value inside the parentheses into the formula wherever we see the input variable.
### Finding the Domain of a Composite Function
As we discussed previously, the domain of a composite function such as
is dependent on the domain of
and the domain of
It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as
Let us assume we know the domains of the functions
and
separately. If we write the composite function for an input
as
we can see right away that
must be a member of the domain of
in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that
must be a member of the domain of
otherwise the second function evaluation in
cannot be completed, and the expression is still undefined. Thus the domain of
consists of only those inputs in the domain of
that produce outputs from
belonging to the domain of
Note that the domain of
composed with
is the set of all
such that
is in the domain of
and
is in the domain of
### Decomposing a Composite Function into its Component Functions
In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.
### Key Equation
### Key Concepts
1. We can perform algebraic operations on functions. See .
2. When functions are composed, the output of the first (inner) function becomes the input of the second (outer) function.
3. The function produced by composing two functions is a composite function. See and .
4. The order of function composition must be considered when interpreting the meaning of composite functions. See .
5. A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
6. A composite function can be evaluated from a table. See .
7. A composite function can be evaluated from a graph. See .
8. A composite function can be evaluated from a formula. See .
9. The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See and .
10. Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
11. Functions can often be decomposed in more than one way. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, use each pair of functions to find
and
Simplify your answers.
For the following exercises, use each set of functions to find
Simplify your answers.
For the following exercises, find functions
and
so the given function can be expressed as
### Graphical
For the following exercises, use the graphs of
shown in , and
shown in , to evaluate the expressions.
For the following exercises, use graphs of
shown in ,
shown in , and
shown in , to evaluate the expressions.
### Numeric
For the following exercises, use the function values for
shown in to evaluate each expression.
For the following exercises, use the function values for
shown in to evaluate the expressions.
For the following exercises, use each pair of functions to find
and
For the following exercises, use the functions
and
to evaluate or find the composite function as indicated.
### Extensions
For the following exercises, use
and
For the following exercises, let
and
For the following exercises, find the composition when
for all
and
### Real-World Applications
|
# Functions
## Transformation of Functions
We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.
### Graphing Functions Using Vertical and Horizontal Shifts
Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.
### Identifying Vertical Shifts
One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function
the function
is shifted vertically
units. See for an example.
To help you visualize the concept of a vertical shift, consider that
Therefore,
is equivalent to
Every unit of
is replaced by
so the
value increases or decreases depending on the value of
The result is a shift upward or downward.
### Identifying Horizontal Shifts
We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift, shown in .
For example, if
then
is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in
### Combining Vertical and Horizontal Shifts
Now that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output (
) axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input (
) axis values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down and right or left.
### Graphing Functions Using Reflections about the Axes
Another transformation that can be applied to a function is a reflection over the x- or y-axis. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. The reflections are shown in .
Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the x-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y-axis.
### Determining Even and Odd Functions
Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions
or
will result in the original graph. We say that these types of graphs are symmetric about the y-axis. Functions whose graphs are symmetric about the y-axis are called even functions.
If the graphs of
or
were reflected over both axes, the result would be the original graph, as shown in .
We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function.
Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example,
is neither even nor odd. Also, the only function that is both even and odd is the constant function
### Graphing Functions Using Stretches and Compressions
Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.
We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.
### Vertical Stretches and Compressions
When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression. shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.
### Horizontal Stretches and Compressions
Now we consider changes to the inside of a function. When we multiply a function’s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a horizontal stretch; if the constant is greater than 1, we get a horizontal compression of the function.
Given a function
the form
results in a horizontal stretch or compression. Consider the function
Observe . The graph of
is a horizontal stretch of the graph of the function
by a factor of 2. The graph of
is a horizontal compression of the graph of the function
by a factor of .
### Performing a Sequence of Transformations
When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.
When we see an expression such as
which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of
we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.
Horizontal transformations are a little trickier to think about. When we write
for example, we have to think about how the inputs to the function
relate to the inputs to the function
Suppose we know
What input to
would produce that output? In other words, what value of
will allow
We would need
To solve for
we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.
This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. We can work around this by factoring inside the function.
Let’s work through an example.
We can factor out a 2.
Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.
### Key Equations
### Key Concepts
1. A function can be shifted vertically by adding a constant to the output. See and .
2. A function can be shifted horizontally by adding a constant to the input. See , , and .
3. Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See .
4. Vertical and horizontal shifts are often combined. See and .
5. A vertical reflection reflects a graph about the
axis. A graph can be reflected vertically by multiplying the output by –1.
6. A horizontal reflection reflects a graph about the
axis. A graph can be reflected horizontally by multiplying the input by –1.
7. A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See .
8. A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See .
9. A function presented as an equation can be reflected by applying transformations one at a time. See .
10. Even functions are symmetric about the
axis, whereas odd functions are symmetric about the origin.
11. Even functions satisfy the condition
12. Odd functions satisfy the condition
13. A function can be odd, even, or neither. See .
14. A function can be compressed or stretched vertically by multiplying the output by a constant. See , , and .
15. A function can be compressed or stretched horizontally by multiplying the input by a constant. See , , and .
16. The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, describe how the graph of the function is a transformation of the graph of the original function
For the following exercises, determine the interval(s) on which the function is increasing and decreasing.
### Graphical
For the following exercises, use the graph of
shown in to sketch a graph of each transformation of
For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.
### Numeric
For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.
For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.
For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.
For the following exercises, determine whether the function is odd, even, or neither.
For the following exercises, describe how the graph of each function is a transformation of the graph of the original function
For the following exercises, write a formula for the function
that results when the graph of a given toolkit function is transformed as described.
For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.
For the following exercises, use the graph in to sketch the given transformations. |
# Functions
## Absolute Value Functions
Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate absolute value functions.
### Understanding Absolute Value
Recall that in its basic form
the absolute value function, is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.
### Graphing an Absolute Value Function
The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in .
shows the graph of
The graph of
has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at
for this transformed function.
### Solving an Absolute Value Equation
Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as
we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.
Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.
An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,
### Solving an Absolute Value Inequality
Absolute value equations may not always involve equalities. Instead, we may need to solve an equation within a range of values. We would use an absolute value inequality to solve such an equation. An absolute value inequality is an equation of the form
where an expression
(and possibly but not usually
) depends on a variable
Solving the inequality means finding the set of all
that satisfy the inequality. Usually this set will be an interval or the union of two intervals.
There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two functions. The advantage of the algebraic approach is it yields solutions that may be difficult to read from the graph.
For example, we know that all numbers within 200 units of 0 may be expressed as
Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of values
such that the distance between
and 600 is less than 200. We represent the distance between
and 600 as
This means our returns would be between $400 and $800.
Sometimes an absolute value inequality problem will be presented to us in terms of a shifted and/or stretched or compressed absolute value function, where we must determine for which values of the input the function’s output will be negative or positive.
### Key Concepts
1. The absolute value function is commonly used to measure distances between points. See .
2. Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See .
3. The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See .
4. In an absolute value equation, an unknown variable is the input of an absolute value function.
5. If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See .
6. An absolute value equation may have one solution, two solutions, or no solutions. See .
7. An absolute value inequality is similar to an absolute value equation but takes the form
It can be solved by determining the boundaries of the solution set and then testing which segments are in the set. See .
8. Absolute value inequalities can also be solved graphically. See .
### Section Exercise
### Verbal
### Algebraic
For the following exercises, solve the equations below and express the answer using set notation.
For the following exercises, find the x- and y-intercepts of the graphs of each function.
For the following exercises, solve each inequality and write the solution in interval notation.
### Graphical
For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.
For the following exercises, graph the given functions by hand.
### Technology
For the following exercises, graph each function using a graphing utility. Specify the viewing window.
### Extensions
For the following exercises, solve the inequality.
### Real-World Applications
|
# Functions
## Inverse Functions
A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.
If some physical machines can run in two directions, we might ask whether some of the function “machines” we have been studying can also run backwards. provides a visual representation of this question. In this section, we will consider the reverse nature of functions.
### Verifying That Two Functions Are Inverse Functions
Betty is traveling to Milan for a fashion show and wants to know what the temperature will be. She is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, Betty wants to convert 75 degrees Fahrenheit to degrees Celsius, using the formula
and substitutes 75 for
to calculate
Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, Betty gets the week’s weather forecast from for Milan, and wants to convert all of the temperatures to degrees Fahrenheit.
At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for
after substituting a value for
For example, to convert 26 degrees Celsius, she could write
After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.
The formula for which Betty is searching corresponds to the idea of an inverse function, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.
Given a function
we represent its inverse as
read as
inverse of
The raised
is part of the notation. It is not an exponent; it does not imply a power of
. In other words,
does not mean
because
is the reciprocal of
and not the inverse.
The “exponent-like” notation comes from an analogy between function composition and multiplication: just as
(1 is the identity element for multiplication) for any nonzero number
so
equals the identity function, that is,
This holds for all
in the domain of
Informally, this means that inverse functions “undo” each other. However, just as zero does not have a reciprocal, some functions do not have inverses.
Given a function
we can verify whether some other function
is the inverse of
by checking whether either
or
is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)
For example,
and
are inverse functions.
and
A few coordinate pairs from the graph of the function
are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function
are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.
### Finding Domain and Range of Inverse Functions
The outputs of the function
are the inputs to
so the range of
is also the domain of
Likewise, because the inputs to
are the outputs of
the domain of
is the range of
We can visualize the situation as in .
When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of
is
because a square “undoes” a square root; but the square is only the inverse of the square root on the domain
since that is the range of
We can look at this problem from the other side, starting with the square (toolkit quadratic) function
If we want to construct an inverse to this function, we run into a problem, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and –3. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the “inverse” is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.
In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function
with its domain limited to
which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).
If
on
then the inverse function is
1. The domain of
= range of
=
2. The domain of
= range of
=
### Finding and Evaluating Inverse Functions
Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.
### Inverting Tabular Functions
Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.
Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.
### Evaluating the Inverse of a Function, Given a Graph of the Original Function
We saw in Functions and Function Notation that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function’s graph.
### Finding Inverses of Functions Represented by Formulas
Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula— for example,
as a function of
we can often find the inverse function by solving to obtain
as a function of
### Finding Inverse Functions and Their Graphs
Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function
restricted to the domain
on which this function is one-to-one, and graph it as in .
Restricting the domain to
makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.
We already know that the inverse of the toolkit quadratic function is the square root function, that is,
What happens if we graph both
and
on the same set of axes, using the
axis for the input to both
We notice a distinct relationship: The graph of
is the graph of
reflected about the diagonal line
which we will call the identity line, shown in .
This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.
### Key Concepts
1. If
is the inverse of
then
See , , and .
2. Each of the toolkit functions has an inverse. See .
3. For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
4. A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
5. For a tabular function, exchange the input and output rows to obtain the inverse. See .
6. The inverse of a function can be determined at specific points on its graph. See .
7. To find the inverse of a formula, solve the equation
for
as a function of
Then exchange the labels
and
See , , and .
8. The graph of an inverse function is the reflection of the graph of the original function across the line
See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find
for each function.
For the following exercises, find a domain on which each function
is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of
restricted to that domain.
For the following exercises, use function composition to verify that
and
are inverse functions.
### Graphical
For the following exercises, use a graphing utility to determine whether each function is one-to-one.
For the following exercises, determine whether the graph represents a one-to-one function.
For the following exercises, use the graph of
shown in .
For the following exercises, use the graph of the one-to-one function shown in .
### Numeric
For the following exercises, evaluate or solve, assuming that the function
is one-to-one.
For the following exercises, use the values listed in to evaluate or solve.
### Technology
For the following exercises, find the inverse function. Then, graph the function and its inverse.
### Real-World Applications
### Chapter Review Exercises
### Functions and Function Notation
For the following exercises, determine whether the relation is a function.
For the following exercises, evaluate the function at the indicated values:
For the following exercises, determine whether the functions are one-to-one.
For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.
For the following exercises, graph the functions.
For the following exercises, use to approximate the values.
For the following exercises, use the function
to find the values.
### Domain and Range
For the following exercises, find the domain of each function, expressing answers using interval notation.
### Rates of Change and Behavior of Graphs
For the following exercises, find the average rate of change of the functions from
For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.
### Composition of Functions
For the following exercises, find
and
for each pair of functions.
For the following exercises, find
and the domain for
for each pair of functions.
For the following exercises, express each function
as a composition of two functions
and
where
### Transformation of Functions
For the following exercises, sketch a graph of the given function.
For the following exercises, sketch the graph of the function
if the graph of the function
is shown in .
For the following exercises, write the equation for the standard function represented by each of the graphs below.
For the following exercises, determine whether each function below is even, odd, or neither.
For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.
### Absolute Value Functions
For the following exercises, write an equation for the transformation of
For the following exercises, graph the absolute value function.
For the following exercises, solve the absolute value equation.
For the following exercises, solve the inequality and express the solution using interval notation.
### Inverse Functions
For the following exercises, find
for each function.
For the following exercise, find a domain on which the function
is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of
restricted to that domain.
For the following exercises, use a graphing utility to determine whether each function is one-to-one.
### Practice Test
For the following exercises, determine whether each of the following relations is a function.
For the following exercises, evaluate the function
at the given input.
For the following exercises, use the functions
to find the composite functions.
For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.
For the following exercises, determine whether the functions are even, odd, or neither.
For the following exercises, find the inverse of the function.
For the following exercises, use the graph of
shown in .
For the following exercises, use the graph of the piecewise function shown in .
For the following exercises, use the values listed in . |
# Linear Functions
## Introduction to Linear Functions
Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour.http://www.guinnessworldrecords.com/records-3000/fastest-growing-plant/ In a twenty-four hour period, this bamboo plant grows about 36 inches, or an incredible 3 feet! A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function.
Recall from Functions and Function Notation that a function is a relation that assigns to every element in the domain exactly one element in the range. Linear functions are a specific type of function that can be used to model many real-world applications, such as plant growth over time. In this chapter, we will explore linear functions, their graphs, and how to relate them to data. |
# Linear Functions
## Linear Functions
Just as with the growth of a bamboo plant, there are many situations that involve constant change over time. Consider, for example, the first commercial maglev train in the world, the Shanghai MagLev Train (). It carries passengers comfortably for a 30-kilometer trip from the airport to the subway station in only eight minutes.http://www.chinahighlights.com/shanghai/transportation/maglev-train.htm
Suppose a maglev train were to travel a long distance, and that the train maintains a constant speed of 83 meters per second for a period of time once it is 250 meters from the station. How can we analyze the train’s distance from the station as a function of time? In this section, we will investigate a kind of function that is useful for this purpose, and use it to investigate real-world situations such as the train’s distance from the station at a given point in time.
### Representing Linear Functions
The function describing the train’s motion is a linear function, which is defined as a function with a constant rate of change, that is, a polynomial of degree 1. There are several ways to represent a linear function, including word form, function notation, tabular form, and graphical form. We will describe the train’s motion as a function using each method.
### Representing a Linear Function in Word Form
Let’s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.
1. The train’s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at constant speed.
The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by 1 second, the corresponding distance (output) increases by 83 meters. The train began moving at this constant speed at a distance of 250 meters from the station.
### Representing a Linear Function in Function Notation
Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where
is the input value,
is the rate of change, and
is the initial value of the dependent variable.
In the example of the train, we might use the notation
in which the total distance
is a function of the time
The rate,
is 83 meters per second. The initial value of the dependent variable
is the original distance from the station, 250 meters. We can write a generalized equation to represent the motion of the train.
### Representing a Linear Function in Tabular Form
A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in . From the table, we can see that the distance changes by 83 meters for every 1 second increase in time.
### Representing a Linear Function in Graphical Form
Another way to represent linear functions is visually, using a graph. We can use the function relationship from above,
to draw a graph, represented in . Notice the graph is a line. When we plot a linear function, the graph is always a line.
The rate of change, which is constant, determines the slant, or slope of the line. The point at which the input value is zero is the vertical intercept, or , of the line. We can see from the graph in that the y-intercept in the train example we just saw is
and represents the distance of the train from the station when it began moving at a constant speed.
Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line
Ask yourself what numbers can be input to the function, that is, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product.
### Determining whether a Linear Function Is Increasing, Decreasing, or Constant
The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant. For an increasing function, as with the train example, the output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in (a). For a decreasing function, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in (b). If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in (c).
### Calculating and Interpreting Slope
In the examples we have seen so far, we have had the slope provided for us. However, we often need to calculate the slope given input and output values. Given two values for the input,
and
and two corresponding values for the output,
and
—which can be represented by a set of points,
and
—we can calculate the slope
as follows
where is the vertical displacement and is the horizontal displacement. Note in function notation two corresponding values for the output and for the function
and so we could equivalently write
indicates how the slope of the line between the points,
and
is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.
### Writing the Point-Slope Form of a Linear Equation
Up until now, we have been using the slope-intercept form of a linear equation to describe linear functions. Here, we will learn another way to write a linear function, the point-slope form.
The point-slope form is derived from the slope formula.
Keep in mind that the slope-intercept form and the point-slope form can be used to describe the same function. We can move from one form to another using basic algebra. For example, suppose we are given an equation in point-slope form,
. We can convert it to the slope-intercept form as shown.
Therefore, the same line can be described in slope-intercept form as
### Writing the Equation of a Line Using a Point and the Slope
The point-slope form is particularly useful if we know one point and the slope of a line. Suppose, for example, we are told that a line has a slope of 2 and passes through the point
We know that
and that
and
We can substitute these values into the general point-slope equation.
If we wanted to then rewrite the equation in slope-intercept form, we apply algebraic techniques.
Both equations,
and
describe the same line. See .
### Writing the Equation of a Line Using Two Points
The point-slope form of an equation is also useful if we know any two points through which a line passes. Suppose, for example, we know that a line passes through the points
and
We can use the coordinates of the two points to find the slope.
Now we can use the slope we found and the coordinates of one of the points to find the equation for the line. Let use (0, 1) for our point.
As before, we can use algebra to rewrite the equation in the slope-intercept form.
Both equations describe the line shown in .
### Writing and Interpreting an Equation for a Linear Function
Now that we have written equations for linear functions in both the slope-intercept form and the point-slope form, we can choose which method to use based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function
in .
We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let’s choose
and
We can use these points to calculate the slope.
Now we can substitute the slope and the coordinates of one of the points into the point-slope form.
If we want to rewrite the equation in the slope-intercept form, we would find
If we wanted to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y-axis when the output value is 7. Therefore,
We now have the initial value and the slope so we can substitute and into the slope-intercept form of a line.
So the function is
and the linear equation would be
### Modeling Real-World Problems with Linear Functions
In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.
### Key Equations
### Key Concepts
1. The ordered pairs given by a linear function represent points on a line.
2. Linear functions can be represented in words, function notation, tabular form, and graphical form. See .
3. The rate of change of a linear function is also known as the slope.
4. An equation in the slope-intercept form of a line includes the slope and the initial value of the function.
5. The initial value, or y-intercept, is the output value when the input of a linear function is zero. It is the y-value of the point at which the line crosses the y-axis.
6. An increasing linear function results in a graph that slants upward from left to right and has a positive slope.
7. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.
8. A constant linear function results in a graph that is a horizontal line.
9. Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant. See .
10. The slope of a linear function can be calculated by dividing the difference between y-values by the difference in corresponding x-values of any two points on the line. See and .
11. The slope and initial value can be determined given a graph or any two points on the line.
12. One type of function notation is the slope-intercept form of an equation.
13. The point-slope form is useful for finding a linear equation when given the slope of a line and one point. See .
14. The point-slope form is also convenient for finding a linear equation when given two points through which a line passes. See .
15. The equation for a linear function can be written if the slope and initial value are known. See , , and .
16. A linear function can be used to solve real-world problems. See and .
17. A linear function can be written from tabular form. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the equation of the curve can be written as a linear function.
For the following exercises, determine whether each function is increasing or decreasing.
For the following exercises, find the slope of the line that passes through the two given points.
For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.
### Graphical
For the following exercises, find the slope of the lines graphed.
For the following exercises, write an equation for the lines graphed.
### Numeric
For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.
### Technology
### Extensions
### Real-World Applications
|
# Linear Functions
## Graphs of Linear Functions
Two competing telephone companies offer different payment plans. The two plans charge the same rate per long distance minute, but charge a different monthly flat fee. A consumer wants to determine whether the two plans will ever cost the same amount for a given number of long distance minutes used. The total cost of each payment plan can be represented by a linear function. To solve the problem, we will need to compare the functions. In this section, we will consider methods of comparing functions using graphs.
### Graphing Linear Functions
In Linear Functions, we saw that that the graph of a linear function is a straight line. We were also able to see the points of the function as well as the initial value from a graph. By graphing two functions, then, we can more easily compare their characteristics.
There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by using the y-intercept and slope. And the third is by using transformations of the identity function
### Graphing a Function by Plotting Points
To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values. The input values and corresponding output values form coordinate pairs. We then plot the coordinate pairs on a grid. In general, we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph. For example, given the function, we might use the input values 1 and 2. Evaluating the function for an input value of 1 yields an output value of 2, which is represented by the point Evaluating the function for an input value of 2 yields an output value of 4, which is represented by the point Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error.
### Graphing a Function Using y-intercept and Slope
Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its y-intercept, which is the point at which the input value is zero. To find the , we can set in the equation.
The other characteristic of the linear function is its slope which is a measure of its steepness. Recall that the slope is the rate of change of the function. The slope of a function is equal to the ratio of the change in outputs to the change in inputs. Another way to think about the slope is by dividing the vertical difference, or rise, by the horizontal difference, or run. We encountered both the y-intercept and the slope in Linear Functions.
Let’s consider the following function.
The slope is Because the slope is positive, we know the graph will slant upward from left to right. The y-intercept is the point on the graph when The graph crosses the y-axis at Now we know the slope and the y-intercept. We can begin graphing by plotting the point We know that the slope is rise over run, From our example, we have which means that the rise is 1 and the run is 2. So starting from our y-intercept we can rise 1 and then run 2, or run 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in .
### Graphing a Function Using Transformations
Another option for graphing is to use transformations of the identity function . A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.
### Vertical Stretch or Compression
In the equation the is acting as the vertical stretch or compression of the identity function. When is negative, there is also a vertical reflection of the graph. Notice in that multiplying the equation of by stretches the graph of by a factor of units if and compresses the graph of by a factor of units if This means the larger the absolute value of the steeper the slope.
### Vertical Shift
In the acts as the vertical shift, moving the graph up and down without affecting the slope of the line. Notice in that adding a value of to the equation of shifts the graph of a total of units up if is positive and units down if is negative.
Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.
### Writing the Equation for a Function from the Graph of a Line
Recall that in Linear Functions, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at . We can see right away that the graph crosses the y-axis at the point so this is the y-intercept.
Then we can calculate the slope by finding the rise and run. We can choose any two points, but let’s look at the point
To get from this point to the y-intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be
Substituting the slope and y-intercept into the slope-intercept form of a line gives
### Finding the x-intercept of a Line
So far, we have been finding the y-intercepts of a function: the point at which the graph of the function crosses the y-axis. A function may also have an -intercept, which is the x-coordinate of the point where the graph of the function crosses the x-axis. In other words, it is the input value when the output value is zero.
To find the x-intercept, set a function equal to zero and solve for the value of For example, consider the function shown.
Set the function equal to 0 and solve for
The graph of the function crosses the x-axis at the point
### Describing Horizontal and Vertical Lines
There are two special cases of lines on a graph—horizontal and vertical lines. A horizontal line indicates a constant output, or y-value. In , we see that the output has a value of 2 for every input value. The change in outputs between any two points, therefore, is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use in the equation the equation simplifies to In other words, the value of the function is a constant. This graph represents the function
A vertical line indicates a constant input, or x-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.
Notice that a vertical line, such as the one in , has an x-intercept, but no y-intercept unless it’s the line This graph represents the line
### Determining Whether Lines are Parallel or Perpendicular
The two lines in are parallel lines: they will never intersect. Notice that they have exactly the same steepness, which means their slopes are identical. The only difference between the two lines is the y-intercept. If we shifted one line vertically toward the y-intercept of the other, they would become the same line.
We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the y-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel.
Unlike parallel lines, perpendicular lines do intersect. Their intersection forms a right, or 90-degree, angle. The two lines in are perpendicular.
Perpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line. The product of a number and its reciprocal is 1. So, if
and
are negative reciprocals of one another, they can be multiplied together to yield –1.
To find the reciprocal of a number, divide 1 by the number. So the reciprocal of 8 is and the reciprocal of is 8. To find the negative reciprocal, first find the reciprocal and then change the sign.
As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor vertical. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.
The product of the slopes is –1.
### Writing the Equation of a Line Parallel or Perpendicular to a Given Line
If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line.
### Writing Equations of Parallel Lines
Suppose for example, we are given the following equation.
We know that the slope of the line formed by the function is 3. We also know that the y-intercept is Any other line with a slope of 3 will be parallel to So the lines formed by all of the following functions will be parallel to
Suppose then we want to write the equation of a line that is parallel to and passes through the point We already know that the slope is 3. We just need to determine which value for will give the correct line. We can begin with the point-slope form of an equation for a line, and then rewrite it in the slope-intercept form.
So is parallel to and passes through the point
### Writing Equations of Perpendicular Lines
We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the following function:
The slope of the line is 2, and its negative reciprocal is Any function with a slope of will be perpendicular to So the lines formed by all of the following functions will be perpendicular to
As before, we can narrow down our choices for a particular perpendicular line if we know that it passes through a given point. Suppose then we want to write the equation of a line that is perpendicular to and passes through the point We already know that the slope is Now we can use the point to find the y-intercept by substituting the given values into the slope-intercept form of a line and solving for
The equation for the function with a slope of and a y-intercept of 2 is
So is perpendicular to and passes through the point Be aware that perpendicular lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature.
### Solving a System of Linear Equations Using a Graph
A system of linear equations includes two or more linear equations. The graphs of two lines will intersect at a single point if they are not parallel. Two parallel lines can also intersect if they are coincident, which means they are the same line and they intersect at every point. For two lines that are not parallel, the single point of intersection will satisfy both equations and therefore represent the solution to the system.
To find this point when the equations are given as functions, we can solve for an input value so that In other words, we can set the formulas for the lines equal to one another, and solve for the input that satisfies the equation.
### Key Concepts
1. Linear functions may be graphed by plotting points or by using the y-intercept and slope. See and .
2. Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See .
3. The y-intercept and slope of a line may be used to write the equation of a line.
4. The x-intercept is the point at which the graph of a linear function crosses the x-axis. See and .
5. Horizontal lines are written in the form, See .
6. Vertical lines are written in the form, See .
7. Parallel lines have the same slope.
8. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See .
9. A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the x- and y-values of the given point into the equation, and using the that results. Similarly, the point-slope form of an equation can also be used. See .
10. A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope. See and .
11. A system of linear equations may be solved setting the two equations equal to one another and solving for The y-value may be found by evaluating either one of the original equations using this x-value.
12. A system of linear equations may also be solved by finding the point of intersection on a graph. See and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
For the following exercises, find the x- and y-intercepts of each equation
For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?
### Graphical
For the following exercises, match the given linear equation with its graph in .
For the following exercises, sketch a line with the given features.
For the following exercises, sketch the graph of each equation.
For the following exercises,, write the equation of the line shown in the graph.
For the following exercises, find the point of intersection of each pair of lines if it exists. If it does not exist, indicate that there is no point of intersection.
### Extensions
For the following exercises, use the functions
### Real-World Applications
|
# Linear Functions
## Modeling with Linear Functions
Elan is a college student who plans to spend a summer in Seattle. Elan has saved $3,500 for the trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent the situation? What would be the x-intercept, and what can Elan learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models.
### Identifying Steps to Model and Solve Problems
When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them:
1. Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
2. Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
3. Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
4. Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
5. When needed, write a formula for the function.
6. Solve or evaluate the function using the formula.
7. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
8. Clearly convey your result using appropriate units, and answer in full sentences when necessary.
### Building Linear Models
Now let’s take a look at the student in Seattle. In Elan's situation, there are two changing quantities: time and money. The amount of money they have remaining while on vacation depends on how long they stay. We can use this information to define our variables, including units.
1. Output:
money remaining, in dollars
2. Input:
time, in weeks
So, the amount of money remaining depends on the number of weeks:
We can also identify the initial value and the rate of change.
1. Initial Value: They saved $3,500, so $3,500 is the initial value for
2. Rate of Change: They anticipate spending $400 each week, so –$400 per week is the rate of change, or slope.
Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because they are spending money each week.
The rate of change is constant, so we can start with the linear model
Then we can substitute the intercept and slope provided.
To find the
intercept, we set the output to zero, and solve for the input.
The
intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Elan will have no money left after 8.75 weeks.
When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before Elan saved $3,500, but the scenario discussed poses the question once they saved $3,500 because this is when the trip and subsequent spending starts. It is also likely that this model is not valid after the
intercept, unless Elan will use a credit card and go into debt. The domain represents the set of input values, so the reasonable domain for this function is
In the above example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model.
### Using a Given Intercept to Build a Model
Some real-world problems provide the
intercept, which is the constant or initial value. Once the
intercept is known, the
intercept can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The
intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. We can then use the slope-intercept form and the given information to develop a linear model.
Now we can set the function equal to 0, and solve for
to find the
intercept.
The
intercept is the number of months it takes her to reach a balance of $0. The
-intercept is 4 months, so it will take Hannah four months to pay off her loan.
### Using a Given Input and Output to Build a Model
Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.
### Using a Diagram to Model a Problem
It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.
### Building Systems of Linear Models
Real-world situations including two or more linear functions may be modeled with a system of linear equations. Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in .
1. We can use the same problem strategies that we would use for any type of function.
2. When modeling and solving a problem, identify the variables and look for key values, including the slope and y-intercept. See .
3. Draw a diagram, where appropriate. See and .
4. Check for reasonableness of the answer.
5. Linear models may be built by identifying or calculating the slope and using the y-intercept.
6. The x-intercept may be found by setting
which is setting the expression
equal to 0.
7. The point of intersection of a system of linear equations is the point where the x- and y-values are the same. See .
8. A graph of the system may be used to identify the points where one line falls below (or above) the other line.
### Verbal
### Algebraic
For the following exercises, consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped to 4,700. Assume this trend continues.
For the following exercises, consider this scenario: A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.
For the following exercises, consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.
For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.
For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were afflicted.
### Graphical
For the following exercises, use the graph in , which shows the profit,
in thousands of dollars, of a company in a given year,
where
represents the number of years since 1980.
For the following exercises, use the graph in , which shows the profit,
in thousands of dollars, of a company in a given year,
where
represents the number of years since 1980.
### Numeric
For the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown in . Assume that the house values are changing linearly.
For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in . Assume that the house values are changing linearly.
### Real-World Applications
|
# Linear Functions
## Fitting Linear Models to Data
A professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders if there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a diagram that relates the age of each student to the exam score received. In this section, we will examine one such diagram known as a scatter plot.
### Drawing and Interpreting Scatter Plots
A scatter plot is a graph of plotted points that may show a relationship between two sets of data. If the relationship is from a linear model, or a model that is nearly linear, the professor can draw conclusions using his knowledge of linear functions. shows a sample scatter plot.
Notice this scatter plot does not indicate a linear relationship. The points do not appear to follow a trend. In other words, there does not appear to be a relationship between the age of the student and the score on the final exam.
### Finding the Line of Best Fit
Once we recognize a need for a linear function to model that data, the natural follow-up question is “what is that linear function?” One way to approximate our linear function is to sketch the line that seems to best fit the data. Then we can extend the line until we can verify the y-intercept. We can approximate the slope of the line by extending it until we can estimate the
### Recognizing Interpolation or Extrapolation
While the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as interpolation when we predict a value inside the domain and range of the data. The process of extrapolation is used when we predict a value outside the domain and range of the data.
compares the two processes for the cricket-chirp data addressed in . We can see that interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44. Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or greater than 44.
There is a difference between making predictions inside the domain and range of values for which we have data and outside that domain and range. Predicting a value outside of the domain and range has its limitations. When our model no longer applies after a certain point, it is sometimes called model breakdown. For example, predicting a cost function for a period of two years may involve examining the data where the input is the time in years and the output is the cost. But if we try to extrapolate a cost when that is in 50 years, the model would not apply because we could not account for factors fifty years in the future.
### Finding the Line of Best Fit Using a Graphing Utility
While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data valuesTechnically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values.. One such technique is called least squares regression and can be computed by many graphing calculators, spreadsheet software, statistical software, and many web-based calculatorsFor example, http://www.shodor.org/unchem/math/lls/leastsq.html. Least squares regression is one means to determine the line that best fits the data, and here we will refer to this method as linear regression.
### Distinguishing Between Linear and Non-Linear Models
As we saw above with the cricket-chirp model, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the correlation coefficient, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a ”diagnostic on” selection to find the correlation coefficient, which mathematicians label as The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.
We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense for the relationship between the value of and the graph of the data, shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.
### Predicting with a Regression Line
Once we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make predictions. As we learned above, a regression line is a line that is closest to the data in the scatter plot, which means that only one such line is a best fit for the data.
### Key Concepts
1. Scatter plots show the relationship between two sets of data. See .
2. Scatter plots may represent linear or non-linear models.
3. The line of best fit may be estimated or calculated, using a calculator or statistical software. See .
4. Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. See .
5. The correlation coefficient, indicates the degree of linear relationship between data. See .
6. A regression line best fits the data. See .
7. The least squares regression line is found by minimizing the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related?
### Graphical
For the following exercises, match each scatterplot with one of the four specified correlations in and .
For the following exercises, draw a best-fit line for the plotted data.
### Numeric
### Technology
For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.
### Extensions
For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span, (number of units sold, profit) for specific recorded years:
.
### Real-World Applications
For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years:
For the following exercises, consider this scenario: The profit of a company increased steadily over a ten-year span. The following ordered pairs show the number of units sold in hundreds and the profit in thousands of over the ten year span, (number of units sold, profit) for specific recorded years:
.
For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs show dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span (number of units sold, profit) for specific recorded years:
### Chapter Review Exercises
### Linear Functions
### Graphs of Linear Functions
For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
For the following exercises, find the x- and y- intercepts of the given equation
For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?
### Modeling with Linear Functions
For the following exercises, use the graph in showing the profit, in thousands of dollars, of a company in a given year, where represents years since 1980.
For the following exercise, consider this scenario: In 2004, a school population was 1,700. By 2012 the population had grown to 2,500.
For the following exercises, consider this scenario: In 2000, the moose population in a park was measured to be 6,500. By 2010, the population was measured to be 12,500. Assume the population continues to change linearly.
For the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in . Assume that the house values are changing linearly.
### Fitting Linear Models to Data
For the following exercises, consider the data in , which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year.
For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded years:
### Practice Test
For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
For the following exercises, use the graph in , showing the profit, , in thousands of dollars, of a company in a given year, , where represents years since 1980.
For the following exercises, use , which shows the percent of unemployed persons 25 years or older who are college graduates in a particular city, by year.
For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years:
|
# Polynomial and Rational Functions
## Introduction to Polynomial and Rational Functions
You don't need to dive very deep to feel the effects of pressure. As a person in their neighborhood pool moves eight, ten, twelve feet down, they often feel pain in their ears as a result of water and air pressure differentials. Pressure plays a much greater role at ocean diving depths.
Scuba and free divers are constantly negotiating the effects of pressure in order to experience enjoyable, safe, and productive dives. Gases in a person's respiratory system and diving apparatus interact according to certain physical properties, which upon discovery and evaluation are collectively known as the gas laws. Some are conceptually simple, such as the inverse relationship regarding pressure and volume, and others are more complex. While their formulas seem more straightforward than many you will encounter in this chapter, the gas laws are generally polynomial expressions. |
# Polynomial and Rational Functions
## Complex Numbers
The study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. For example, we still have no solution to equations such as
Our best guesses might be +2 or –2. But if we test +2 in this equation, it does not work. If we test –2, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.
### Expressing Square Roots of Negative Numbers as Multiples of i
We know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary number
is defined as the square root of negative 1.
So, using properties of radicals,
We can write the square root of any negative number as a multiple of
Consider the square root of –25.
We use
and not
because the principal root of
is the positive root.
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written
where
is the real part and
is the imaginary part. For example,
is a complex number. So, too, is
Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.
### Plotting a Complex Number on the Complex Plane
We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs
where
represents the coordinate for the horizontal axis and
represents the coordinate for the vertical axis.
Let’s consider the number
The real part of the complex number is
and the imaginary part is
We plot the ordered pair
to represent the complex number
as shown in .
### Adding and Subtracting Complex Numbers
Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.
### Multiplying Complex Numbers
Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.
### Multiplying a Complex Number by a Real Number
Let’s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,
### Multiplying Complex Numbers Together
Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get
Because
we have
To simplify, we combine the real parts, and we combine the imaginary parts.
### Dividing Complex Numbers
Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of
is
Note that complex conjugates have a reciprocal relationship: The complex conjugate of
is
and the complex conjugate of
is
Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.
Suppose we want to divide
by
where neither
nor
equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.
Multiply the numerator and denominator by the complex conjugate of the denominator.
Apply the distributive property.
Simplify, remembering that
### Simplifying Powers of i
The powers of
are cyclic. Let’s look at what happens when we raise
to increasing powers.
We can see that when we get to the fifth power of
it is equal to the first power. As we continue to multiply
by itself for increasing powers, we will see a cycle of 4. Let’s examine the next 4 powers of
### Key Concepts
1. The square root of any negative number can be written as a multiple of
See .
2. To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See .
3. Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See .
4. Complex numbers can be multiplied and divided.
5. To multiply complex numbers, distribute just as with polynomials. See , , and .
6. To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. See , , and .
7. The powers of
are cyclic, repeating every fourth one. See .
### Verbal
### Algebraic
For the following exercises, evaluate the algebraic expressions.
### Graphical
For the following exercises, determine the number of real and nonreal solutions for each quadratic function shown.
For the following exercises, plot the complex numbers on the complex plane.
### Numeric
For the following exercises, perform the indicated operation and express the result as a simplified complex number.
### Technology
For the following exercises, use a calculator to help answer the questions.
### Extensions
For the following exercises, evaluate the expressions, writing the result as a simplified complex number. |
# Polynomial and Rational Functions
## Quadratic Functions
Curved antennas, such as the ones shown in , are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.
In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.
### Recognizing Characteristics of Parabolas
The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry. These features are illustrated in .
The y-intercept is the point at which the parabola crosses the y-axis. The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of
at which
### Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions
The general form of a quadratic function presents the function in the form
where
and
are real numbers and
If
the parabola opens upward. If
the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.
The axis of symmetry is defined by
If we use the quadratic formula,
to solve
for the
intercepts, or zeros, we find the value of
halfway between them is always
the equation for the axis of symmetry.
represents the graph of the quadratic function written in general form as
In this form,
and
Because
the parabola opens upward. The axis of symmetry is
This also makes sense because we can see from the graph that the vertical line
divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance,
The
intercepts, those points where the parabola crosses the
axis, occur at
and
The standard form of a quadratic function presents the function in the form
where
is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.
As with the general form, if
the parabola opens upward and the vertex is a minimum. If
the parabola opens downward, and the vertex is a maximum. represents the graph of the quadratic function written in standard form as
Since
in this example,
In this form,
and
Because
the parabola opens downward. The vertex is at
The standard form is useful for determining how the graph is transformed from the graph of
is the graph of this basic function.
If
the graph shifts upward, whereas if
the graph shifts downward. In ,
so the graph is shifted 4 units upward. If
the graph shifts toward the right and if
the graph shifts to the left. In ,
so the graph is shifted 2 units to the left. The magnitude of
indicates the stretch of the graph. If
the point associated with a particular
value shifts farther from the x-axis, so the graph appears to become narrower, and there is a vertical stretch. But if
the point associated with a particular
value shifts closer to the x-axis, so the graph appears to become wider, but in fact there is a vertical compression. In ,
so the graph becomes narrower.
The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.
For the linear terms to be equal, the coefficients must be equal.
This is the axis of symmetry we defined earlier. Setting the constant terms equal:
In practice, though, it is usually easier to remember that k is the output value of the function when the input is
so
### Finding the Domain and Range of a Quadratic Function
Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y-values greater than or equal to the y-coordinate at the turning point or less than or equal to the y-coordinate at the turning point, depending on whether the parabola opens up or down.
### Determining the Maximum and Minimum Values of Quadratic Functions
The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the parabola. We can see the maximum and minimum values in .
There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.
### Finding the x- and y-Intercepts of a Quadratic Function
Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the
intercept of a quadratic by evaluating the function at an input of zero, and we find the
intercepts at locations where the output is zero. Notice in that the number of
intercepts can vary depending upon the location of the graph.
### Rewriting Quadratics in Standard Form
In , the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.
### Key Equations
### Key Concepts
1. A polynomial function of degree two is called a quadratic function.
2. The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
3. The axis of symmetry is the vertical line passing through the vertex. The zeros, or
intercepts, are the points at which the parabola crosses the
axis. The
intercept is the point at which the parabola crosses the
axis. See , , and .
4. Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See .
5. The vertex can be found from an equation representing a quadratic function. See .
6. The domain of a quadratic function is all real numbers. The range varies with the function. See .
7. A quadratic function’s minimum or maximum value is given by the
value of the vertex.
8. The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See and .
9. Some quadratic equations must be solved by using the quadratic formula. See .
10. The vertex and the intercepts can be identified and interpreted to solve real-world problems. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, rewrite the quadratic functions in standard form and give the vertex.
For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.
For the following exercises, determine the domain and range of the quadratic function.
For the following exercises, solve the equations over the complex numbers.
For the following exercises, use the vertex
and a point on the graph
to find the general form of the equation of the quadratic function.
### Graphical
For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.
For the following exercises, write the equation for the graphed function.
### Numeric
For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.
### Technology
For the following exercises, use a calculator to find the answer.
### Extensions
For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.
For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.
### Real-World Applications
|
# Polynomial and Rational Functions
## Power Functions and Polynomial Functions
Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in .
The population can be estimated using the function
where
represents the bird population on the island
years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.
### Identifying Power Functions
In order to better understand the bird problem, we need to understand a specific type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.)
As an example, consider functions for area or volume. The function for the area of a circle with radius
is
and the function for the volume of a sphere with radius
is
Both of these are examples of power functions because they consist of a coefficient,
or
multiplied by a variable
raised to a power.
### Identifying End Behavior of Power Functions
shows the graphs of
and
which are all power functions with even, positive integer powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.
To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol
for positive infinity and
for negative infinity. When we say that “
approaches infinity,” which can be symbolically written as
we are describing a behavior; we are saying that
is increasing without bound.
With the even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as
approaches positive or negative infinity, the
values increase without bound. In symbolic form, we could write
shows the graphs of
which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.
These examples illustrate that functions of the form
reveal symmetry of one kind or another. First, in we see that even functions of the form
are symmetric about the
axis. In we see that odd functions of the form
are symmetric about the origin.
For these odd power functions, as
approaches negative infinity,
decreases without bound. As
approaches positive infinity,
increases without bound. In symbolic form we write
The behavior of the graph of a function as the input values get very small (
) and get very large (
) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.
shows the end behavior of power functions in the form
where
is a non-negative integer depending on the power and the constant.
### Identifying Polynomial Functions
An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius
of the spill depends on the number of weeks
that have passed. This relationship is linear.
We can combine this with the formula for the area
of a circle.
Composing these functions gives a formula for the area in terms of weeks.
Multiplying gives the formula.
This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.
### Identifying the Degree and Leading Coefficient of a Polynomial Function
Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.
### Identifying End Behavior of Polynomial Functions
Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as
gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree. See .
### Identifying Local Behavior of Polynomial Functions
In addition to the end behavior of polynomial functions, we are also interested in what happens in the “middle” of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.
We are also interested in the intercepts. As with all functions, the y-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one y-intercept
The x-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one x-intercept. See .
### Comparing Smooth and Continuous Graphs
The degree of a polynomial function helps us to determine the number of
intercepts and the number of turning points. A polynomial function of
degree is the product of
factors, so it will have at most
roots or zeros, or
intercepts. The graph of the polynomial function of degree
must have at most
turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.
A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.
### Key Equations
### Key Concepts
1. A power function is a variable base raised to a number power. See .
2. The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
3. The end behavior depends on whether the power is even or odd. See and .
4. A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See .
5. The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See .
6. The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See and .
7. A polynomial of degree
will have at most
x-intercepts and at most
turning points. See , , , , and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, identify the function as a power function, a polynomial function, or neither.
For the following exercises, find the degree and leading coefficient for the given polynomial.
For the following exercises, determine the end behavior of the functions.
For the following exercises, find the intercepts of the functions.
### Graphical
For the following exercises, determine the least possible degree of the polynomial function shown.
For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.
### Numeric
For the following exercises, make a table to confirm the end behavior of the function.
### Technology
For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.
### Extensions
For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.
### Real-World Applications
For the following exercises, use the written statements to construct a polynomial function that represents the required information. |
# Polynomial and Rational Functions
## Graphs of Polynomial Functions
The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in .
The revenue can be modeled by the polynomial function
where
represents the revenue in millions of dollars and
represents the year, with
corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.
### Recognizing Characteristics of Graphs of Polynomial Functions
Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. shows a graph that represents a polynomial function and a graph that represents a function that is not a polynomial.
### Using Factoring to Find Zeros of Polynomial Functions
Recall that if
is a polynomial function, the values of
for which
are called zeros of
If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros.
We can use this method to find
intercepts because at the
intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:
1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring.
2. The polynomial is given in factored form.
3. Technology is used to determine the intercepts.
### Identifying Zeros and Their Multiplicities
Graphs behave differently at various
intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.
Suppose, for example, we graph the function
Notice in that the behavior of the function at each of the
intercepts is different.
The
intercept
is the solution of equation
The graph passes directly through the
intercept at
The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line—it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.
The
intercept
is the repeated solution of equation
The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept.
The factor is repeated, that is, the factor
appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor,
has multiplicity 2 because the factor
occurs twice.
The
intercept
is the repeated solution of factor
The graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic—with the same S-shape near the intercept as the toolkit function
We call this a triple zero, or a zero with multiplicity 3.
For zeros with even multiplicities, the graphs touch or are tangent to the
axis. For zeros with odd multiplicities, the graphs cross or intersect the
axis. See for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.
For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the
axis.
For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the
axis.
### Determining End Behavior
As we have already learned, the behavior of a graph of a polynomial function of the form
will either ultimately rise or fall as
increases without bound and will either rise or fall as
decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say –100 or –1,000.
Recall that we call this behavior the end behavior of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function,
is an even power function, as
increases or decreases without bound,
increases without bound. When the leading term is an odd power function, as
decreases without bound,
also decreases without bound; as
increases without bound,
also increases without bound. If the leading term is negative, it will change the direction of the end behavior. summarizes all four cases.
### Understanding the Relationship between Degree and Turning Points
In addition to the end behavior, recall that we can analyze a polynomial function’s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function
in . The graph has three turning points.
This function
is a 4th degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.
### Graphing Polynomial Functions
We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.
### Using the Intermediate Value Theorem
In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x-axis, we can confirm that there is a zero between them. Consider a polynomial function
whose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbers
and
in the domain of
if
and
then the function
takes on every value between
and
We can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous function
at
lies above the
axis and another point at
lies below the
axis, there must exist a third point between
and
where the graph crosses the
axis. Call this point
This means that we are assured there is a solution
where
In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the
axis. shows that there is a zero between
and
### Writing Formulas for Polynomial Functions
Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an
intercept where each factor is equal to zero, we can form a function that will pass through a set of
intercepts by introducing a corresponding set of factors.
### Using Local and Global Extrema
With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.
Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a global maximum or a global minimum. These are also referred to as the absolute maximum and absolute minimum values of the function.
### Key Concepts
1. Polynomial functions of degree 2 or more are smooth, continuous functions. See .
2. To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See , , and .
3. Another way to find the
intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the
axis. See .
4. The multiplicity of a zero determines how the graph behaves at the
intercepts. See .
5. The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
6. The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
7. The end behavior of a polynomial function depends on the leading term.
8. The graph of a polynomial function changes direction at its turning points.
9. A polynomial function of degree
has at most
turning points. See .
10. To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most
turning points. See and .
11. Graphing a polynomial function helps to estimate local and global extremas. See .
12. The Intermediate Value Theorem tells us that if
have opposite signs, then there exists at least one value
between
and
for which
See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the
or t-intercepts of the polynomial functions.
For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.
For the following exercises, find the zeros and give the multiplicity of each.
### Graphical
For the following exercises, graph the polynomial functions. Note
and
intercepts, multiplicity, and end behavior.
For the following exercises, use the graphs to write the formula for a polynomial function of least degree.
For the following exercises, use the graph to identify zeros and multiplicity.
For the following exercises, use the given information about the polynomial graph to write the equation.
### Technology
For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.
### Extensions
For the following exercises, use the graphs to write a polynomial function of least degree.
### Real-World Applications
For the following exercises, write the polynomial function that models the given situation. |
# Polynomial and Rational Functions
## Dividing Polynomials
The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width 40 m, and height 30 m.National Park Service. "Lincoln Memorial Building Statistics." http://www.nps.gov/linc/historyculture/lincoln-memorial-building-statistics.htm. Accessed 4/3/2014 We can easily find the volume using elementary geometry.
So the volume is 73,800 cubic meters
Suppose we knew the volume, length, and width. We could divide to find the height.
As we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any or all of the measurements contain variable expressions. For example, suppose the volume of a rectangular solid is given by the polynomial
The length of the solid is given by
the width is given by
To find the height of the solid, we can use polynomial division, which is the focus of this section.
### Using Long Division to Divide Polynomials
We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let’s divide 178 by 3 using long division.
Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.
We call this the Division Algorithm and will discuss it more formally after looking at an example.
Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide
by
using the long division algorithm, it would look like this:
We have found
or
We can identify the dividend, the divisor, the quotient, and the remainder.
Writing the result in this manner illustrates the Division Algorithm.
### Using Synthetic Division to Divide Polynomials
As we’ve seen, long division of polynomials can involve many steps and be quite cumbersome. Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.
To illustrate the process, recall the example at the beginning of the section.
Divide
by
using the long division algorithm.
The final form of the process looked like this:
There is a lot of repetition in the table. If we don’t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.
Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the “divisor” to –2, multiply and add. The process starts by bringing down the leading coefficient.
We then multiply it by the “divisor” and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is
and the remainder is
The process will be made more clear in .
### Using Polynomial Division to Solve Application Problems
Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.
### Key Equations
### Key Concepts
1. Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See and .
2. The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
3. Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form
See , , and .
4. Polynomial division can be used to solve application problems, including area and volume. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, use long division to divide. Specify the quotient and the remainder.
For the following exercises, use synthetic division to find the quotient.
For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.
### Graphical
For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.
For the following exercises, use synthetic division to find the quotient and remainder.
### Technology
For the following exercises, use a calculator with CAS to answer the questions.
### Extensions
For the following exercises, use synthetic division to determine the quotient involving a complex number.
### Real-World Applications
For the following exercises, use the given length and area of a rectangle to express the width algebraically.
For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.
For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically. |
# Polynomial and Rational Functions
## Zeros of Polynomial Functions
A new bakery offers decorated, multi-tiered cakes for display and cutting at Quinceañera and wedding celebrations, as well as sheet cakes to serve most of the guests.. The bakery wants the volume of a small sheet cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?
This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.
### Evaluating a Polynomial Using the Remainder Theorem
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by
the remainder may be found quickly by evaluating the polynomial function at
that is,
Let’s walk through the proof of the theorem.
Recall that the Division Algorithm states that, given a polynomial dividend
and a non-zero polynomial divisor
where the degree of
is less than or equal to the degree of
there exist unique polynomials
and
such that
If the divisor,
is
this takes the form
Since the divisor
is linear, the remainder will be a constant,
And, if we evaluate this for
we have
In other words,
is the remainder obtained by dividing
by
### Using the Factor Theorem to Solve a Polynomial Equation
The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm tells us
If
is a zero, then the remainder
is
and
or
Notice, written in this form,
is a factor of
We can conclude if
is a zero of
then
is a factor of
Similarly, if
is a factor of
then the remainder of the Division Algorithm
is 0. This tells us that
is a zero.
This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree
in the complex number system will have
zeros. We can use the Factor Theorem to completely factor a polynomial into the product of
factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.
### Using the Rational Zero Theorem to Find Rational Zeros
Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial. But first we need a pool of rational numbers to test. The Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial
Consider a quadratic function with two zeros,
and
By the Factor Theorem, these zeros have factors associated with them. Let us set each factor equal to 0, and then construct the original quadratic function absent its stretching factor.
Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4.
We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros.
### Finding the Zeros of Polynomial Functions
The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function.
### Using the Fundamental Theorem of Algebra
Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. This theorem forms the foundation for solving polynomial equations.
Suppose
is a polynomial function of degree four, and
The Fundamental Theorem of Algebra states that there is at least one complex solution, call it
By the Factor Theorem, we can write
as a product of
and a polynomial quotient. Since
is linear, the polynomial quotient will be of degree three. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. It will have at least one complex zero, call it
So we can write the polynomial quotient as a product of
and a new polynomial quotient of degree two. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. There will be four of them and each one will yield a factor of
### Using the Linear Factorization Theorem to Find Polynomials with Given Zeros
A vital implication of the Fundamental Theorem of Algebra, as we stated above, is that a polynomial function of degree
will have
zeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the polynomial function into
factors. The Linear Factorization Theorem tells us that a polynomial function will have the same number of factors as its degree, and that each factor will be in the form
where
is a complex number.
Let
be a polynomial function with real coefficients, and suppose
is a zero of
Then, by the Factor Theorem,
is a factor of
For
to have real coefficients,
must also be a factor of
This is true because any factor other than
when multiplied by
will leave imaginary components in the product. Only multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. In other words, if a polynomial function
with real coefficients has a complex zero
then the complex conjugate
must also be a zero of
This is called the Complex Conjugate Theorem.
### Using Descartes’ Rule of Signs
There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order, Descartes’ Rule of Signs tells us of a relationship between the number of sign changes in
and the number of positive real zeros. For example, the polynomial function below has one sign change.
This tells us that the function must have 1 positive real zero.
There is a similar relationship between the number of sign changes in
and the number of negative real zeros.
In this case,
has 3 sign changes. This tells us that
could have 3 or 1 negative real zeros.
### Solving Real-World Applications
We have now introduced a variety of tools for solving polynomial equations. Let’s use these tools to solve the bakery problem from the beginning of the section.
### Key Concepts
1. To find
determine the remainder of the polynomial
when it is divided by
See .
2.
is a zero of
if and only if
is a factor of
See .
3. Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See and .
4. When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
5. Synthetic division can be used to find the zeros of a polynomial function. See .
6. According to the Fundamental Theorem, every polynomial function has at least one complex zero. See .
7. Every polynomial function with degree greater than 0 has at least one complex zero.
8. Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form
where
is a complex number. See .
9. The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
10. The number of negative real zeros of a polynomial function is either the number of sign changes of
or less than the number of sign changes by an even integer. See .
11. Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, use the Remainder Theorem to find the remainder.
For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.
For the following exercises, use the Rational Zero Theorem to find all real zeros.
For the following exercises, find all complex solutions (real and non-real).
### Graphical
For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination.
### Numeric
For the following exercises, list all possible rational zeros for the functions.
### Technology
For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.
### Extensions
For the following exercises, construct a polynomial function of least degree possible using the given information.
### Real-World Applications
For the following exercises, find the dimensions of the box described.
For the following exercises, find the dimensions of the right circular cylinder described. |
# Polynomial and Rational Functions
## Rational Functions
Suppose we know that the cost of making a product is dependent on the number of items,
produced. This is given by the equation
If we want to know the average cost for producing
items, we would divide the cost function by the number of items,
The average cost function, which yields the average cost per item for
items produced, is
Many other application problems require finding an average value in a similar way, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power.
In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
### Using Arrow Notation
We have seen the graphs of the basic reciprocal function and the squared reciprocal function from our study of toolkit functions. Examine these graphs, as shown in , and notice some of their features.
Several things are apparent if we examine the graph of
1. On the left branch of the graph, the curve approaches the x-axis
2. As the graph approaches
from the left, the curve drops, but as we approach zero from the right, the curve rises.
3. Finally, on the right branch of the graph, the curves approaches the x-axis
To summarize, we use arrow notation to show that
or
is approaching a particular value. See .
### Local Behavior of
Let’s begin by looking at the reciprocal function,
We cannot divide by zero, which means the function is undefined at
so zero is not in the domain. As the input values approach zero from the left side (becoming very small, negative values), the function values decrease without bound (in other words, they approach negative infinity). We can see this behavior in .
We write in arrow notation
As the input values approach zero from the right side (becoming very small, positive values), the function values increase without bound (approaching infinity). We can see this behavior in .
We write in arrow notation
See .
This behavior creates a vertical asymptote, which is a vertical line that the graph approaches but never crosses. In this case, the graph is approaching the vertical line
as the input becomes close to zero. See .
### End Behavior of
As the values of
approach infinity, the function values approach 0. As the values of
approach negative infinity, the function values approach 0. See . Symbolically, using arrow notation
Based on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it seems to level off as the inputs become large. This behavior creates a horizontal asymptote, a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line
See .
### Solving Applied Problems Involving Rational Functions
In , we shifted a toolkit function in a way that resulted in the function
This is an example of a rational function. A rational function is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.
### Finding the Domains of Rational Functions
A vertical asymptote represents a value at which a rational function is undefined, so that value is not in the domain of the function. A reciprocal function cannot have values in its domain that cause the denominator to equal zero. In general, to find the domain of a rational function, we need to determine which inputs would cause division by zero.
### Identifying Vertical Asymptotes of Rational Functions
By looking at the graph of a rational function, we can investigate its local behavior and easily see whether there are asymptotes. We may even be able to approximate their location. Even without the graph, however, we can still determine whether a given rational function has any asymptotes, and calculate their location.
### Vertical Asymptotes
The vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not common to the factors in the numerator. Vertical asymptotes occur at the zeros of such factors.
### Removable Discontinuities
Occasionally, a graph will contain a hole: a single point where the graph is not defined, indicated by an open circle. We call such a hole a removable discontinuity.
For example, the function
may be re-written by factoring the numerator and the denominator.
Notice that
is a common factor to the numerator and the denominator. The zero of this factor,
is the location of the removable discontinuity. Notice also that
is not a factor in both the numerator and denominator. The zero of this factor,
is the vertical asymptote. See .
### Identifying Horizontal Asymptotes of Rational Functions
While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.
There are three distinct outcomes when checking for horizontal asymptotes:
Case 1: If the degree of the denominator > degree of the numerator, there is a horizontal asymptote at
In this case, the end behavior is
This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function
and the outputs will approach zero, resulting in a horizontal asymptote at
See . Note that this graph crosses the horizontal asymptote.
Case 2: If the degree of the denominator < degree of the numerator by one, we get a slant asymptote.
In this case, the end behavior is
This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function
As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of
looks like a diagonal line, and since
will behave similarly to
it will approach a line close to
This line is a slant asymptote.
To find the equation of the slant asymptote, divide
The quotient is
and the remainder is 2. The slant asymptote is the graph of the line
See .
Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at
where
and
are the leading coefficients of
and
for
In this case, the end behavior is
This tells us that as the inputs grow large, this function will behave like the function
which is a horizontal line. As
resulting in a horizontal asymptote at
See . Note that this graph crosses the horizontal asymptote.
Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.
It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the function
with end behavior
the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.
### Graphing Rational Functions
In , we see that the numerator of a rational function reveals the x-intercepts of the graph, whereas the denominator reveals the vertical asymptotes of the graph. As with polynomials, factors of the numerator may have integer powers greater than one. Fortunately, the effect on the shape of the graph at those intercepts is the same as we saw with polynomials.
The vertical asymptotes associated with the factors of the denominator will mirror one of the two toolkit reciprocal functions. When the degree of the factor in the denominator is odd, the distinguishing characteristic is that on one side of the vertical asymptote the graph heads towards positive infinity, and on the other side the graph heads towards negative infinity. See .
When the degree of the factor in the denominator is even, the distinguishing characteristic is that the graph either heads toward positive infinity on both sides of the vertical asymptote or heads toward negative infinity on both sides. See .
For example, the graph of
is shown in .
1. At the x-intercept
corresponding to the
factor of the numerator, the graph bounces, consistent with the quadratic nature of the factor.
2. At the x-intercept
corresponding to the
factor of the numerator, the graph passes through the axis as we would expect from a linear factor.
3. At the vertical asymptote
corresponding to the
factor of the denominator, the graph heads towards positive infinity on both sides of the asymptote, consistent with the behavior of the function
4. At the vertical asymptote
corresponding to the
factor of the denominator, the graph heads towards positive infinity on the left side of the asymptote and towards negative infinity on the right side.
### Writing Rational Functions
Now that we have analyzed the equations for rational functions and how they relate to a graph of the function, we can use information given by a graph to write the function. A rational function written in factored form will have an x-intercept where each factor of the numerator is equal to zero. (An exception occurs in the case of a removable discontinuity.) As a result, we can form a numerator of a function whose graph will pass through a set of x-intercepts by introducing a corresponding set of factors. Likewise, because the function will have a vertical asymptote where each factor of the denominator is equal to zero, we can form a denominator that will produce the vertical asymptotes by introducing a corresponding set of factors.
### Key Equations
### Key Concepts
1. We can use arrow notation to describe local behavior and end behavior of the toolkit functions
and
See .
2. A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. See .
3. Application problems involving rates and concentrations often involve rational functions. See .
4. The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. See .
5. The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. See .
6. A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. See .
7. A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. See , , , and .
8. Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. See .
9. If a rational function has x-intercepts at
vertical asymptotes at
and no
then the function can be written in the form
See
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the domain of the rational functions.
For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.
For the following exercises, find the x- and y-intercepts for the functions.
For the following exercises, describe the local and end behavior of the functions.
For the following exercises, find the slant asymptote of the functions.
### Graphical
For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.
For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.
For the following exercises, write an equation for a rational function with the given characteristics.
For the following exercises, use the graphs to write an equation for the function.
### Numeric
For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote
### Technology
For the following exercises, use a calculator to graph
Use the graph to solve
### Extensions
For the following exercises, identify the removable discontinuity.
### Real-World Applications
For the following exercises, express a rational function that describes the situation.
For the following exercises, use the given rational function to answer the question.
For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question. |
# Polynomial and Rational Functions
## Inverses and Radical Functions
Park rangers and other trail managers may construct rock piles, stacks, or other arrangements, usually called cairns, to mark trails or other landmarks. (Rangers and environmental scientists discourage hikers from doing the same, in order to avoid confusion and preserve the habitats of plants and animals.) A cairn in the form of a mound of gravel is in the shape of a cone with the height equal to twice the radius.
The volume is found using a formula from elementary geometry.
We have written the volume
in terms of the radius
However, in some cases, we may start out with the volume and want to find the radius. For example: A customer purchases 100 cubic feet of gravel to construct a cone shape mound with a height twice the radius. What are the radius and height of the new cone? To answer this question, we use the formula
This function is the inverse of the formula for
in terms of
In this section, we will explore the inverses of polynomial and rational functions and in particular the radical functions we encounter in the process.
### Finding the Inverse of a Polynomial Function
Two functions
and
are inverse functions if for every coordinate pair in
there exists a corresponding coordinate pair in the inverse function,
In other words, the coordinate pairs of the inverse functions have the input and output interchanged.
For a function to have an inverse, it must be one-to-one.
For example, suppose the Sustainability Club builds a water runoff collector in the shape of a parabolic trough as shown in . We can use the information in the figure to find the surface area of the water in the trough as a function of the depth of the water.
Because it will be helpful to have an equation for the parabolic cross-sectional shape, we will impose a coordinate system at the cross section, with
measured horizontally and
measured vertically, with the origin at the vertex of the parabola. See .
From this we find an equation for the parabolic shape. We placed the origin at the vertex of the parabola, so we know the equation will have form
Our equation will need to pass through the point (6, 18), from which we can solve for the stretch factor
Our parabolic cross section has the equation
We are interested in the surface area of the water, so we must determine the width at the top of the water as a function of the water depth. For any depth
the width will be given by
so we need to solve the equation above for
and find the inverse function. However, notice that the original function is not one-to-one, and indeed, given any output there are two inputs that produce the same output, one positive and one negative.
To find an inverse, we can restrict our original function to a limited domain on which it is one-to-one. In this case, it makes sense to restrict ourselves to positive
values. On this domain, we can find an inverse by solving for the input variable:
This is not a function as written. Since we are limiting ourselves to positive
values in the original function, we can eliminate the negative solution, which gives us the inverse function we’re looking for.
Because
is the distance from the center of the parabola to either side, the entire width of the water at the top will be
The trough is 3 feet (36 inches) long, so the surface area will then be:
This example illustrates two important points:
1. When finding the inverse of a quadratic, we have to limit ourselves to a domain on which the function is one-to-one.
2. The inverse of a quadratic function is a square root function. Both are toolkit functions and different types of power functions.
Functions involving roots are often called radical functions. While it is not possible to find an inverse of most polynomial functions, some basic polynomials do have inverses. Such functions are called invertible functions, and we use the notation
Warning:
is not the same as the reciprocal of the function
This use of “–1” is reserved to denote inverse functions. To denote the reciprocal of a function
we would need to write
An important relationship between inverse functions is that they “undo” each other. If
is the inverse of a function
then
is the inverse of the function
In other words, whatever the function
does to
undoes it—and vice-versa. More formally, we write
and
### Restricting the Domain to Find the Inverse of a Polynomial Function
So far, we have been able to find the inverse functions of cubic functions without having to restrict their domains. However, as we know, not all cubic polynomials are one-to-one. Some functions that are not one-to-one may have their domain restricted so that they are one-to-one, but only over that domain. The function over the restricted domain would then have an inverse function. Since quadratic functions are not one-to-one, we must restrict their domain in order to find their inverses.
### Solving Applications of Radical Functions
Notice that the functions from previous examples were all polynomials, and their inverses were radical functions. If we want to find the inverse of a radical function, we will need to restrict the domain of the answer because the range of the original function is limited.
###
Radical functions are common in physical models, as we saw in the section opener. We now have enough tools to be able to solve the problem posed at the start of the section.
### Determining the Domain of a Radical Function Composed with Other Functions
When radical functions are composed with other functions, determining domain can become more complicated.
### Finding Inverses of Rational Functions
As with finding inverses of quadratic functions, it is sometimes desirable to find the inverse of a rational function, particularly of rational functions that are the ratio of linear functions, such as in concentration applications.
### Key Concepts
1. The inverse of a quadratic function is a square root function.
2. If
is the inverse of a function
then
is the inverse of the function
See .
3. While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See .
4. To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See and .
5. When finding the inverse of a radical function, we need a restriction on the domain of the answer. See and .
6. Inverse and radical and functions can be used to solve application problems. See and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the inverse of the function on the given domain.
For the following exercises, find the inverse of the functions.
For the following exercises, find the inverse of the functions.
### Graphical
For the following exercises, find the inverse of the function and graph both the function and its inverse.
For the following exercises, use a graph to help determine the domain of the functions.
### Technology
For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.
### Extensions
For the following exercises, find the inverse of the functions with
positive real numbers.
### Real-World Applications
For the following exercises, determine the function described and then use it to answer the question. |
# Polynomial and Rational Functions
## Modeling Using Variation
A pre-owned car dealer has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn $736. As she considers the offer, she takes into account the typical price of the dealer's cars, the overall market, and how many she can reasonably expect to sell. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.
### Solving Direct Variation Problems
In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formula
tells us her earnings,
come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. See .
Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called direct variation. Each variable in this type of relationship varies directly with the other.
represents the data for Nicole’s potential earnings. We say that earnings vary directly with the sales price of the car. The formula
is used for direct variation. The value
is a nonzero constant greater than zero and is called the constant of variation. In this case,
and
### Solving Inverse Variation Problems
Water temperature in an ocean varies inversely to the water’s depth. Between the depths of 250 feet and 500 feet, the formula
gives us the temperature in degrees Fahrenheit at a depth in feet below Earth’s surface. Consider the Atlantic Ocean, which covers 22% of Earth’s surface. At a certain location, at the depth of 500 feet, the temperature may be 28°F.
If we create , we observe that, as the depth increases, the water temperature decreases.
We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be inversely proportional and each term varies inversely with the other. Inversely proportional relationships are also called inverse variations.
For our example, depicts the inverse variation. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula
for inverse variation in this case uses
### Solving Problems Involving Joint Variation
Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable
cost, varies jointly with the number of students,
and the distance,
### Key Equations
### Key Concepts
1. A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See .
2. Two variables that are directly proportional to one another will have a constant ratio.
3. A relationship where one quantity is a constant divided by another quantity is called inverse variation. See .
4. Two variables that are inversely proportional to one another will have a constant multiple. See .
5. In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, write an equation describing the relationship of the given variables.
### Numeric
For the following exercises, use the given information to find the unknown value.
### Technology
For the following exercises, use a calculator to graph the equation implied by the given variation.
### Extensions
For the following exercises, use Kepler’s Law, which states that the square of the time,
required for a planet to orbit the Sun varies directly with the cube of the mean distance,
that the planet is from the Sun.
### Real-World Applications
For the following exercises, use the given information to answer the questions.
### Chapter Review Exercises
You have reached the end of Chapter 3: Polynomial and Rational Functions. Let’s review some of the Key Terms, Concepts and Equations you have learned.
### Complex Numbers
Perform the indicated operation with complex numbers.
Solve the following equations over the complex number system.
### Quadratic Functions
For the following exercises, write the quadratic function in standard form. Then, give the vertex and axes intercepts. Finally, graph the function.
For the following problems, find the equation of the quadratic function using the given information.
Answer the following questions.
### Power Functions and Polynomial Functions
For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.
For the following exercises, determine end behavior of the polynomial function.
### Graphs of Polynomial Functions
For the following exercises, find all zeros of the polynomial function, noting multiplicities.
For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.
### Dividing Polynomials
For the following exercises, use long division to find the quotient and remainder.
For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.
### Zeros of Polynomial Functions
For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.
For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.
### Rational Functions
For the following rational functions, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph.
For the following exercises, find the slant asymptote.
### Inverses and Radical Functions
For the following exercises, find the inverse of the function with the domain given.
### Modeling Using Variation
For the following exercises, find the unknown value.
For the following exercises, solve the application problem.
### Chapter Test
Perform the indicated operation or solve the equation.
Give the degree and leading coefficient of the following polynomial function.
Determine the end behavior of the polynomial function.
Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.
Given information about the graph of a quadratic function, find its equation.
Solve the following application problem.
Find all zeros of the following polynomial functions, noting multiplicities.
Based on the graph, determine the zeros of the function and multiplicities.
Use long division to find the quotient.
Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.
Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.
Given the following information about a polynomial function, find the function.
Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.
For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.
Find the slant asymptote of the rational function.
Find the inverse of the function.
Find the unknown value.
Solve the following application problem. |
# Exponential and Logarithmic Functions
## Introduction to Exponential and Logarithmic Functions
Focus in on a square centimeter of your skin. Look closer. Closer still. If you could look closely enough, you would see hundreds of thousands of microscopic organisms. They are bacteria, and they are not only on your skin, but in your mouth, nose, and even your intestines. In fact, the bacterial cells in your body at any given moment outnumber your own cells. But that is no reason to feel bad about yourself. While some bacteria can cause illness, many are healthy and even essential to the body.
Bacteria commonly reproduce through a process called binary fission, during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours, as opposed to days or years.Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. http://textbookofbacteriology.net/growth_3.html.
For simplicity’s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! And if we were to extrapolate the table to twenty-four hours, we would have over 16 million!
In this chapter, we will explore exponential functions, which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions, which are closely related to exponential functions. Both types of functions have numerous real-world applications when it comes to modeling and interpreting data. |
# Exponential and Logarithmic Functions
## Exponential Functions
### Learning Objectives
1. Find the value of a function (exponential). (IA 3.5.3)
2. Graph exponential functions. (IA 10.2.1)
### Objective 1: Find the value of a function (exponential). (IA 3.5.3)
### Practice Makes Perfect
Find the value of an exponential function.
### Objective 2: Graph exponential functions. (IA 10.2.1)
### Practice Makes Perfect
Graph exponential functions.
The number e, e ≈ 2.718281827, is like the number π in that we use a symbol to represent it because its decimal representation never stops or repeats. The irrational number e is called the natural base or Euler's number after the Swiss mathematician Leonhard Euler.
The exponential function whose base is e, is called the natural exponential function.
### Practice Makes Perfect
India is the second most populous country in the world with a population of about
billion people in 2021. The population is growing at a rate of about
each yearhttp://www.worldometers.info/world-population/. Accessed February 24, 2014.. If this rate continues, the population of India will exceed China’s population by the year
When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions, which model this kind of rapid growth.
### Identifying Exponential Functions
When exploring linear growth, we observed a constant rate of change—a constant number by which the output increased for each unit increase in input. For example, in the equation
the slope tells us the output increases by 3 each time the input increases by 1. The scenario in the India population example is different because we have a percent change per unit time (rather than a constant change) in the number of people.
### Defining an Exponential Function
A study found that the percent of the population who are vegans in the United States doubled from 2009 to 2011. In 2011, 2.5% of the population was vegan, adhering to a diet that does not include any animal products—no meat, poultry, fish, dairy, or eggs. If this rate continues, vegans will make up 10% of the U.S. population in 2015, 40% in 2019, and 80% in 2021.
What exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.
For us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth. We will construct two functions. The first function is exponential. We will start with an input of 0, and increase each input by 1. We will double the corresponding consecutive outputs. The second function is linear. We will start with an input of 0, and increase each input by 1. We will add 2 to the corresponding consecutive outputs. See .
From we can infer that for these two functions, exponential growth dwarfs linear growth.
1. Exponential growth refers to the original value from the range increases by the same percentage over equal increments found in the domain.
2. Linear growth refers to the original value from the range increases by the same amount over equal increments found in the domain.
Apparently, the difference between “the same percentage” and “the same amount” is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding 2 to the output whenever the input was increased by one.
The general form of the exponential function is
where
is any nonzero number,
is a positive real number not equal to 1.
1. If
the function grows at a rate proportional to its size.
2. If
the function decays at a rate proportional to its size.
Let’s look at the function
from our example. We will create a table () to determine the corresponding outputs over an interval in the domain from
to
Let us examine the graph of
by plotting the ordered pairs we observe on the table in , and then make a few observations.
Let’s define the behavior of the graph of the exponential function
and highlight some its key characteristics.
1. the domain is
2. the range is
3. as
4. as
5.
is always increasing,
6. the graph of
will never touch the x-axis because base two raised to any exponent never has the result of zero.
7.
is the horizontal asymptote.
8. the y-intercept is 1.
### Evaluating Exponential Functions
Recall that the base of an exponential function must be a positive real number other than
Why do we limit the base
to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:
1. Let
and
Then
which is not a real number.
Why do we limit the base to positive values other than
Because base
results in the constant function. Observe what happens if the base is
1. Let
Then
for any value of
To evaluate an exponential function with the form
we simply substitute
with the given value, and calculate the resulting power. For example:
Let
What is
To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:
Let
What is
Note that if the order of operations were not followed, the result would be incorrect:
### Defining Exponential Growth
Because the output of exponential functions increases very rapidly, the term “exponential growth” is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.
In more general terms, we have an exponential function, in which a constant base is raised to a variable exponent. To differentiate between linear and exponential functions, let’s consider two companies, A and B. Company A has 100 stores and expands by opening 50 new stores a year, so its growth can be represented by the function
Company B has 100 stores and expands by increasing the number of stores by 50% each year, so its growth can be represented by the function
A few years of growth for these companies are illustrated in .
The graphs comparing the number of stores for each company over a five-year period are shown in . We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.
Notice that the domain for both functions is
and the range for both functions is
After year 1, Company B always has more stores than Company A.
Now we will turn our attention to the function representing the number of stores for Company B,
In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and
represents the growth factor. Generalizing further, we can write this function as
where 100 is the initial value,
is called the base, and
is called the exponent.
### Finding Equations of Exponential Functions
In the previous examples, we were given an exponential function, which we then evaluated for a given input. Sometimes we are given information about an exponential function without knowing the function explicitly. We must use the information to first write the form of the function, then determine the constants
and
and evaluate the function.
### Applying the Compound-Interest Formula
Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest. The term compounding refers to interest earned not only on the original value, but on the accumulated value of the account.
The annual percentage rate (APR) of an account, also called the nominal rate, is the yearly interest rate earned by an investment account. The term nominal is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.
We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time
principal
APR
and number of compounding periods in a year
For example, observe , which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.
### Evaluating Functions with Base e
As we saw earlier, the amount earned on an account increases as the compounding frequency increases. shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.
Examine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies, listed in .
These values appear to be approaching a limit as
increases without bound. In fact, as
gets larger and larger, the expression
approaches a number used so frequently in mathematics that it has its own name: the letter
This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.
### Investigating Continuous Growth
So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use
as the base are called continuous growth or decay models. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.
### Key Equations
### Key Concepts
1. An exponential function is defined as a function with a positive constant other than
raised to a variable exponent. See .
2. A function is evaluated by solving at a specific value. See and .
3. An exponential model can be found when the growth rate and initial value are known. See .
4. An exponential model can be found when the two data points from the model are known. See .
5. An exponential model can be found using two data points from the graph of the model. See .
6. An exponential model can be found using two data points from the graph and a calculator. See .
7. The value of an account at any time
can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known. See .
8. The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. See .
9. The number
is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is
10. Scientific and graphing calculators have the key
or
for calculating powers of
See .
11. Continuous growth or decay models are exponential models that use
as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known. See and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, identify whether the statement represents an exponential function. Explain.
For the following exercises, consider this scenario: For each year
the population of a forest of trees is represented by the function
In a neighboring forest, the population of the same type of tree is represented by the function
(Round answers to the nearest whole number.)
For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.
For the following exercises, find the formula for an exponential function that passes through the two points given.
For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.
For the following exercises, use the compound interest formula,
For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.
### Numeric
For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
### Technology
For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.
### Extensions
### Real-World Applications
|
# Exponential and Logarithmic Functions
## Graphs of Exponential Functions
### Learning Objectives
1. Graph exponential functions (IA 10.2.1).
2. Function transformations (exponential) (CA 3.5.1-3.5.5).
### Objective 1: Graph exponential functions (IA 10.2.1).
### Practice Makes Perfect
### Objective 2: Function transformations (exponential). (CA 3.5.1-3.5.5)
Vertical and Horizontal Shifts: Given a function , a new function where is a constant, is a vertical shift of the function . All the output values change by k units. If k is a positive, the graph will shift up. If k is negative, the graph will shift down.
Given a function , a new function , where h is a constant, is a horizontal shift of the function . If h is positive, the graph will shift right. If h is negative, the graph will shift left.
### Practice Makes Perfect
Function transformations (exponential).
As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.
### Graphing Exponential Functions
Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form
whose base is greater than one. We’ll use the function
Observe how the output values in change as the input increases by
Each output value is the product of the previous output and the base,
We call the base
the constant ratio. In fact, for any exponential function with the form
is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of
Notice from the table that
1. the output values are positive for all values of
2. as
increases, the output values increase without bound; and
3. as
decreases, the output values grow smaller, approaching zero.
shows the exponential growth function
The domain of
is all real numbers, the range is
and the horizontal asymptote is
To get a sense of the behavior of exponential decay, we can create a table of values for a function of the form
whose base is between zero and one. We’ll use the function
Observe how the output values in change as the input increases by
Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio
Notice from the table that
1. the output values are positive for all values of
2. as
increases, the output values grow smaller, approaching zero; and
3. as
decreases, the output values grow without bound.
shows the exponential decay function,
The domain of
is all real numbers, the range is
and the horizontal asymptote is
### Graphing Transformations of Exponential Functions
Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function
without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.
### Graphing a Vertical Shift
The first transformation occurs when we add a constant
to the parent function
giving us a vertical shift
units in the same direction as the sign. For example, if we begin by graphing a parent function,
we can then graph two vertical shifts alongside it, using
the upward shift,
and the downward shift,
Both vertical shifts are shown in .
Observe the results of shifting
vertically:
1. The domain,
remains unchanged.
2. When the function is shifted up
units to
3. When the function is shifted down
units to
### Graphing a Horizontal Shift
The next transformation occurs when we add a constant
to the input of the parent function
giving us a horizontal shift
units in the opposite direction of the sign. For example, if we begin by graphing the parent function
we can then graph two horizontal shifts alongside it, using
the shift left,
and the shift right,
Both horizontal shifts are shown in .
Observe the results of shifting
horizontally:
1. The domain,
remains unchanged.
2. The asymptote,
remains unchanged.
3. The y-intercept shifts such that:
### Graphing a Stretch or Compression
While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function
by a constant
For example, if we begin by graphing the parent function
we can then graph the stretch, using
to get
as shown on the left in , and the compression, using
to get
as shown on the right in .
### Graphing Reflections
In addition to shifting, compressing, and stretching a graph, we can also reflect it about the x-axis or the y-axis. When we multiply the parent function
by
we get a reflection about the x-axis. When we multiply the input by
we get a reflection about the y-axis. For example, if we begin by graphing the parent function
we can then graph the two reflections alongside it. The reflection about the x-axis,
is shown on the left side of , and the reflection about the y-axis
is shown on the right side of .
### Summarizing Translations of the Exponential Function
Now that we have worked with each type of translation for the exponential function, we can summarize them in to arrive at the general equation for translating exponential functions.
### Key Equations
### Key Concepts
1. The graph of the function
has a y-intercept at
domain
range
and horizontal asymptote
See .
2. If
the function is increasing. The left tail of the graph will approach the asymptote
and the right tail will increase without bound.
3. If
the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote
4. The equation
represents a vertical shift of the parent function
5. The equation
represents a horizontal shift of the parent function
See .
6. Approximate solutions of the equation
can be found using a graphing calculator. See .
7. The equation
where
represents a vertical stretch if
or compression if
of the parent function
See .
8. When the parent function
is multiplied by
the result,
is a reflection about the x-axis. When the input is multiplied by
the result,
is a reflection about the y-axis. See .
9. All translations of the exponential function can be summarized by the general equation
See .
10. Using the general equation
we can write the equation of a function given its description. See .
### Section Exercises
### Verbal
### Algebraic
### Graphical
For the following exercises, graph the function and its reflection about the y-axis on the same axes, and give the y-intercept.
For the following exercises, graph each set of functions on the same axes.
For the following exercises, match each function with one of the graphs in .
For the following exercises, use the graphs shown in . All have the form
For the following exercises, graph the function and its reflection about the x-axis on the same axes.
For the following exercises, graph the transformation of
Give the horizontal asymptote, the domain, and the range.
For the following exercises, describe the end behavior of the graphs of the functions.
For the following exercises, start with the graph of
Then write a function that results from the given transformation.
For the following exercises, each graph is a transformation of
Write an equation describing the transformation.
For the following exercises, find an exponential equation for the graph.
### Numeric
For the following exercises, evaluate the exponential functions for the indicated value of
### Technology
For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.
### Extensions
|
# Exponential and Logarithmic Functions
## Logarithmic Functions
### Learning Objectives
1. Convert between exponential and logarithmic form. (IA 10.3.1)
2. Evaluate logarithmic functions. (IA 10.3.2)
### Objective 1: Convert between exponential and logarithmic form. (IA 10.3.1)
### Practice Makes Perfect
Since the equations and are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth, let’s take a close look at the equations. Notice the positions of the exponent and base.
If we remember the logarithm is the exponent, it makes the conversion easier. You may want to repeat, “base to the exponent gives us the number.”
### Practice Makes Perfect
Convert between exponential and logarithmic form.
Remember these logarithmic notations to help complete the following:
Common Logarithm
Natural Logarithm
### Objective 2: Evaluate logarithmic functions (IA 10.3.2).
We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential form.
### Practice Makes Perfect
In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homeshttp://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013.. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013. like those shown in . Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scalehttp://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013. whereas the Japanese earthquake registered a 9.0.http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.
The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is
times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.
### Converting from Logarithmic to Exponential Form
In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is
where
represents the difference in magnitudes on the Richter Scale. How would we solve for
We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve
We know that
and
so it is clear that
must be some value between 2 and 3, since
is increasing. We can examine a graph, as in , to better estimate the solution.
Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in passes the horizontal line test. The exponential function
is one-to-one, so its inverse,
is also a function. As is the case with all inverse functions, we simply interchange
and
and solve for
to find the inverse function. To represent
as a function of
we use a logarithmic function of the form
The base
logarithm of a number is the exponent by which we must raise
to get that number.
We read a logarithmic expression as, “The logarithm with base
of
is equal to
” or, simplified, “log base
of
is
” We can also say, “
raised to the power of
is
” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since
we can write
We read this as “log base 2 of 32 is 5.”
We can express the relationship between logarithmic form and its corresponding exponential form as follows:
Note that the base
is always positive.
Because logarithm is a function, it is most correctly written as
using parentheses to denote function evaluation, just as we would with
However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as
Note that many calculators require parentheses around the
We can illustrate the notation of logarithms as follows:
Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means
and
are inverse functions.
### Converting from Exponential to Logarithmic Form
To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base
exponent
and output
Then we write
### Evaluating Logarithms
Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider
We ask, “To what exponent must
be raised in order to get 8?” Because we already know
it follows that
Now consider solving
and
mentally.
1. We ask, “To what exponent must 7 be raised in order to get 49?” We know
Therefore,
2. We ask, “To what exponent must 3 be raised in order to get 27?” We know
Therefore,
Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate
mentally.
1. We ask, “To what exponent must
be raised in order to get
” We know
and
so
Therefore,
### Using Common Logarithms
Sometimes you may see a logarithm written without a base. When you see one written this way, you need to look at the expression before evaluating it. It may be that the base you use doesn't matter. If you find it in computer science, it often means
. However, in mathematics it almost always means the common logarithm of 10. In other words, the expression
often means
Currently, we use
as the common logarithm,
as the binary logarithm, and
as the natural logarithm. Writing
without specifying a base is now considered bad form, despite being frequently found in older materials.
### Using Natural Logarithms
The most frequently used base for logarithms is
Base
logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base
logarithm,
has its own notation,
Most values of
can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base,
For other natural logarithms, we can use the
key that can be found on most scientific calculators. We can also find the natural logarithm of any power of
using the inverse property of logarithms.
### Key Equations
### Key Concepts
1. The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
2. Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See .
3. Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See .
4. Logarithmic functions with base
can be evaluated mentally using previous knowledge of powers of
See and .
5. Common logarithms can be evaluated mentally using previous knowledge of powers of
See .
6. When common logarithms cannot be evaluated mentally, a calculator can be used. See .
7. Real-world exponential problems with base
can be rewritten as a common logarithm and then evaluated using a calculator. See .
8. Natural logarithms can be evaluated using a calculator .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, rewrite each equation in exponential form.
For the following exercises, rewrite each equation in logarithmic form.
For the following exercises, solve for
by converting the logarithmic equation to exponential form.
For the following exercises, use the definition of common and natural logarithms to simplify.
### Numeric
For the following exercises, evaluate the base
logarithmic expression without using a calculator.
For the following exercises, evaluate the common logarithmic expression without using a calculator.
For the following exercises, evaluate the natural logarithmic expression without using a calculator.
### Technology
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.
### Extensions
### Real-World Applications
|
# Exponential and Logarithmic Functions
## Graphs of Logarithmic Functions
### Learning Objectives
1. Find the domain and range of a relation and a function. (IA 3.5.1)
2. Graph Logarithmic functions. (IA 10.3.3)
### Objective 1: Find the domain and range of a relation and a function. (IA 3.5.1)
### Practice Makes Perfect
Find the domain and range of a relation and a function.
### Objective 2: Graph Logarithmic functions. (IA 10.3.3)
To graph a logarithmic function , it is easiest to convert the equation to its exponential form, . Generally, when we look for ordered pairs for the graph of a function, we usually choose an x-value and then determine its corresponding y-value. In this case you may find it easier to choose y-values and then determine its corresponding x-value.
### Practice Makes Perfect
Graph Logarithmic functions
In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect.
To illustrate, suppose we invest
in an account that offers an annual interest rate of
compounded continuously. We already know that the balance in our account for any year
can be found with the equation
But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? shows this point on the logarithmic graph.
In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.
### Finding the Domain of a Logarithmic Function
Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.
Recall that the exponential function is defined as
for any real number
and constant
where
1. The domain of
is
2. The range of
is
In the last section we learned that the logarithmic function
is the inverse of the exponential function
So, as inverse functions:
1. The domain of
is the range of
2. The range of
is the domain of
Transformations of the parent function
behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections.
In Graphs of Exponential Functions we saw that certain transformations can change the range of
Similarly, applying transformations to the parent function
can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero.
For example, consider
This function is defined for any values of
such that the argument, in this case
is greater than zero. To find the domain, we set up an inequality and solve for
In interval notation, the domain of
is
### Graphing Logarithmic Functions
Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function
along with all its transformations: shifts, stretches, compressions, and reflections.
We begin with the parent function
Because every logarithmic function of this form is the inverse of an exponential function with the form
their graphs will be reflections of each other across the line
To illustrate this, we can observe the relationship between the input and output values of
and its equivalent
in .
Using the inputs and outputs from , we can build another table to observe the relationship between points on the graphs of the inverse functions
and
See .
As we’d expect, the x- and y-coordinates are reversed for the inverse functions. shows the graph of
and
Observe the following from the graph:
1.
has a y-intercept at
and
has an x- intercept at
2. The domain of
is the same as the range of
3. The range of
is the same as the domain of
### Graphing Transformations of Logarithmic Functions
As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function
without loss of shape.
### Graphing a Horizontal Shift of f(x) = log(x)
When a constant
is added to the input of the parent function
the result is a horizontal shift
units in the opposite direction of the sign on
To visualize horizontal shifts, we can observe the general graph of the parent function
and for
alongside the shift left,
and the shift right,
See .
### Graphing a Vertical Shift of y = log(x)
When a constant
is added to the parent function
the result is a vertical shift
units in the direction of the sign on
To visualize vertical shifts, we can observe the general graph of the parent function
alongside the shift up,
and the shift down,
See .
### Graphing Stretches and Compressions of y = log(x)
When the parent function
is multiplied by a constant
the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set
and observe the general graph of the parent function
alongside the vertical stretch,
and the vertical compression,
See .
### Graphing Reflections of f(x) = log(x)
When the parent function
is multiplied by
the result is a reflection about the x-axis. When the input is multiplied by
the result is a reflection about the y-axis. To visualize reflections, we restrict
and observe the general graph of the parent function
alongside the reflection about the x-axis,
and the reflection about the y-axis,
### Summarizing Translations of the Logarithmic Function
Now that we have worked with each type of translation for the logarithmic function, we can summarize each in to arrive at the general equation for translating exponential functions.
### Key Equations
### Key Concepts
1. To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for
See and
2. The graph of the parent function
has an x-intercept at
domain
range
vertical asymptote
and
See .
3. The equation
shifts the parent function
horizontally
See .
4. The equation
shifts the parent function
vertically
See .
5. For any constant
the equation
See and .
6. When the parent function
is multiplied by
the result is a reflection about the x-axis. When the input is multiplied by
the result is a reflection about the y-axis.
See .
7. All translations of the logarithmic function can be summarized by the general equation
See .
8. Given an equation with the general form
we can identify the vertical asymptote
for the transformation. See .
9. Using the general equation
we can write the equation of a logarithmic function given its graph. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, state the domain and range of the function.
For the following exercises, state the domain and the vertical asymptote of the function.
For the following exercises, state the domain, vertical asymptote, and end behavior of the function.
For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.
### Graphical
For the following exercises, match each function in with the letter corresponding to its graph.
For the following exercises, match each function in with the letter corresponding to its graph.
For the following exercises, sketch the graphs of each pair of functions on the same axis.
For the following exercises, match each function in with the letter corresponding to its graph.
For the following exercises, sketch the graph of the indicated function.
For the following exercises, write a logarithmic equation corresponding to the graph shown.
### Technology
For the following exercises, use a graphing calculator to find approximate solutions to each equation.
### Extensions
|
# Exponential and Logarithmic Functions
## Logarithmic Properties
### Learning Objectives
1. Simplify expressions using the properties for exponents. (IA 5.2.1)
2. Use the properties of logarithms. (IA 10.4.1)
### Objective 1: Simplify expressions using the properties for exponents (IA 5.2.1)
### The Product Property
Simplify expressions using the properties for exponents.
To multiply powers with the same base we need to ________ exponents.
This leads us to the Product Property
### The Quotient Property
Simplify
To divide powers with the same base we need to __________ exponents.
This leads us to the Quotient Property
### The Power Property
Simplify
To raise a power to a power we need to __________ exponents.
This leads us to the Power Property .
We will also use these other properties:
### Practice Makes Perfect
Simplify expressions using the properties for exponents.
### Objective 2: Use the properties of logarithms (IA 10.4.1).
### Practice Makes Perfect
In chemistry, pH is used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14. Substances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be basic. Our bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is acidic and what is basic, consider the following pH levels of some common substances:
1. Battery acid: 0.8
2. Stomach acid: 2.7
3. Orange juice: 3.3
4. Pure water: 7 (at 25° C)
5. Human blood: 7.35
6. Fresh coconut: 7.8
7. Sodium hydroxide (lye): 14
To determine whether a solution is acidic or basic, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where
is the concentration of hydrogen ion in the solution
The equivalence of
and
is one of the logarithm properties we will examine in this section.
### Using the Product Rule for Logarithms
Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.
For example,
since
And
since
Next, we have the inverse property.
For example, to evaluate
we can rewrite the logarithm as
and then apply the inverse property
to get
To evaluate
we can rewrite the logarithm as
and then apply the inverse property
to get
Finally, we have the one-to-one property.
We can use the one-to-one property to solve the equation
for
Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for
But what about the equation
The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining terms on the left side of the equation.
Recall that we use the product rule of exponents to combine the product of powers by adding exponents:
We have a similar property for logarithms, called the product rule for logarithms, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.
Given any real number
and positive real numbers
and
where
we will show
Let
and
In exponential form, these equations are
and
It follows that
Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider
Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:
### Using the Quotient Rule for Logarithms
For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to combine the quotient of exponents by subtracting:
The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.
Given any real number
and positive real numbers
and
where
we will show
Let
and
In exponential form, these equations are
and
It follows that
For example, to expand
we must first express the quotient in lowest terms. Factoring and canceling we get,
Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.
### Using the Power Rule for Logarithms
We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as
One method is as follows:
Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,
### Expanding Logarithmic Expressions
Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.” Sometimes we apply more than one rule in order to simplify an expression. For example:
We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:
We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.
With practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. Remember, however, that we can only do this with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm.
### Condensing Logarithmic Expressions
We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.
### Using the Change-of-Base Formula for Logarithms
Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 or
we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.
To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms.
Given any positive real numbers
and
where
and
we show
Let
By exponentiating both sides with base
, we arrive at an exponential form, namely
It follows that
For example, to evaluate
using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.
### Key Equations
### Key Concepts
1. We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See .
2. We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See .
3. We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See , , and .
4. We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. See , , and .
5. The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See , , , and .
6. We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. See .
7. The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and
as the quotient of natural or common logs. That way a calculator can be used to evaluate. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
For the following exercises, condense to a single logarithm if possible.
For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
For the following exercises, condense each expression to a single logarithm using the properties of logarithms.
For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.
For the following exercises, suppose
and
Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of
and
Show the steps for solving.
### Numeric
For the following exercises, use properties of logarithms to evaluate without using a calculator.
For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.
### Extensions
|
# Exponential and Logarithmic Functions
## Exponential and Logarithmic Equations
### Learning Objectives
1. Solve Exponential Equations. (IA 10.2.2)
2. Solve Logarithmic Equations. (IA 10.3.4)
### Objective 1: Solve Exponential Equations. (IA 10.2.2)
Equations that include an exponential expression are called exponential equations. There are two types of exponential equations: those with the common base on each side, and those without a common base.
Type 1: Possible common base on each side: Use properties of exponents to rewrite each side with a common base. Use base-exponent property to set exponents equal to each other and solve for x.
Type 2: No possible common base: Use properties of exponents to rewrite each side in terms of one exponential expression. Take the log or ln of each side and use the power rule to bring down the power. Solve the remaining equation for x.
### Practice Makes Perfect
Solve. Find the exact answer and then approximate it to three decimal places.
### Objective 2: Solving Logarithmic Equations. (IA 10.3.4)
There are two types of logarithmic equations: those with log terms on just one side of the equation or those with log terms on each side of the equation. Since the domain of logarithmic functions is positive numbers only, make sure to check the solutions.
Type 1: Log terms on one side of the equation: Use properties of logs to rewrite a side with just one log term. Convert to exponential notation and solve for x.
If then .
Type 2: Log terms on both sides of equation: First, use log properties to rewrite each side in terms of a single log expression, if necessary. Use the one-to-one property of logarithmic equality to set arguments equal to one another. Solve the resulting equation for x.
### Practice Makes Perfect
Don’t forget to check your solutions.
In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions.
Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.
### Using Like Bases to Solve Exponential Equations
The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers
and
where
if and only if
In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.
For example, consider the equation
To solve for
we use the division property of exponents to rewrite the right side so that both sides have the common base,
Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for
:
### Rewriting Equations So All Powers Have the Same Base
Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.
For example, consider the equation
We can rewrite both sides of this equation as a power of
Then we apply the rules of exponents, along with the one-to-one property, to solve for
### Solving Exponential Equations Using Logarithms
Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since
is equivalent to
we may apply logarithms with the same base on both sides of an exponential equation.
### Equations Containing e
One common type of exponential equations are those with base
This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base
on either side, we can use the natural logarithm to solve it.
### Extraneous Solutions
Sometimes the methods used to solve an equation introduce an extraneous solution, which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.
### Using the Definition of a Logarithm to Solve Logarithmic Equations
We have already seen that every logarithmic equation
is equivalent to the exponential equation
We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.
For example, consider the equation
To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for
### Using the One-to-One Property of Logarithms to Solve Logarithmic Equations
As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers
and any positive real number
where
For example,
So, if
then we can solve for
and we get
To check, we can substitute
into the original equation:
In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown.
For example, consider the equation
To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for
To check the result, substitute
into
### Solving Applied Problems Using Exponential and Logarithmic Equations
In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.
One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life. lists the half-life for several of the more common radioactive substances.
We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:
where
1.
is the amount initially present
2.
is the half-life of the substance
3.
is the time period over which the substance is studied
4.
is the amount of the substance present after time
### Key Equations
### Key Concepts
1. We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
2. When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See .
3. When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See , , and .
4. When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See .
5. We can solve exponential equations with base
by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See and .
6. After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See .
7. When given an equation of the form
where
is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation
and solve for the unknown. See and .
8. We can also use graphing to solve equations with the form
We graph both equations
and
on the same coordinate plane and identify the solution as the x-value of the intersecting point. See .
9. When given an equation of the form
where
and
are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation
for the unknown. See .
10. Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, use like bases to solve the exponential equation.
For the following exercises, use logarithms to solve.
For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.
For the following exercises, use the definition of a logarithm to solve the equation.
For the following exercises, use the one-to-one property of logarithms to solve.
For the following exercises, solve each equation for
### Graphical
For the following exercises, solve the equation for
if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.
For the following exercises, solve for the indicated value, and graph the situation showing the solution point.
### Technology
For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.
For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.
### Extensions
|
# Exponential and Logarithmic Functions
## Exponential and Logarithmic Models
### Learning Objectives
1. Use exponential models in applications. (IA 10.2.3)
2. Use logarithmic models in applications. (IA 10.3.5)
### Objective 1: Use exponential models in applications. (IA 10.2.3)
### Using exponential models
Exponential functions model many situations. If you have a savings account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, P, is invested at an interest rate, r, for t years, the new balance, A, will depend on how often the interest is compounded.
### Exponential Growth and Decay
Other topics that are modeled by exponential functions involve growth and decay. Both also use the formula we used for the growth of money. For growth and decay, generally we use as the original amount instead of calling it the principal. We see that exponential growth has a positive rate of growth and exponential decay has a negative rate of growth.
### Practice Makes Perfect
### Objective 2: Use logarithmic models in applications. (IA 10.3.5)
### Decibel Level of Sound
There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The10-12 in the formula represents the intensity of sound that is barely audible.
The magnitude of an earthquake is measured by a logarithmic scale called the Richter scale. The model is where is the intensity of the shock wave. This model provides a way to measure earthquake intensity.
### Practice Makes Perfect
Use logarithmic models in applications.
We have already explored some basic applications of exponential and logarithmic functions. In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling.
### Modeling Exponential Growth and Decay
In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth, we may choose the exponential growth function:
where is equal to the value at time zero, is Euler’s constant, and is a positive constant that determines the rate (percentage) of growth. We may use the exponential growth function in applications involving doubling time, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time. In some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model.
On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model. Again, we have the form where is the starting value, and is Euler’s constant. Now is a negative constant that determines the rate of decay. We may use the exponential decay model when we are calculating half-life, or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.
In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential growth and decay graphs have a distinctive shape, as we can see in and . It is important to remember that, although parts of each of the two graphs seem to lie on the x-axis, they are really a tiny distance above the x-axis.
Exponential growth and decay often involve very large or very small numbers. To describe these numbers, we often use orders of magnitude. The order of magnitude is the power of ten, when the number is expressed in scientific notation, with one digit to the left of the decimal. For example, the distance to the nearest star, Proxima Centauri, measured in kilometers, is 40,113,497,200,000 kilometers. Expressed in scientific notation, this is So, we could describe this number as having order of magnitude
### Half-Life
We now turn to exponential decay. One of the common terms associated with exponential decay, as stated above, is half-life, the length of time it takes an exponentially decaying quantity to decrease to half its original amount. Every radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is called radioactive decay.
To find the half-life of a function describing exponential decay, solve the following equation:
We find that the half-life depends only on the constant and not on the starting quantity
The formula is derived as follows
Since the time, is positive, must, as expected, be negative. This gives us the half-life formula
### Radiocarbon Dating
The formula for radioactive decay is important in radiocarbon dating, which is used to calculate the approximate date a plant or animal died. Radiocarbon dating was discovered in 1949 by Willard Libby, who won a Nobel Prize for his discovery. It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air. It is believed to be accurate to within about 1% error for plants or animals that died within the last 60,000 years.
Carbon-14 is a radioactive isotope of carbon that has a half-life of 5,730 years. It occurs in small quantities in the carbon dioxide in the air we breathe. Most of the carbon on Earth is carbon-12, which has an atomic weight of 12 and is not radioactive. Scientists have determined the ratio of carbon-14 to carbon-12 in the air for the last 60,000 years, using tree rings and other organic samples of known dates—although the ratio has changed slightly over the centuries.
As long as a plant or animal is alive, the ratio of the two isotopes of carbon in its body is close to the ratio in the atmosphere. When it dies, the carbon-14 in its body decays and is not replaced. By comparing the ratio of carbon-14 to carbon-12 in a decaying sample to the known ratio in the atmosphere, the date the plant or animal died can be approximated.
Since the half-life of carbon-14 is 5,730 years, the formula for the amount of carbon-14 remaining after years is
where
This formula is derived as follows:
To find the age of an object, we solve this equation for
Out of necessity, we neglect here the many details that a scientist takes into consideration when doing carbon-14 dating, and we only look at the basic formula. The ratio of carbon-14 to carbon-12 in the atmosphere is approximately 0.0000000001%. Let be the ratio of carbon-14 to carbon-12 in the organic artifact or fossil to be dated, determined by a method called liquid scintillation. From the equation we know the ratio of the percentage of carbon-14 in the object we are dating to the initial amount of carbon-14 in the object when it was formed is We solve this equation for to get
### Calculating Doubling Time
For decaying quantities, we determined how long it took for half of a substance to decay. For growing quantities, we might want to find out how long it takes for a quantity to double. As we mentioned above, the time it takes for a quantity to double is called the doubling time.
Given the basic exponential growth equation
doubling time can be found by solving for when the original quantity has doubled, that is, by solving
The formula is derived as follows:
Thus the doubling time is
### Using Newton’s Law of Cooling
Exponential decay can also be applied to temperature. When a hot object is left in surrounding air that is at a lower temperature, the object’s temperature will decrease exponentially, leveling off as it approaches the surrounding air temperature. On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the temperature of the surrounding air. Unless the room temperature is zero, this will correspond to a vertical shift of the generic exponential decay function. This translation leads to Newton’s Law of Cooling, the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature
This formula is derived as follows:
### Using Logistic Growth Models
Exponential growth cannot continue forever. Exponential models, while they may be useful in the short term, tend to fall apart the longer they continue. Consider an aspiring writer who writes a single line on day one and plans to double the number of lines she writes each day for a month. By the end of the month, she must write over 17 billion lines, or one-half-billion pages. It is impractical, if not impossible, for anyone to write that much in such a short period of time. Eventually, an exponential model must begin to approach some limiting value, and then the growth is forced to slow. For this reason, it is often better to use a model with an upper bound instead of an exponential growth model, though the exponential growth model is still useful over a short term, before approaching the limiting value.
The logistic growth model is approximately exponential at first, but it has a reduced rate of growth as the output approaches the model’s upper bound, called the carrying capacity. For constants
and
the logistic growth of a population over time
is represented by the model
The graph in shows how the growth rate changes over time. The graph increases from left to right, but the growth rate only increases until it reaches its point of maximum growth rate, at which point the rate of increase decreases.
### Choosing an Appropriate Model for Data
Now that we have discussed various mathematical models, we need to learn how to choose the appropriate model for the raw data we have. Many factors influence the choice of a mathematical model, among which are experience, scientific laws, and patterns in the data itself. Not all data can be described by elementary functions. Sometimes, a function is chosen that approximates the data over a given interval. For instance, suppose data were gathered on the number of homes bought in the United States from the years 1960 to 2013. After plotting these data in a scatter plot, we notice that the shape of the data from the years 2000 to 2013 follow a logarithmic curve. We could restrict the interval from 2000 to 2010, apply regression analysis using a logarithmic model, and use it to predict the number of home buyers for the year 2015.
Three kinds of functions that are often useful in mathematical models are linear functions, exponential functions, and logarithmic functions. If the data lies on a straight line, or seems to lie approximately along a straight line, a linear model may be best. If the data is non-linear, we often consider an exponential or logarithmic model, though other models, such as quadratic models, may also be considered.
In choosing between an exponential model and a logarithmic model, we look at the way the data curves. This is called the concavity. If we draw a line between two data points, and all (or most) of the data between those two points lies above that line, we say the curve is concave down. We can think of it as a bowl that bends downward and therefore cannot hold water. If all (or most) of the data between those two points lies below the line, we say the curve is concave up. In this case, we can think of a bowl that bends upward and can therefore hold water. An exponential curve, whether rising or falling, whether representing growth or decay, is always concave up away from its horizontal asymptote. A logarithmic curve is always concave away from its vertical asymptote. In the case of positive data, which is the most common case, an exponential curve is always concave up, and a logarithmic curve always concave down.
A logistic curve changes concavity. It starts out concave up and then changes to concave down beyond a certain point, called a point of inflection.
After using the graph to help us choose a type of function to use as a model, we substitute points, and solve to find the parameters. We reduce round-off error by choosing points as far apart as possible.
### Expressing an Exponential Model in Base
While powers and logarithms of any base can be used in modeling, the two most common bases are
and
In science and mathematics, the base
is often preferred. We can use laws of exponents and laws of logarithms to change any base to base
### Key Equations
### Key Concepts
1. The basic exponential function is
If
we have exponential growth; if
we have exponential decay.
2. We can also write this formula in terms of continuous growth as
where
is the starting value. If
is positive, then we have exponential growth when
and exponential decay when
See .
3. In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See .
4. We can find the age,
of an organic artifact by measuring the amount,
of carbon-14 remaining in the artifact and using the formula
to solve for
See .
5. Given a substance’s doubling time or half-time, we can find a function that represents its exponential growth or decay. See .
6. We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. See .
7. We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors. See .
8. We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. See .
9. Any exponential function with the form
can be rewritten as an equivalent exponential function with the form
where
See .
### Section Exercises
### Verbal
### Numeric
For the following exercises, use the logistic growth model
### Technology
For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.
For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in
years is modeled by the equation
### Extensions
### Real-World Applications
For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.
For the following exercises, use this scenario: A tumor is injected with
grams of Iodine-125, which has a decay rate of
per day.
For the following exercises, use this scenario: A biologist recorded a count of
bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes.
For the following exercises, use this scenario: A pot of warm soup with an internal temperature of
Fahrenheit was taken off the stove to cool in a
room. After fifteen minutes, the internal temperature of the soup was
For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of
and is allowed to cool in a
room. After half an hour, the internal temperature of the turkey is
For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.
For the following exercises, use this scenario: The equation
models the number of people in a town who have heard a rumor after t days.
For the following exercise, choose the correct answer choice. |
# Exponential and Logarithmic Functions
## Fitting Exponential Models to Data
### Learning Objectives
1. Draw and interpret scatter diagrams (linear, exponential, logarithmic). (CA 4.3.1)
2. Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions. (CA 4.3.4)
### Objective 1: Draw and interpret scatter diagrams (linear, exponential, logarithmic). (CA 4.3.1)
A Scatter Plot is a graph of plotted points that may show a relationship between the variables in a set of data.
### Practice Makes Perfect
Draw and interpret scatter diagrams ( linear, exponential, logarithmic).
### Objective 2: Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions. (CA 4.3.4)
We can find a linear function that fits the data in the previous problem by “eyeballing” a line that seems to fit. But while estimating a line works relatively well, technology can help us find a line that fits the data as perfect as possible.
This line is called the Least Squares Regression Line or Linear Regression Model.
A regression line is a line that is closest to the data in the scatter plot, which means that such a line is a best fit for the data.
Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions.
### Practice Makes Perfect
Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions.
In previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points exactly. In this section, we use a modeling technique called regression analysis to find a curve that models data collected from real-world observations. With regression analysis, we don’t expect all the points to lie perfectly on the curve. The idea is to find a model that best fits the data. Then we use the model to make predictions about future events.
Do not be confused by the word model. In mathematics, we often use the terms function, equation, and model interchangeably, even though they each have their own formal definition. The term model is typically used to indicate that the equation or function approximates a real-world situation.
We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review. Take a moment to rethink each of these functions, reflect on the work we’ve done so far, and then explore the ways regression is used to model real-world phenomena.
### Building an Exponential Model from Data
As we’ve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that’s not the whole story. It’s the way data increase or decrease that helps us determine whether it is best modeled by an exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let’s review exponential growth and decay.
Recall that exponential functions have the form
or
When performing regression analysis, we use the form most commonly used on graphing utilities,
Take a moment to reflect on the characteristics we’ve already learned about the exponential function
(assume
1.
must be greater than zero and not equal to one.
2. The initial value of the model is
As part of the results, your calculator will display a number known as the correlation coefficient, labeled by the variable
or
(You may have to change the calculator’s settings for these to be shown.) The values are an indication of the “goodness of fit” of the regression equation to the data. We more commonly use the value of
instead of
but the closer either value is to 1, the better the regression equation approximates the data.
### Building a Logarithmic Model from Data
Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best.
Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we’ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic regression analysis, we use the form of the logarithmic function most commonly used on graphing utilities,
For this function
1. All input values,
must be greater than zero.
2. The point
is on the graph of the model.
3. If
the model is increasing. Growth increases rapidly at first and then steadily slows over time.
4. If
the model is decreasing. Decay occurs rapidly at first and then steadily slows over time.
### Building a Logistic Model from Data
Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or limiting value. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.
It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic regression analysis, we use the form most commonly used on graphing utilities:
Recall that:
1.
is the initial value of the model.
2. when
the model increases rapidly at first until it reaches its point of maximum growth rate,
At that point, growth steadily slows and the function becomes asymptotic to the upper bound
3.
is the limiting value, sometimes called the carrying capacity, of the model.
### Key Concepts
1. Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.
2. We use the command “ExpReg” on a graphing utility to fit function of the form
to a set of data points. See .
3. Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.
4. We use the command “LnReg” on a graphing utility to fit a function of the form
to a set of data points. See .
5. Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit.
6. We use the command “Logistic” on a graphing utility to fit a function of the form
to a set of data points. See .
### Section Exercises
### Verbal
### Graphical
For the following exercises, match the given function of best fit with the appropriate scatterplot in through . Answer using the letter beneath the matching graph.
### Numeric
### Technology
For the following exercises, use this scenario: The population
of a koi pond over
months is modeled by the function
For the following exercises, use this scenario: The population
of an endangered species habitat for wolves is modeled by the function
where
is given in years.
For the following exercises, refer to .
For the following exercises, refer to .
For the following exercises, refer to .
For the following exercises, refer to .
For the following exercises, refer to .
For the following exercises, refer to .
### Extensions
### Chapter Review Exercises
### Exponential Functions
### Graphs of Exponential Functions
### Logarithmic Functions
### Graphs of Logarithmic Functions
### Logarithmic Properties
### Exponential and Logarithmic Equations
### Exponential and Logarithmic Models
For the following exercises, use this scenario: A doctor prescribes
milligrams of a therapeutic drug that decays by about
each hour.
For the following exercises, use this scenario: A soup with an internal temperature of
Fahrenheit was taken off the stove to cool in a
room. After fifteen minutes, the internal temperature of the soup was
For the following exercises, use this scenario: The equation
models the number of people in a school who have heard a rumor after
days.
For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.
### Fitting Exponential Models to Data
For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.
### Practice Test
For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. |
# Trigonometric Functions
## Introduction to Trigonometric Functions
Life is dense with phenomena that repeat in regular intervals. Each day, for example, the tides rise and fall in response to the gravitational pull of the moon.Hamacher, D.W., Tapim, A., Passi, S., and Barsa, J. (2018). And as a result of the motion of the moon itself, the tides occur with different strengths. Throughout history, many Indigenous peoples have used this regularity to build cultural narratives and direct key activities, such as agriculture, hunting, and fishing. Aboriginal people in the Torres Straight area (the northern tip) of Australia used the tidal peaks to determine the best times to fish. Their elders explain that the stronger spring tides stirred up sediment and obscured fish vision, leaving them more likely to take in lures and resulting in a larger catch.
In mathematics, a function that repeats its values in regular intervals is known as a periodic function. The graphs of such functions show a general shape reflective of a pattern that keeps repeating. This means the graph of the function has the same output at exactly the same place in every cycle. And this translates to all the cycles of the function having exactly the same length. So, if we know all the details of one full cycle of a true periodic function, then we know the state of the function’s outputs at all times, future and past. In this chapter, we will investigate various examples of periodic functions. |
# Trigonometric Functions
## Angles
A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles.
### Drawing Angles in Standard Position
Properly defining an angle first requires that we define a ray. A ray consists of one point on a line and all points extending in one direction from that point. The first point is called the endpoint of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in can be named as ray EF, or in symbol form
An angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the two rays are the sides of the angle. The angle in is formed from
and
Angles can be named using a point on each ray and the vertex, such as angle DEF, or in symbol form
Greek letters are often used as variables for the measure of an angle. is a list of Greek letters commonly used to represent angles, and a sample angle is shown in .
Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex as in .
As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The measure of an angle is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One degree is
of a circular rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol °. For example, 90 degrees = 90°.
To formalize our work, we will begin by drawing angles on an x-y coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. See .
If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle.
Drawing an angle in standard position always starts the same way—draw the initial side along the positive x-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by 360°. For example, to draw a 90° angle, we calculate that
So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive x-axis. To draw a 360° angle, we calculate that
So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive x-axis. In this case, the initial side and the terminal side overlap. See .
Since we define an angle in standard position by its initial side, we have a special type of angle whose terminal side lies on an axis, a quadrantal angle. This type of angle can have a measure of 0°, 90°, 180°, 270° or 360°. See .
### Converting Between Degrees and Radians
Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.
The circumference of a circle is
If we divide both sides of this equation by
we create the ratio of the circumference to the radius, which is always
regardless of the length of the radius. So the circumference of any circle is
times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in .
This brings us to our new angle measure. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals
times the radius, a full circular rotation is
radians. So
See . Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out.
### Relating Arc Lengths to Radius
An arc length
is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.
This ratio, called the radian measure, is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length
to the radius
See .
If
then
To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is
where
is the radius. The smaller circle then has circumference
and the larger has circumference
Now we draw a 45° angle on the two circles, as in .
Notice what happens if we find the ratio of the arc length divided by the radius of the circle.
Since both ratios are
the angle measures of both circles are the same, even though the arc length and radius differ.
### Using Radians
Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in , suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.
Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360°. We can also track one rotation around a circle by finding the circumference,
and for the unit circle
These two different ways to rotate around a circle give us a way to convert from degrees to radians.
### Identifying Special Angles Measured in Radians
In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in . Memorizing these angles will be very useful as we study the properties associated with angles.
Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in , which are shown in . Be sure you can verify each of these measures.
### Converting between Radians and Degrees
Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion.
This proportion shows that the measure of angle
in degrees divided by 180 equals the measure of angle
in radians divided by
Or, phrased another way, degrees is to 180 as radians is to
### Finding Coterminal Angles
Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of 0° to 360°, or 0 to
It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.
It is possible for more than one angle to have the same terminal side. Look at . The angle of 140° is a positive angle, measured counterclockwise. The angle of –220° is a negative angle, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles. Every angle greater than 360° or less than 0° is coterminal with an angle between 0° and 360°, and it is often more convenient to find the coterminal angle within the range of 0° to 360° than to work with an angle that is outside that range.
Any angle has infinitely many coterminal angles because each time we add 360° to that angle—or subtract 360° from it—the resulting value has a terminal side in the same location. For example, 100° and 460° are coterminal for this reason, as is −260°. Recognizing that any angle has infinitely many coterminal angles explains the repetitive shape in the graphs of trigonometric functions.
An angle’s reference angle is the measure of the smallest, positive, acute angle
formed by the terminal side of the angle
and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See for examples of reference angles for angles in different quadrants.
### Finding Coterminal Angles Measured in Radians
We can find coterminal angles measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.
### Determining the Length of an Arc
Recall that the radian measure
of an angle was defined as the ratio of the arc length
of a circular arc to the radius
of the circle,
From this relationship, we can find arc length along a circle, given an angle.
### Finding the Area of a Sector of a Circle
In addition to arc length, we can also use angles to find the area of a sector of a circle. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius
can be found using the formula
If the two radii form an angle of
measured in radians, then
is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the area of a sector is the fraction
multiplied by the entire area. (Always remember that this formula only applies if
is in radians.)
### Use Linear and Angular Speed to Describe Motion on a Circular Path
In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed is speed along a straight path and can be determined by the distance it moves along (its displacement) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or
inches, every second. So the linear speed of the point is
in./s. The equation for linear speed is as follows where
is linear speed,
is displacement, and
is time.
Angular speed results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as
90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where
(read as omega) is angular speed,
is the angle traversed, and
is time.
Combining the definition of angular speed with the arc length equation,
we can find a relationship between angular and linear speeds. The angular speed equation can be solved for
giving
Substituting this into the arc length equation gives:
Substituting this into the linear speed equation gives:
Water wheels have been used for thousands of years to transfer the power of flowing water to other devices. The image below depicts the design of the the 3rd century Roman water wheel in Hierapolis, a city in what is now Turkey. Water turned the wheel, which in turn rotated a crank connected to two saws used to cut blocks. These design elements were used in water wheel applications throughout the world, and even provided the underlying principle for the steam engine, invented about 1500 years later.
### Key Equations
### Key Concepts
1. An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The amount of rotation determines the measure of the angle.
2. An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis. A positive angle is measured counterclockwise from the initial side and a negative angle is measured clockwise.
3. To draw an angle in standard position, draw the initial side along the positive x-axis and then place the terminal side according to the fraction of a full rotation the angle represents. See .
4. In addition to degrees, the measure of an angle can be described in radians. See .
5. To convert between degrees and radians, use the proportion
See and .
6. Two angles that have the same terminal side are called coterminal angles.
7. We can find coterminal angles by adding or subtracting 360° or
See and .
8. Coterminal angles can be found using radians just as they are for degrees. See .
9. The length of a circular arc is a fraction of the circumference of the entire circle. See .
10. The area of sector is a fraction of the area of the entire circle. See .
11. An object moving in a circular path has both linear and angular speed.
12. The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a unit of time. See .
13. The linear speed of an object traveling along a circular path is the distance it travels in a unit of time. See .
### Section Exercises
### Verbal
### Graphical
For the following exercises, draw an angle in standard position with the given measure.
For the following exercises, refer to . Round to two decimal places.
For the following exercises, refer to . Round to two decimal places.
### Algebraic
For the following exercises, convert angles in radians to degrees.
For the following exercises, convert angles in degrees to radians.
For the following exercises, use to given information to find the length of a circular arc. Round to two decimal places.
For the following exercises, use the given information to find the area of the sector. Round to four decimal places.
For the following exercises, find the angle between 0° and 360° that is coterminal to the given angle.
For the following exercises, find the angle between 0 and
in radians that is coterminal to the given angle.
### Real-World Applications
### Extensions
|
# Trigonometric Functions
## Unit Circle: Sine and Cosine Functions
Looking for a thrill? Then consider a ride on the Ain Dubai, the world's tallest Ferris wheel. Located in Dubai, the most populous city and the financial and tourism hub of the United Arab Emirates, the wheel soars to 820 feet, about 1.5 tenths of a mile. Described as an observation wheel, riders enjoy spectacular views of the Burj Khalifa (the world's tallest building) and the Palm Jumeirah (a human-made archipelago home to over 10,000 people and 20 resorts) as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs.
### Finding Function Values for the Sine and Cosine
To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in . The angle (in radians) that
intercepts forms an arc of length
Using the formula
and knowing that
we see that for a unit circle,
Recall that the x- and y-axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.
For any angle
we can label the intersection of the terminal side and the unit circle as by its coordinates,
The coordinates
and
will be the outputs of the trigonometric functions
and
respectively. This means
and
### Defining Sine and Cosine Functions
Now that we have our unit circle labeled, we can learn how the
coordinates relate to the arc length and angle. The sine function relates a real number
to the y-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle
equals the y-value of the endpoint on the unit circle of an arc of length
In , the sine is equal to
Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the y-coordinate of the corresponding point on the unit circle.
The cosine function of an angle
equals the x-value of the endpoint on the unit circle of an arc of length
In , the cosine is equal to
Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses:
is the same as
and
is the same as
Likewise,
is a commonly used shorthand notation for
Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.
### Finding Sines and Cosines of Angles on an Axis
For quadrantral angles, the corresponding point on the unit circle falls on the x- or y-axis. In that case, we can easily calculate cosine and sine from the values of
and
### The Pythagorean Identity
Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is
Because
and
we can substitute for
and
to get
This equation,
is known as the Pythagorean Identity. See .
We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.
### Finding Sines and Cosines of Special Angles
We have already learned some properties of the special angles, such as the conversion from radians to degrees. We can also calculate sines and cosines of the special angles using the Pythagorean Identity and our knowledge of triangles.
### Finding Sines and Cosines of 45° Angles
First, we will look at angles of
or
as shown in . A
triangle is an isosceles triangle, so the x- and y-coordinates of the corresponding point on the circle are the same. Because the x- and y-values are the same, the sine and cosine values will also be equal.
At
, which is 45 degrees, the radius of the unit circle bisects the first quadrantal angle. This means the radius lies along the line
A unit circle has a radius equal to 1. So, the right triangle formed below the line
has sides
and
and a radius = 1. See
From the Pythagorean Theorem we get
Substituting
we get
Combining like terms we get
And solving for
we get
In quadrant I,
At
or 45 degrees,
If we then rationalize the denominators, we get
Therefore, the
coordinates of a point on a circle of radius
at an angle of
are
### Finding Sines and Cosines of 30° and 60° Angles
Next, we will find the cosine and sine at an angle of
or
. First, we will draw a triangle inside a circle with one side at an angle of
and another at an angle of
as shown in . If the resulting two right triangles are combined into one large triangle, notice that all three angles of this larger triangle will be
as shown in .
Because all the angles are equal, the sides are also equal. The vertical line has length
and since the sides are all equal, we can also conclude that
or
Since
,
And since
in our unit circle,
Using the Pythagorean Identity, we can find the cosine value.
The
coordinates for the point on a circle of radius
at an angle of
are
At
(60°), the radius of the unit circle, 1, serves as the hypotenuse of a 30-60-90 degree right triangle,
as shown in . Angle
has measure
At point
we draw an angle
with measure of
We know the angles in a triangle sum to
so the measure of angle
is also
Now we have an equilateral triangle. Because each side of the equilateral triangle
is the same length, and we know one side is the radius of the unit circle, all sides must be of length 1.
The measure of angle
is 30°. So, if double, angle
is 60°.
is the perpendicular bisector of
so it cuts
in half. This means that
is
the radius, or
Notice that
is the x-coordinate of point
which is at the intersection of the 60° angle and the unit circle. This gives us a triangle
with hypotenuse of 1 and side
of length
From the Pythagorean Theorem, we get
Substituting
we get
Solving for
we get
Since
has the terminal side in quadrant I where the y-coordinate is positive, we choose
the positive value.
At
(60°), the
coordinates for the point on a circle of radius
at an angle of
are
so we can find the sine and cosine.
We have now found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle. summarizes these values.
shows the common angles in the first quadrant of the unit circle.
### Using a Calculator to Find Sine and Cosine
To find the cosine and sine of angles other than the special angles, we turn to a computer or calculator. Be aware: Most calculators can be set into “degree” or “radian” mode, which tells the calculator the units for the input value. When we evaluate
on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the cosine of 30 radians if the calculator is in radian mode.
### Identifying the Domain and Range of Sine and Cosine Functions
Now that we can find the sine and cosine of an angle, we need to discuss their domains and ranges. What are the domains of the sine and cosine functions? That is, what are the smallest and largest numbers that can be inputs of the functions? Because angles smaller than 0 and angles larger than
can still be graphed on the unit circle and have real values of
and
there is no lower or upper limit to the angles that can be inputs to the sine and cosine functions. The input to the sine and cosine functions is the rotation from the positive x-axis, and that may be any real number.
What are the ranges of the sine and cosine functions? What are the least and greatest possible values for their output? We can see the answers by examining the unit circle, as shown in . The bounds of the x-coordinate are
The bounds of the y-coordinate are also
Therefore, the range of both the sine and cosine functions is
### Finding Reference Angles
We have discussed finding the sine and cosine for angles in the first quadrant, but what if our angle is in another quadrant? For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value. Because the sine value is the y-coordinate on the unit circle, the other angle with the same sine will share the same y-value, but have the opposite x-value. Therefore, its cosine value will be the opposite of the first angle’s cosine value.
Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. The angle with the same cosine will share the same x-value but will have the opposite y-value. Therefore, its sine value will be the opposite of the original angle’s sine value.
As shown in , angle
has the same sine value as angle
the cosine values are opposites. Angle
has the same cosine value as angle
the sine values are opposites.
Recall that an angle’s reference angle is the acute angle,
formed by the terminal side of the angle
and the horizontal axis. A reference angle is always an angle between
and
or
and
radians. As we can see from , for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.
### Using Reference Angles
Now let’s take a moment to reconsider the Ferris wheel introduced at the beginning of this section. Suppose a rider snaps a photograph while stopped twenty feet above ground level. The rider then rotates three-quarters of the way around the circle. What is the rider’s new elevation? To answer questions such as this one, we need to evaluate the sine or cosine functions at angles that are greater than 90 degrees or at a negative angle. Reference angles make it possible to evaluate trigonometric functions for angles outside the first quadrant. They can also be used to find
coordinates for those angles. We will use the reference angle of the angle of rotation combined with the quadrant in which the terminal side of the angle lies.
### Using Reference Angles to Evaluate Trigonometric Functions
We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The absolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the quadrant of the original angle. The cosine will be positive or negative depending on the sign of the x-values in that quadrant. The sine will be positive or negative depending on the sign of the y-values in that quadrant.
### Using Reference Angles to Find Coordinates
Now that we have learned how to find the cosine and sine values for special angles in the first quadrant, we can use symmetry and reference angles to fill in cosine and sine values for the rest of the special angles on the unit circle. They are shown in . Take time to learn the
coordinates of all of the major angles in the first quadrant.
In addition to learning the values for special angles, we can use reference angles to find
coordinates of any point on the unit circle, using what we know of reference angles along with the identities
First we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the reference angle, and give them the signs corresponding to the y- and x-values of the quadrant.
### Key Equations
### Key Concepts
1. Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.
2. Using the unit circle, the sine of an angle
equals the y-value of the endpoint on the unit circle of an arc of length
whereas the cosine of an angle
equals the x-value of the endpoint. See .
3. The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See .
4. When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. See .
5. Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See .
6. The domain of the sine and cosine functions is all real numbers.
7. The range of both the sine and cosine functions is
8. The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
9. The signs of the sine and cosine are determined from the x- and y-values in the quadrant of the original angle.
10. An angle’s reference angle is the size angle,
formed by the terminal side of the angle
and the horizontal axis. See .
11. Reference angles can be used to find the sine and cosine of the original angle. See .
12. Reference angles can also be used to find the coordinates of a point on a circle. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by lies.
For the following exercises, find the exact value of each trigonometric function.
### Numeric
For the following exercises, state the reference angle for the given angle.
For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the special angles on the unit circle, use a calculator and round to three decimal places.
For the following exercises, find the requested value.
### Graphical
For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of
### Technology
For the following exercises, use a graphing calculator to evaluate.
### Extensions
For the following exercises, evaluate.
### Real-World Applications
For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point
that is, on the due north position. Assume the carousel revolves counter clockwise. |
# Trigonometric Functions
## The Other Trigonometric Functions
A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is
or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. We have already defined the sine and cosine functions of an angle. Though sine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of six trigonometric functions. In this section, we will investigate the remaining functions.
### Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent
To define the remaining functions, we will once again draw a unit circle with a point
corresponding to an angle of
as shown in . As with the sine and cosine, we can use the
coordinates to find the other functions.
The first function we will define is the tangent. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. In , the tangent of angle
is equal to
Because the y-value is equal to the sine of
and the x-value is equal to the cosine of
the tangent of angle
can also be defined as
The tangent function is abbreviated as
The remaining three functions can all be expressed as reciprocals of functions we have already defined.
1. The secant function is the reciprocal of the cosine function. In , the secant of angle
is equal to
The secant function is abbreviated as
2. The cotangent function is the reciprocal of the tangent function. In , the cotangent of angle
is equal to
The cotangent function is abbreviated as
3. The cosecant function is the reciprocal of the sine function. In , the cosecant of angle
is equal to
The cosecant function is abbreviated as
Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting
equal to the cosine and
equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in .
### Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent
We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x- and y-values in the original quadrant. shows which functions are positive in which quadrant.
To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “A,” ll of the six trigonometric functions are positive. In quadrant II, “Smart,” only ine and its reciprocal function, cosecant, are positive. In quadrant III, “Trig,” only angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “Class,” only osine and its reciprocal function, secant, are positive.
### Using Even and Odd Trigonometric Functions
To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.
Consider the function
shown in . The graph of the function is symmetrical about the y-axis. All along the curve, any two points with opposite x-values have the same function value. This matches the result of calculation:
and so on. So
is an even function, a function such that two inputs that are opposites have the same output. That means
Now consider the function
shown in . The graph is not symmetrical about the y-axis. All along the graph, any two points with opposite x-values also have opposite y-values. So
is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means
We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in . The sine of the positive angle is
The sine of the negative angle is −y. The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in .
### Recognizing and Using Fundamental Identities
We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.
### Alternate Forms of the Pythagorean Identity
We can use these fundamental identities to derive alternative forms of the Pythagorean Identity,
One form is obtained by dividing both sides by
The other form is obtained by dividing both sides by
As we discussed in the chapter opening, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or
will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.
Other functions can also be periodic. For example, the lengths of months repeat every four years. If
represents the length time, measured in years, and
represents the number of days in February, then
This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.
### Evaluating Trigonometric Functions with a Calculator
We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.
Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.
If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor
to convert the degrees to radians. To find the secant of
we could press
or
### Key Equations
### Key Concepts
1. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle.
2. The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
3. The six trigonometric functions can be found from a point on the unit circle. See .
4. Trigonometric functions can also be found from an angle. See .
5. Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See .
6. A function is said to be even if
and odd if
7. Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
8. Even and odd properties can be used to evaluate trigonometric functions. See .
9. The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
10. Identities can be used to evaluate trigonometric functions. See and .
11. Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See .
12. The trigonometric functions repeat at regular intervals.
13. The period
of a repeating function
is the smallest interval such that
for any value of
14. The values of trigonometric functions of special angles can be found by mathematical analysis.
15. To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the exact value of each expression.
For the following exercises, use reference angles to evaluate the expression.
### Graphical
For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.
### Technology
For the following exercises, use a graphing calculator to evaluate.
### Extensions
For the following exercises, use identities to evaluate the expression.
For the following exercises, use identities to simplify the expression.
### Real-World Applications
|
# Trigonometric Functions
## Right Triangle Trigonometry
We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle:
In this section, we will see another way to define trigonometric functions using properties of right triangles.
### Using Right Triangles to Evaluate Trigonometric Functions
In earlier sections, we used a unit circle to define the trigonometric functions. In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of
is its value at
radians. First, we need to create our right triangle. shows a point on a unit circle of radius 1. If we drop a vertical line segment from the point
to the x-axis, we have a right triangle whose vertical side has length
and whose horizontal side has length
We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.
We know
Likewise, we know
These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using
coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of
we will call the side between the given angle and the right angle the adjacent side to angle
(Adjacent means “next to.”) Instead of
we will call the side most distant from the given angle the opposite side from angle
And instead of
we will call the side of a right triangle opposite the right angle the hypotenuse. These sides are labeled in .
### Understanding Right Triangle Relationships
Given a right triangle with an acute angle of
A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.”
### Relating Angles and Their Functions
When working with right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in . The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.
We will be asked to find all six trigonometric functions for a given angle in a triangle. Our strategy is to find the sine, cosine, and tangent of the angles first. Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.
### Finding Trigonometric Functions of Special Angles Using Side Lengths
We have already discussed the trigonometric functions as they relate to the special angles on the unit circle. Now, we can use those relationships to evaluate triangles that contain those special angles. We do this because when we evaluate the special angles in trigonometric functions, they have relatively friendly values, values that contain either no or just one square root in the ratio. Therefore, these are the angles often used in math and science problems. We will use multiples of
and
however, remember that when dealing with right triangles, we are limited to angles between
Suppose we have a
triangle, which can also be described as a
triangle. The sides have lengths in the relation
The sides of a
triangle, which can also be described as a
triangle, have lengths in the relation
These relations are shown in .
We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.
### Using Equal Cofunction of Complements
If we look more closely at the relationship between the sine and cosine of the special angles relative to the unit circle, we will notice a pattern. In a right triangle with angles of
and
we see that the sine of
namely
is also the cosine of
while the sine of
namely
is also the cosine of
See
This result should not be surprising because, as we see from , the side opposite the angle of
is also the side adjacent to
so
and
are exactly the same ratio of the same two sides,
and
Similarly,
and
are also the same ratio using the same two sides,
and
The interrelationship between the sines and cosines of
and
also holds for the two acute angles in any right triangle, since in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other. Since the three angles of a triangle add to
and the right angle is
the remaining two angles must also add up to
That means that a right triangle can be formed with any two angles that add to
—in other words, any two complementary angles. So we may state a cofunction identity: If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This identity is illustrated in .
Using this identity, we can state without calculating, for instance, that the sine of
equals the cosine of
and that the sine of
equals the cosine of
We can also state that if, for a certain angle
then
as well.
### Using Trigonometric Functions
In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.
### Using Right Triangle Trigonometry to Solve Applied Problems
Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. See .
### Key Equations
### Key Concepts
1. We can define trigonometric functions as ratios of the side lengths of a right triangle. See .
2. The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. See .
3. We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. See .
4. Any two complementary angles could be the two acute angles of a right triangle.
5. If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. See .
6. We can use trigonometric functions of an angle to find unknown side lengths.
7. Select the trigonometric function representing the ratio of the unknown side to the known side. See .
8. Right-triangle trigonometry permits the measurement of inaccessible heights and distances.
9. The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, use cofunctions of complementary angles.
For the following exercises, find the lengths of the missing sides if side
is opposite angle
side
is opposite angle
and side
is the hypotenuse.
### Graphical
For the following exercises, use to evaluate each trigonometric function of angle
For the following exercises, use to evaluate each trigonometric function of angle
For the following exercises, solve for the unknown sides of the given triangle.
### Technology
For the following exercises, use a calculator to find the length of each side to four decimal places.
### Extensions
### Real-World Applications
### Review Exercises
### Angles
For the following exercises, convert the angle measures to degrees.
For the following exercises, convert the angle measures to radians.
For the following exercises, find the angle between 0° and 360° that is coterminal with the given angle.
For the following exercises, find the angle between 0 and
in radians that is coterminal with the given angle.
For the following exercises, draw the angle provided in standard position on the Cartesian plane.
### Unit Circle: Sine and Cosine Functions
### The Other Trigonometric Functions
For the following exercises, find the exact value of the given expression.
For the following exercises, use reference angles to evaluate the given expression.
### Right Triangle Trigonometry
For the following exercises, use side lengths to evaluate.
For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.
For the following exercises, use to evaluate each trigonometric function.
For the following exercises, solve for the unknown sides of the given triangle.
### Practice Test
|
# Periodic Functions
## Introduction to Periodic Functions
The sun has played a core role in many religions. The ancient Egyptian culture portrayed the sun god, Ra (sometimes written as Re), as undertaking a two-part daily journey, with one portion in the sky (day) and the other through the underworld (night). Surya, the Hindu sun god, traces a similar path through the sky on a chariot pulled by seven horses. While their origins and associated narratives are quite different, both Ra and Surya are primary deities and seen as creators and preservers of life. In many Native American cultures, the sun is core to spiritual and religious practice, but is not always a deity. The Sun Dance, practiced differently by many Native American tribes, was a ceremony that generally paid homage to the sun and, in many cases, tested or expressed the strength of the tribe's people.
As one of the most most prominent natural phenomena and with its close association to giving life, the sun was an obvious subject for reverence. And its regularity, even in ancient times, made it the primary determinant of time. Each day, the sun rises in an easterly direction, approaches some maximum height relative to the celestial equator, and sets in a westerly direction. The celestial equator is an imaginary line that divides the visible universe into two halves in much the same way Earth’s equator is an imaginary line that divides the planet into two halves. The exact path the sun appears to follow depends on the exact location on Earth, but each location observes a predictable pattern over time.
The pattern of the sun’s motion throughout the course of a year is a periodic function. Creating a visual representation of a periodic function in the form of a graph can help us analyze the properties of the function. In this chapter, we will investigate graphs of sine, cosine, and other trigonometric functions. |
# Periodic Functions
## Graphs of the Sine and Cosine Functions
White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.
Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.
### Graphing Sine and Cosine Functions
Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. We can create a table of values and use them to sketch a graph. lists some of the values for the sine function on a unit circle.
Plotting the points from the table and continuing along the x-axis gives the shape of the sine function. See .
Notice how the sine values are positive between 0 and
which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between
and
which correspond to the values of the sine function in quadrants III and IV on the unit circle. See .
Now let’s take a similar look at the cosine function. Again, we can create a table of values and use them to sketch a graph. lists some of the values for the cosine function on a unit circle.
As with the sine function, we can plots points to create a graph of the cosine function as in .
Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval
In both graphs, the shape of the graph repeats after
which means the functions are periodic with a period of
A periodic function is a function for which a specific horizontal shift, P, results in a function equal to the original function:
for all values of
in the domain of
When this occurs, we call the smallest such horizontal shift with
the period of the function. shows several periods of the sine and cosine functions.
Looking again at the sine and cosine functions on a domain centered at the y-axis helps reveal symmetries. As we can see in , the sine function is symmetric about the origin. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because
Now we can clearly see this property from the graph.
shows that the cosine function is symmetric about the y-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that
### Investigating Sinusoidal Functions
As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal functions are
### Determining the Period of Sinusoidal Functions
Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.
In the general formula,
is related to the period by
If
then the period is less than
and the function undergoes a horizontal compression, whereas if
then the period is greater than
and the function undergoes a horizontal stretch. For example,
so the period is
which we knew. If
then
so the period is
and the graph is compressed. If
then
so the period is
and the graph is stretched. Notice in how the period is indirectly related to
### Determining Amplitude
Returning to the general formula for a sinusoidal function, we have analyzed how the variable
relates to the period. Now let’s turn to the variable
so we can analyze how it is related to the amplitude, or greatest distance from rest.
represents the vertical stretch factor, and its absolute value
is the amplitude. The local maxima will be a distance
above the horizontal midline of the graph, which is the line
because
in this case, the midline is the x-axis. The local minima will be the same distance below the midline. If
the function is stretched. For example, the amplitude of
is twice the amplitude of
If
the function is compressed. compares several sine functions with different amplitudes.
### Analyzing Graphs of Variations of y = sin x and y = cos x
Now that we understand how
and
relate to the general form equation for the sine and cosine functions, we will explore the variables
and
Recall the general form:
The value
for a sinusoidal function is called the phase shift, or the horizontal displacement of the basic sine or cosine function. If
the graph shifts to the right. If
the graph shifts to the left. The greater the value of
the more the graph is shifted. shows that the graph of
shifts to the right by
units, which is more than we see in the graph of
which shifts to the right by
units.
While
relates to the horizontal shift,
indicates the vertical shift from the midline in the general formula for a sinusoidal function. See . The function
has its midline at
Any value of
other than zero shifts the graph up or down. compares
with
which is shifted 2 units up on a graph.
### Graphing Variations of y = sin x and y = cos x
Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.
Instead of focusing on the general form equations
we will let
and
and work with a simplified form of the equations in the following examples.
### Using Transformations of Sine and Cosine Functions
We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function.
### Key Equations
### Key Concepts
1. Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of
2. The function
is odd, so its graph is symmetric about the origin. The function
is even, so its graph is symmetric about the y-axis.
3. The graph of a sinusoidal function has the same general shape as a sine or cosine function.
4. In the general formula for a sinusoidal function, the period is
See .
5. In the general formula for a sinusoidal function,
represents amplitude. If
the function is stretched, whereas if
the function is compressed. See .
6. The value
in the general formula for a sinusoidal function indicates the phase shift. See .
7. The value
in the general formula for a sinusoidal function indicates the vertical shift from the midline. See .
8. Combinations of variations of sinusoidal functions can be detected from an equation. See .
9. The equation for a sinusoidal function can be determined from a graph. See and .
10. A function can be graphed by identifying its amplitude and period. See and .
11. A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See .
12. Sinusoidal functions can be used to solve real-world problems. See , , and .
### Section Exercises
### Verbal
### Graphical
For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for
Round answers to two decimal places if necessary.
For the following exercises, graph one full period of each function, starting at
For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for
State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.
### Algebraic
For the following exercises, let
For the following exercises, let
### Technology
### Real-World Applications
|
# Periodic Functions
## Graphs of the Other Trigonometric Functions
We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a fire truck parked next to a warehouse. The rotating light from the truck would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and other trigonometric functions.
### Analyzing the Graph of y = tan x
We will begin with the graph of the tangent function, plotting points as we did for the sine and cosine functions. Recall that
The period of the tangent function is
because the graph repeats itself on intervals of
where
is a constant. If we graph the tangent function on
to
we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.
We can determine whether tangent is an odd or even function by using the definition of tangent.
Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in .
These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when
we can use a table to look for a trend. Because
and
we will evaluate
at radian measures
as shown in .
As
approaches
the outputs of the function get larger and larger. Because
is an odd function, we see the corresponding table of negative values in .
We can see that, as
approaches
the outputs get smaller and smaller. Remember that there are some values of
for which
For example,
and
At these values, the tangent function is undefined, so the graph of
has discontinuities at
At these values, the graph of the tangent has vertical asymptotes. represents the graph of
The tangent is positive from 0 to
and from
to
corresponding to quadrants I and III of the unit circle.
### Graphing Variations of y = tan x
As with the sine and cosine functions, the tangent function can be described by a general equation.
We can identify horizontal and vertical stretches and compressions using values of
and
The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.
Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant
### Graphing One Period of a Stretched or Compressed Tangent Function
We can use what we know about the properties of the tangent function to quickly sketch a graph of any stretched and/or compressed tangent function of the form
We focus on a single period of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function’s domain if we wish. Our limited domain is then the interval
and the graph has vertical asymptotes at
where
On
the graph will come up from the left asymptote at
cross through the origin, and continue to increase as it approaches the right asymptote at
To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use
because
### Graphing One Period of a Shifted Tangent Function
Now that we can graph a tangent function that is stretched or compressed, we will add a vertical and/or horizontal (or phase) shift. In this case, we add
and
to the general form of the tangent function.
The graph of a transformed tangent function is different from the basic tangent function
in several ways:
### Analyzing the Graphs of y = sec x and y = cscx
The secant was defined by the reciprocal identity
Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at
etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.
We can graph
by observing the graph of the cosine function because these two functions are reciprocals of one another. See . The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.
The secant graph has vertical asymptotes at each value of
where the cosine graph crosses the x-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.
Note that, because cosine is an even function, secant is also an even function. That is,
As we did for the tangent function, we will again refer to the constant
as the stretching factor, not the amplitude.
Similar to the secant, the cosecant is defined by the reciprocal identity
Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at
etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.
We can graph
by observing the graph of the sine function because these two functions are reciprocals of one another. See . The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the cosecant function increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.
The cosecant graph has vertical asymptotes at each value of
where the sine graph crosses the x-axis; we show these in the graph below with dashed vertical lines.
Note that, since sine is an odd function, the cosecant function is also an odd function. That is,
The graph of cosecant, which is shown in , is similar to the graph of secant.
### Graphing Variations of y = sec x and y= csc x
For shifted, compressed, and/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the cosecant function in the same way as for the secant and other functions.The equations become the following.
### Analyzing the Graph of y = cot x
The last trigonometric function we need to explore is cotangent. The cotangent is defined by the reciprocal identity
Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at
etc. Since the output of the tangent function is all real numbers, the output of the cotangent function is also all real numbers.
We can graph
by observing the graph of the tangent function because these two functions are reciprocals of one another. See . Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.
The cotangent graph has vertical asymptotes at each value of
where
we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent,
has vertical asymptotes at all values of
where
and
at all values of
where
has its vertical asymptotes.
### Graphing Variations of y = cot x
We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.
### Using the Graphs of Trigonometric Functions to Solve Real-World Problems
Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let’s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a fire truck and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function.
### Key Equations
### Key Concepts
1. The tangent function has period
2.
is a tangent with vertical and/or horizontal stretch/compression and shift. See , , and .
3. The secant and cosecant are both periodic functions with a period of
gives a shifted, compressed, and/or stretched secant function graph. See and .
4.
gives a shifted, compressed, and/or stretched cosecant function graph. See and .
5. The cotangent function has period
and vertical asymptotes at
6. The range of cotangent is
and the function is decreasing at each point in its range.
7. The cotangent is zero at
8.
is a cotangent with vertical and/or horizontal stretch/compression and shift. See and .
9. Real-world scenarios can be solved using graphs of trigonometric functions. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, match each trigonometric function with one of the following graphs.
For the following exercises, find the period and horizontal shift of each of the functions.
For the following exercises, evaluate the transformed functions.
For the following exercises, rewrite each expression such that the argument
is positive.
### Graphical
For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.
For the following exercises, find and graph two periods of the periodic function with the given stretching factor,
period, and phase shift.
For the following exercises, find an equation for the graph of each function.
### Technology
For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input
as
### Real-World Applications
|
# Periodic Functions
## Inverse Trigonometric Functions
For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we will explore the inverse trigonometric functions.
### Understanding and Using the Inverse Sine, Cosine, and Tangent Functions
In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in .
For example, if
then we would write
Be aware that
does not mean
The following examples illustrate the inverse trigonometric functions:
1. Since
then
2. Since
then
3. Since
then
In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that, for a one-to-one function, if
then an inverse function would satisfy
Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number 0. shows the graph of the sine function limited to
and the graph of the cosine function limited to
shows the graph of the tangent function limited to
These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an asymptote.
On these restricted domains, we can define the inverse trigonometric functions.
1. The inverse sine function
means
The inverse sine function is sometimes called the arcsine function, and notated
2. The inverse cosine function
means
The inverse cosine function is sometimes called the arccosine function, and notated
3. The inverse tangent function
means
The inverse tangent function is sometimes called the arctangent function, and notated
The graphs of the inverse functions are shown in , , and . Notice that the output of each of these inverse functions is a number, an angle in radian measure. We see that
has domain
and range
has domain
and range
and
has domain of all real numbers and range
To find the domain and range of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line
### Finding the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions
Now that we can identify inverse functions, we will learn to evaluate them. For most values in their domains, we must evaluate the inverse trigonometric functions by using a calculator, interpolating from a table, or using some other numerical technique. Just as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically
(30°),
(45°), and
(60°), and their reflections into other quadrants.
### Using a Calculator to Evaluate Inverse Trigonometric Functions
To evaluate inverse trigonometric functions that do not involve the special angles discussed previously, we will need to use a calculator or other type of technology. Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. These may be labeled, for example, SIN
, ARCSIN, or ASIN.
In the previous chapter, we worked with trigonometry on a right triangle to solve for the sides of a triangle given one side and an additional angle. Using the inverse trigonometric functions, we can solve for the angles of a right triangle given two sides, and we can use a calculator to find the values to several decimal places.
In these examples and exercises, the answers will be interpreted as angles and we will use
as the independent variable. The value displayed on the calculator may be in degrees or radians, so be sure to set the mode appropriate to the application.
### Finding Exact Values of Composite Functions with Inverse Trigonometric Functions
There are times when we need to compose a trigonometric function with an inverse trigonometric function. In these cases, we can usually find exact values for the resulting expressions without resorting to a calculator. Even when the input to the composite function is a variable or an expression, we can often find an expression for the output. To help sort out different cases, let
and
be two different trigonometric functions belonging to the set
and let
and
be their inverses.
### Evaluating Compositions of the Form f(f−1(y)) and f−1(f(x))
For any trigonometric function,
for all
in the proper domain for the given function. This follows from the definition of the inverse and from the fact that the range of
was defined to be identical to the domain of
However, we have to be a little more careful with expressions of the form
### Evaluating Compositions of the Form f−1(g(x))
Now that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a trigonometric function and the inverse of another trigonometric function. We will begin with compositions of the form
For special values of
we can exactly evaluate the inner function and then the outer, inverse function. However, we can find a more general approach by considering the relation between the two acute angles of a right triangle where one is
making the other
Consider the sine and cosine of each angle of the right triangle in .
Because
we have
if
If
is not in this domain, then we need to find another angle that has the same cosine as
and does belong to the restricted domain; we then subtract this angle from
Similarly,
so
if
These are just the function-cofunction relationships presented in another way.
### Evaluating Compositions of the Form f(g−1(x))
To evaluate compositions of the form
where
and
are any two of the functions sine, cosine, or tangent and
is any input in the domain of
we have exact formulas, such as
When we need to use them, we can derive these formulas by using the trigonometric relations between the angles and sides of a right triangle, together with the use of Pythagoras’s relation between the lengths of the sides. We can use the Pythagorean identity,
to solve for one when given the other. We can also use the inverse trigonometric functions to find compositions involving algebraic expressions.
### Key Concepts
1. An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
2. Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
3. For any trigonometric function
if
then
However,
only implies
if
is in the restricted domain of
See .
4. Special angles are the outputs of inverse trigonometric functions for special input values; for example,
See .
5. A calculator will return an angle within the restricted domain of the original trigonometric function. See .
6. Inverse functions allow us to find an angle when given two sides of a right triangle. See .
7. In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example,
See .
8. If the inside function is a trigonometric function, then the only possible combinations are
if
and
if
See and .
9. When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. See .
10. When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, evaluate the expressions.
For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.
For the following exercises, find the angle
in the given right triangle. Round answers to the nearest hundredth.
For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.
For the following exercises, find the exact value of the expression in terms of
with the help of a reference triangle.
### Extensions
For the following exercises, evaluate the expression without using a calculator. Give the exact value.
For the following exercises, find the function if
### Graphical
### Real-World Applications
### Chapter Review Exercises
### Graphs of the Sine and Cosine Functions
For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.
### Graphs of the Other Trigonometric Functions
For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period,
midline equation, and asymptotes.
For the following exercises, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.
For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function:
where the domain is the years since 1980 and the range is the population of the city.
For the following exercises, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.
### Inverse Trigonometric Functions
For the following exercises, find the exact value without the aid of a calculator.
### Chapter Practice Test
For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.
For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.
For the following exercises, find the amplitude, period, phase shift, and midline.
For the following exercises, find the period and horizontal shift of each function.
For the following exercises, graph the functions on the specified window and answer the questions.
For the following exercises, let
For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.
For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.
For the following exercises, find the exact value.
For the following exercises, suppose
Evaluate the following expressions.
For the following exercises, determine whether the equation is true or false. |
# Trigonometric Identities and Equations
## Introduction to Trigonometric Identities and Equations
Math is everywhere, even in places we might not immediately recognize. For example, mathematical relationships describe the transmission of images, light, and sound. The sinusoidal graph in the figure above models music playing on a phone, radio, or computer. Such graphs are described using trigonometric equations and functions. In this chapter, we discuss how to manipulate trigonometric equations algebraically by applying various formulas and trigonometric identities. We will also investigate some of the ways that trigonometric equations are used to model real-life phenomena. |
# Trigonometric Identities and Equations
## Solving Trigonometric Equations with Identities
In espionage movies, we see international spies with multiple passports, each claiming a different identity. However, we know that each of those passports represents the same person. The trigonometric identities act in a similar manner to multiple passports—there are many ways to represent the same trigonometric expression. Just as a spy will choose an Italian passport when traveling to Italy, we choose the identity that applies to the given scenario when solving a trigonometric equation.
In this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify them and how we can use them to simplify trigonometric expressions.
### Verifying the Fundamental Trigonometric Identities
Identities enable us to simplify complicated expressions. They are the basic tools of trigonometry used in solving trigonometric equations, just as factoring, finding common denominators, and using special formulas are the basic tools of solving algebraic equations. In fact, we use algebraic techniques constantly to simplify trigonometric expressions. Basic properties and formulas of algebra, such as the difference of squares formula and the perfect squares formula, will simplify the work involved with trigonometric expressions and equations. We already know that all of the trigonometric functions are related because they all are defined in terms of the unit circle. Consequently, any trigonometric identity can be written in many ways.
To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression until it has been transformed into the same expression as the other side of the equation. Sometimes we have to factor expressions, expand expressions, find common denominators, or use other algebraic strategies to obtain the desired result. In this first section, we will work with the fundamental identities: the Pythagorean Identities, the even-odd identities, the reciprocal identities, and the quotient identities.
We will begin with the Pythagorean Identities (see ), which are equations involving trigonometric functions based on the properties of a right triangle. We have already seen and used the first of these identifies, but now we will also use additional identities.
The second and third identities can be obtained by manipulating the first. The identity
is found by rewriting the left side of the equation in terms of sine and cosine.
Prove:
Similarly,
can be obtained by rewriting the left side of this identity in terms of sine and cosine. This gives
The next set of fundamental identities is the set of even-odd identities. The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle and determine whether the identity is odd or even. (See ).
Recall that an odd function is one in which
for all
in the domain of
The sine function is an odd function because
The graph of an odd function is symmetric about the origin. For example, consider corresponding inputs of
and
The output of
is opposite the output of
Thus,
This is shown in .
Recall that an even function is one in which
The graph of an even function is symmetric about the y-axis. The cosine function is an even function because
For example, consider corresponding inputs
and
The output of
is the same as the output of
Thus,
See .
For all
in the domain of the sine and cosine functions, respectively, we can state the following:
1. Since
sine is an odd function.
2. Since,
cosine is an even function.
The other even-odd identities follow from the even and odd nature of the sine and cosine functions. For example, consider the tangent identity,
We can interpret the tangent of a negative angle as
Tangent is therefore an odd function, which means that
for all
in the domain of the tangent function.
The cotangent identity,
also follows from the sine and cosine identities. We can interpret the cotangent of a negative angle as
Cotangent is therefore an odd function, which means that
for all
in the domain of the cotangent function.
The cosecant function is the reciprocal of the sine function, which means that the cosecant of a negative angle will be interpreted as
The cosecant function is therefore odd.
Finally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is interpreted as
The secant function is therefore even.
To sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions are odd, verifying the even-odd identities.
The next set of fundamental identities is the set of reciprocal identities, which, as their name implies, relate trigonometric functions that are reciprocals of each other. See .
The final set of identities is the set of quotient identities, which define relationships among certain trigonometric functions and can be very helpful in verifying other identities. See .
The reciprocal and quotient identities are derived from the definitions of the basic trigonometric functions.
### Using Algebra to Simplify Trigonometric Expressions
We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.
For example, the equation
resembles the equation
which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations.
Another example is the difference of squares formula,
which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by continually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve.
### Key Equations
### Key Concepts
1. There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem.
2. Graphing both sides of an identity will verify it. See .
3. Simplifying one side of the equation to equal the other side is another method for verifying an identity. See and .
4. The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more complex side of the equation. See .
5. We can create an identity by simplifying an expression and then verifying it. See .
6. Verifying an identity may involve algebra with the fundamental identities. See and .
7. Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the fundamental rules of mathematics. See , , and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, use the fundamental identities to fully simplify the expression.
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.
For the following exercises, verify the identity.
### Extensions
For the following exercises, prove or disprove the identity.
For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression. |
# Trigonometric Identities and Equations
## Sum and Difference Identities
How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long distances.
The trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are special equations or postulates, true for all values input to the equations, and with innumerable applications.
In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the term formula is used synonymously with the word identity.
### Using the Sum and Difference Formulas for Cosine
Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles, which we can review in the unit circle shown in .
We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. See .
First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. See . Point
is at an angle
from the positive x-axis with coordinates
and point
is at an angle of
from the positive x-axis with coordinates
Note the measure of angle
is
Label two more points:
at an angle of
from the positive x-axis with coordinates
and point
with coordinates
Triangle
is a rotation of triangle
and thus the distance from
to
is the same as the distance from
to
We can find the distance from
to
using the distance formula.
Then we apply the Pythagorean Identity and simplify.
Similarly, using the distance formula we can find the distance from
to
Applying the Pythagorean Identity and simplifying we get:
Because the two distances are the same, we set them equal to each other and simplify.
Finally we subtract
from both sides and divide both sides by
Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.
### Using the Sum and Difference Formulas for Sine
The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.
### Using the Sum and Difference Formulas for Tangent
Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.
Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall,
Let’s derive the sum formula for tangent.
We can derive the difference formula for tangent in a similar way.
### Using Sum and Difference Formulas for Cofunctions
Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall from Right Triangle Trigonometry that, if the sum of two positive angles is
those two angles are complements, and the sum of the two acute angles in a right triangle is
so they are also complements. In , notice that if one of the acute angles is labeled as
then the other acute angle must be labeled
Notice also that
opposite over hypotenuse. Thus, when two angles are complementary, we can say that the sine of
equals the cofunction of the complement of
Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.
From these relationships, the cofunction identities are formed.
Notice that the formulas in the table may also be justified algebraically using the sum and difference formulas. For example, using
we can write
### Using the Sum and Difference Formulas to Verify Identities
Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems. Reviewing the general rules from Solving Trigonometric Equations with Identities may help simplify the process of verifying an identity.
### Key Equations
### Key Concepts
1. The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.
2. The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle. See and .
3. The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle. See .
4. The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. See .
5. The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the product of the tangents of the angles. See .
6. The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles. See .
7. The cofunction identities apply to complementary angles and pairs of reciprocal functions. See .
8. Sum and difference formulas are useful in verifying identities. See and .
9. Application problems are often easier to solve by using sum and difference formulas. See and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the exact value.
For the following exercises, rewrite in terms of
and
For the following exercises, simplify the given expression.
For the following exercises, find the requested information.
For the following exercises, find the exact value of each expression.
### Graphical
For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical.
For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think
)
### Technology
For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.
### Extensions
For the following exercises, prove the identities provided.
For the following exercises, prove or disprove the statements. |
# Trigonometric Identities and Equations
## Double-Angle, Half-Angle, and Reduction Formulas
Bicycle ramps made for competition (see ) must vary in height depending on the skill level of the competitors. For advanced competitors, the angle formed by the ramp and the ground should be
such that
The angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one.
### Using Double-Angle Formulas to Find Exact Values
In the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. The double-angle formulas are a special case of the sum formulas, where
Deriving the double-angle formula for sine begins with the sum formula,
If we let
then we have
Deriving the double-angle for cosine gives us three options. First, starting from the sum formula,
and letting
we have
Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more interpretations. The first one is:
The second interpretation is:
Similarly, to derive the double-angle formula for tangent, replacing
in the sum formula gives
### Using Double-Angle Formulas to Verify Identities
Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.
### Use Reduction Formulas to Simplify an Expression
The double-angle formulas can be used to derive the reduction formulas, which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.
We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let’s begin with
Solve for
Next, we use the formula
Solve for
The last reduction formula is derived by writing tangent in terms of sine and cosine:
### Using Half-Angle Formulas to Find Exact Values
The next set of identities is the set of half-angle formulas, which can be derived from the reduction formulas and we can use when we have an angle that is half the size of a special angle. If we replace
with
the half-angle formula for sine is found by simplifying the equation and solving for
Note that the half-angle formulas are preceded by a
sign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in which
terminates.
The half-angle formula for sine is derived as follows:
To derive the half-angle formula for cosine, we have
For the tangent identity, we have
### Key Equations
### Key Concepts
1. Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. See , , , and .
2. Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. See and .
3. Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. See , , and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the exact values of a)
b)
and c)
without solving for
For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.
For the following exercises, simplify to one trigonometric expression.
For the following exercises, find the exact value using half-angle formulas.
For the following exercises, find the exact values of a)
b)
and c)
without solving for
when
For the following exercises, use to find the requested half and double angles.
For the following exercises, simplify each expression. Do not evaluate.
For the following exercises, prove the identity given.
For the following exercises, rewrite the expression with an exponent no higher than 1.
### Technology
For the following exercises, reduce the equations to powers of one, and then check the answer graphically.
For the following exercises, algebraically find an equivalent function, only in terms of
and/or
and then check the answer by graphing both equations.
### Extensions
For the following exercises, prove the identities. |
# Trigonometric Identities and Equations
## Sum-to-Product and Product-to-Sum Formulas
A band marches down the field creating an amazing sound that bolsters the crowd. That sound travels as a wave that can be interpreted using trigonometric functions. For example, represents a sound wave for the musical note A. In this section, we will investigate trigonometric identities that are the foundation of everyday phenomena such as sound waves.
### Expressing Products as Sums
We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the product-to-sum formulas, which express products of trigonometric functions as sums. Let’s investigate the cosine identity first and then the sine identity.
### Expressing Products as Sums for Cosine
We can derive the product-to-sum formula from the sum and difference identities for cosine. If we add the two equations, we get:
Then, we divide by
to isolate the product of cosines:
### Expressing the Product of Sine and Cosine as a Sum
Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for sine. If we add the sum and difference identities, we get:
Then, we divide by 2 to isolate the product of cosine and sine:
### Expressing Products of Sines in Terms of Cosine
Expressing the product of sines in terms of cosine is also derived from the sum and difference identities for cosine. In this case, we will first subtract the two cosine formulas:
Then, we divide by 2 to isolate the product of sines:
Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.
### Expressing Sums as Products
Some problems require the reverse of the process we just used. The sum-to-product formulas allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for sine. Let
and
Then,
Thus, replacing
and
in the product-to-sum formula with the substitute expressions, we have
The other sum-to-product identities are derived similarly.
### Key Equations
### Key Concepts
1. From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.
2. We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. See , , and .
3. We can also derive the sum-to-product identities from the product-to-sum identities using substitution.
4. We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. See .
5. Trigonometric expressions are often simpler to evaluate using the formulas. See .
6. The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side. See and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, rewrite the product as a sum or difference.
For the following exercises, rewrite the sum or difference as a product.
For the following exercises, evaluate the product for the following using a sum or difference of two functions.
For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.
For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.
For the following exercises, prove the identity.
### Numeric
For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.
### Technology
For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.
For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.
### Extensions
For the following exercises, prove the following sum-to-product formulas.
For the following exercises, prove the identity. |
# Trigonometric Identities and Equations
## Solving Trigonometric Equations
Thales of Miletus (circa 625–547 BC) is known as the founder of geometry. The legend is that he calculated the height of the Great Pyramid of Giza in Egypt using the theory of similar triangles, which he developed by measuring the shadow of his staff. Based on proportions, this theory has applications in a number of areas, including fractal geometry, engineering, and architecture. Often, the angle of elevation and the angle of depression are found using similar triangles.
In earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids.
### Solving Linear Trigonometric Equations in Sine and Cosine
Trigonometric equations are, as the name implies, equations that involve trigonometric functions. Similar in many ways to solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are solutions at all. Often we will solve a trigonometric equation over a specified interval. However, just as often, we will be asked to find all possible solutions, and as trigonometric functions are periodic, solutions are repeated within each period. In other words, trigonometric equations may have an infinite number of solutions. Additionally, like rational equations, the domain of the function must be considered before we assume that any solution is valid. The period of both the sine function and the cosine function is
In other words, every
units, the y-values repeat. If we need to find all possible solutions, then we must add
where
is an integer, to the initial solution. Recall the rule that gives the format for stating all possible solutions for a function where the period is
There are similar rules for indicating all possible solutions for the other trigonometric functions. Solving trigonometric equations requires the same techniques as solving algebraic equations. We read the equation from left to right, horizontally, like a sentence. We look for known patterns, factor, find common denominators, and substitute certain expressions with a variable to make solving a more straightforward process. However, with trigonometric equations, we also have the advantage of using the identities we developed in the previous sections.
### Solving Equations Involving a Single Trigonometric Function
When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle (see ). We need to make several considerations when the equation involves trigonometric functions other than sine and cosine. Problems involving the reciprocals of the primary trigonometric functions need to be viewed from an algebraic perspective. In other words, we will write the reciprocal function, and solve for the angles using the function. Also, an equation involving the tangent function is slightly different from one containing a sine or cosine function. First, as we know, the period of tangent is
not
Further, the domain of tangent is all real numbers with the exception of odd integer multiples of
unless, of course, a problem places its own restrictions on the domain.
### Solve Trigonometric Equations Using a Calculator
Not all functions can be solved exactly using only the unit circle. When we must solve an equation involving an angle other than one of the special angles, we will need to use a calculator. Make sure it is set to the proper mode, either degrees or radians, depending on the criteria of the given problem.
### Solving Trigonometric Equations in Quadratic Form
Solving a quadratic equation may be more complicated, but once again, we can use algebra as we would for any quadratic equation. Look at the pattern of the equation. Is there more than one trigonometric function in the equation, or is there only one? Which trigonometric function is squared? If there is only one function represented and one of the terms is squared, think about the standard form of a quadratic. Replace the trigonometric function with a variable such as
or
If substitution makes the equation look like a quadratic equation, then we can use the same methods for solving quadratics to solve the trigonometric equations.
### Solving Trigonometric Equations Using Fundamental Identities
While algebra can be used to solve a number of trigonometric equations, we can also use the fundamental identities because they make solving equations simpler. Remember that the techniques we use for solving are not the same as those for verifying identities. The basic rules of algebra apply here, as opposed to rewriting one side of the identity to match the other side. In the next example, we use two identities to simplify the equation.
### Solving Trigonometric Equations with Multiple Angles
Sometimes it is not possible to solve a trigonometric equation with identities that have a multiple angle, such as
or
When confronted with these equations, recall that
is a horizontal compression by a factor of 2 of the function
On an interval of
we can graph two periods of
as opposed to one cycle of
This compression of the graph leads us to believe there may be twice as many x-intercepts or solutions to
compared to
This information will help us solve the equation.
### Solving Right Triangle Problems
We can now use all of the methods we have learned to solve problems that involve applying the properties of right triangles and the Pythagorean Theorem. We begin with the familiar Pythagorean Theorem,
and model an equation to fit a situation.
### Key Concepts
1. When solving linear trigonometric equations, we can use algebraic techniques just as we do solving algebraic equations. Look for patterns, like the difference of squares, quadratic form, or an expression that lends itself well to substitution. See , , and .
2. Equations involving a single trigonometric function can be solved or verified using the unit circle. See , , and , and .
3. We can also solve trigonometric equations using a graphing calculator. See and .
4. Many equations appear quadratic in form. We can use substitution to make the equation appear simpler, and then use the same techniques we use solving an algebraic quadratic: factoring, the quadratic formula, etc. See , , , and .
5. We can also use the identities to solve trigonometric equation. See , , and .
6. We can use substitution to solve a multiple-angle trigonometric equation, which is a compression of a standard trigonometric function. We will need to take the compression into account and verify that we have found all solutions on the given interval. See .
7. Real-world scenarios can be modeled and solved using the Pythagorean Theorem and trigonometric functions. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find all solutions exactly on the interval
For the following exercises, solve exactly on
For the following exercises, find all exact solutions on
For the following exercises, solve with the methods shown in this section exactly on the interval
For the following exercises, solve exactly on the interval
Use the quadratic formula if the equations do not factor.
For the following exercises, find exact solutions on the interval
Look for opportunities to use trigonometric identities.
### Graphical
For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.
### Technology
For the following exercises, use a calculator to find all solutions to four decimal places.
For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval
Round to four decimal places.
### Extensions
For the following exercises, find all solutions exactly to the equations on the interval
### Real-World Applications
For the following exercises, find a solution to the word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree. |
# Trigonometric Identities and Equations
## Modeling with Trigonometric Functions
Suppose we charted the average daily temperatures in New York City over the course of one year. We would expect to find the lowest temperatures in January and February and highest in July and August. This familiar cycle repeats year after year, and if we were to extend the graph over multiple years, it would resemble a periodic function.
Many other natural phenomena are also periodic. For example, the phases of the moon have a period of approximately 28 days, and birds know to fly south at about the same time each year.
So how can we model an equation to reflect periodic behavior? First, we must collect and record data. We then find a function that resembles an observed pattern. Finally, we make the necessary alterations to the function to get a model that is dependable. In this section, we will take a deeper look at specific types of periodic behavior and model equations to fit data.
### Determining the Amplitude and Period of a Sinusoidal Function
Any motion that repeats itself in a fixed time period is considered periodic motion and can be modeled by a sinusoidal function. The amplitude of a sinusoidal function is the distance from the midline to the maximum value, or from the midline to the minimum value. The midline is the average value. Sinusoidal functions oscillate above and below the midline, are periodic, and repeat values in set cycles. Recall from Graphs of the Sine and Cosine Functions that the period of the sine function and the cosine function is
In other words, for any value of
### Finding Equations and Graphing Sinusoidal Functions
One method of graphing sinusoidal functions is to find five key points. These points will correspond to intervals of equal length representing
of the period. The key points will indicate the location of maximum and minimum values. If there is no vertical shift, they will also indicate x-intercepts. For example, suppose we want to graph the function
We know that the period is
so we find the interval between key points as follows.
Starting with
we calculate the first y-value, add the length of the interval
to 0, and calculate the second y-value. We then add
repeatedly until the five key points are determined. The last value should equal the first value, as the calculations cover one full period. Making a table similar to , we can see these key points clearly on the graph shown in .
### Modeling Periodic Behavior
We will now apply these ideas to problems involving periodic behavior.
### Modeling Harmonic Motion Functions
Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general periodic motion applications cycle through their periods with no outside interference, harmonic motion requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.
### Simple Harmonic Motion
A type of motion described as simple harmonic motion involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When
### Damped Harmonic Motion
In reality, a pendulum does not swing back and forth forever, nor does an object on a spring bounce up and down forever. Eventually, the pendulum stops swinging and the object stops bouncing and both return to equilibrium. Periodic motion in which an energy-dissipating force, or damping factor, acts is known as damped harmonic motion. Friction is typically the damping factor.
In physics, various formulas are used to account for the damping factor on the moving object. Some of these are calculus-based formulas that involve derivatives. For our purposes, we will use formulas for basic damped harmonic motion models.
### Bounding Curves in Harmonic Motion
Harmonic motion graphs may be enclosed by bounding curves. When a function has a varying amplitude, such that the amplitude rises and falls multiple times within a period, we can determine the bounding curves from part of the function.
### Key Equations
### Key Concepts
1. Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts. See and .
2. Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values. See .
3. Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year. See , , and .
4. Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight. See .
5. Damped harmonic motion is a form of periodic behavior affected by a damping factor. Energy dissipating factors, like friction, cause the displacement of the object to shrink. See , , , , and .
6. Bounding curves delineate the graph of harmonic motion with variable maximum and minimum values. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.
### Graphical
For the following exercises, graph the given function, and then find a possible physical process that the equation could model.
### Technology
For the following exercise, construct a function modeling behavior and use a calculator to find desired results.
### Real-World Applications
For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.
For the following exercises, find the amplitude, period, and frequency of the given function.
For the following exercises, construct an equation that models the described behavior.
For the following exercises, construct an equation that models the described behavior.
For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.
### Extensions
For the following exercises, find a function of the form
that fits the given data.
For the following exercises, find a function of the form
that fits the given data.
### Chapter Review Exercises
### Solving Trigonometric Equations with Identities
For the following exercises, find all solutions exactly that exist on the interval
For the following exercises, use basic identities to simplify the expression.
For the following exercises, determine if the given identities are equivalent.
### Sum and Difference Identities
For the following exercises, find the exact value.
For the following exercises, prove the identity.
For the following exercise, simplify the expression.
For the following exercises, find the exact value.
### Double-Angle, Half-Angle, and Reduction Formulas
For the following exercises, find the exact value.
For the following exercises, use to find the desired quantities.
For the following exercises, prove the identity.
For the following exercises, rewrite the expression with no powers.
### Sum-to-Product and Product-to-Sum Formulas
For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.
For the following exercises, evaluate the sum by using a product formula. Write the exact answer.
For the following exercises, change the functions from a product to a sum or a sum to a product.
### Solving Trigonometric Equations
For the following exercises, find all exact solutions on the interval
For the following exercises, find all exact solutions on the interval
For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval
Round to four decimal places.
For the following exercises, graph each side of the equation to find the zeroes on the interval
### Modeling with Trigonometric Equations
For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.
For the following exercises, construct functions that model the described behavior.
For the following exercises, find the amplitude, frequency, and period of the given equations.
For the following exercises, model the described behavior and find requested values.
### Practice Test
For the following exercises, simplify the given expression.
For the following exercises, find the exact value.
For the following exercises, find all exact solutions to the equation on
For the following exercises, prove the identity. |
# Further Applications of Trigonometry
## Introduction to Further Applications of Trigonometry
The world’s largest tree by volume, named General Sherman, stands 274.9 feet tall and resides in Northern California.Source: National Park Service. "The General Sherman Tree." http://www.nps.gov/seki/naturescience/sherman.htm. Accessed April 25, 2014. Just how do scientists know its true height? A common way to measure the height involves determining the angle of elevation, which is formed by the tree and the ground at a point some distance away from the base of the tree. This method is much more practical than climbing the tree and dropping a very long tape measure.
In this chapter, we will explore applications of trigonometry that will enable us to solve many different kinds of problems, including finding the height of a tree. We extend topics we introduced in Trigonometric Functions and investigate applications more deeply and meaningfully. |
# Further Applications of Trigonometry
## Non-right Triangles: Law of Sines
To ensure the safety of over 5,000 U.S. aircraft flying simultaneously during peak times, air traffic controllers monitor and communicate with them after receiving data from the robust radar beacon system. Suppose two radar stations located 20 miles apart each detect an aircraft between them. The angle of elevation measured by the first station is 35 degrees, whereas the angle of elevation measured by the second station is 15 degrees. How can we determine the altitude of the aircraft? We see in that the triangle formed by the aircraft and the two stations is not a right triangle, so we cannot use what we know about right triangles. In this section, we will find out how to solve problems involving non-right triangles.
### Using the Law of Sines to Solve Oblique Triangles
In any triangle, we can draw an altitude, a perpendicular line from one vertex to the opposite side, forming two right triangles. It would be preferable, however, to have methods that we can apply directly to non-right triangles without first having to create right triangles.
Any triangle that is not a right triangle is an oblique triangle. Solving an oblique triangle means finding the measurements of all three angles and all three sides. To do so, we need to start with at least three of these values, including at least one of the sides. We will investigate three possible oblique triangle problem situations:
1. ASA (angle-side-angle) We know the measurements of two angles and the included side. See .
2. AAS (angle-angle-side) We know the measurements of two angles and a side that is not between the known angles. See .
3. SSA (side-side-angle) We know the measurements of two sides and an angle that is not between the known sides. See .
Knowing how to approach each of these situations enables us to solve oblique triangles without having to drop a perpendicular to form two right triangles. Instead, we can use the fact that the ratio of the measurement of one of the angles to the length of its opposite side will be equal to the other two ratios of angle measure to opposite side. Let’s see how this statement is derived by considering the triangle shown in .
Using the right triangle relationships, we know that
and
Solving both equations for
gives two different expressions for
We then set the expressions equal to each other.
Similarly, we can compare the other ratios.
Collectively, these relationships are called the Law of Sines.
Note the standard way of labeling triangles: angle
(alpha) is opposite side
angle
(beta) is opposite side
and angle
(gamma) is opposite side
See .
While calculating angles and sides, be sure to carry the exact values through to the final answer. Generally, final answers are rounded to the nearest tenth, unless otherwise specified.
### Using The Law of Sines to Solve SSA Triangles
We can use the Law of Sines to solve any oblique triangle, but some solutions may not be straightforward. In some cases, more than one triangle may satisfy the given criteria, which we describe as an ambiguous case. Triangles classified as SSA, those in which we know the lengths of two sides and the measurement of the angle opposite one of the given sides, may result in one or two solutions, or even no solution.
### Finding the Area of an Oblique Triangle Using the Sine Function
Now that we can solve a triangle for missing values, we can use some of those values and the sine function to find the area of an oblique triangle. Recall that the area formula for a triangle is given as
where
is base and
is height. For oblique triangles, we must find
before we can use the area formula. Observing the two triangles in , one acute and one obtuse, we can drop a perpendicular to represent the height and then apply the trigonometric property
to write an equation for area in oblique triangles. In the acute triangle, we have
or
However, in the obtuse triangle, we drop the perpendicular outside the triangle and extend the base
to form a right triangle. The angle used in calculation is
or
Thus,
Similarly,
### Solving Applied Problems Using the Law of Sines
The more we study trigonometric applications, the more we discover that the applications are countless. Some are flat, diagram-type situations, but many applications in calculus, engineering, and physics involve three dimensions and motion.
### Key Equations
### Key Concepts
1. The Law of Sines can be used to solve oblique triangles, which are non-right triangles.
2. According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side.
3. There are three possible cases: ASA, AAS, SSA. Depending on the information given, we can choose the appropriate equation to find the requested solution. See .
4. The ambiguous case arises when an oblique triangle can have different outcomes.
5. There are three possible cases that arise from SSA arrangement—a single solution, two possible solutions, and no solution. See and .
6. The Law of Sines can be used to solve triangles with given criteria. See .
7. The general area formula for triangles translates to oblique triangles by first finding the appropriate height value. See .
8. There are many trigonometric applications. They can often be solved by first drawing a diagram of the given information and then using the appropriate equation. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, assume
is opposite side
is opposite side
and
is opposite side
Solve each triangle, if possible. Round each answer to the nearest tenth.
For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle
is opposite side
angle
is opposite side
and angle
is opposite side
For the following exercises, assume
is opposite side
is opposite side
and
is opposite side
Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.
For the following exercises, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.
For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.
### Graphical
For the following exercises, find the length of side
Round to the nearest tenth.
For the following exercises, find the measure of angle
if possible. Round to the nearest tenth.
For the following exercise, solve the triangle. Round each answer to the nearest tenth.
### Extensions
### Real-World Applications
|
# Further Applications of Trigonometry
## Non-right Triangles: Law of Cosines
Suppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles as shown in . How far from port is the boat?
Unfortunately, while the Law of Sines enables us to address many non-right triangle cases, it does not help us with triangles where the known angle is between two known sides, a SAS (side-angle-side) triangle, or when all three sides are known, but no angles are known, a SSS (side-side-side) triangle. In this section, we will investigate another tool for solving oblique triangles described by these last two cases.
### Using the Law of Cosines to Solve Oblique Triangles
The tool we need to solve the problem of the boat’s distance from the port is the Law of Cosines, which defines the relationship among angle measurements and side lengths in oblique triangles. Three formulas make up the Law of Cosines. At first glance, the formulas may appear complicated because they include many variables. However, once the pattern is understood, the Law of Cosines is easier to work with than most formulas at this mathematical level.
Understanding how the Law of Cosines is derived will be helpful in using the formulas. The derivation begins with the Generalized Pythagorean Theorem, which is an extension of the Pythagorean Theorem to non-right triangles. Here is how it works: An arbitrary non-right triangle
is placed in the coordinate plane with vertex
at the origin, side
drawn along the x-axis, and vertex
located at some point
in the plane, as illustrated in . Generally, triangles exist anywhere in the plane, but for this explanation we will place the triangle as noted.
We can drop a perpendicular from
to the x-axis (this is the altitude or height). Recalling the basic trigonometric identities, we know that
In terms of
and
The
point located at
has coordinates
Using the side
as one leg of a right triangle and
as the second leg, we can find the length of hypotenuse
using the Pythagorean Theorem. Thus,
The formula derived is one of the three equations of the Law of Cosines. The other equations are found in a similar fashion.
Keep in mind that it is always helpful to sketch the triangle when solving for angles or sides. In a real-world scenario, try to draw a diagram of the situation. As more information emerges, the diagram may have to be altered. Make those alterations to the diagram and, in the end, the problem will be easier to solve.
### Solving Applied Problems Using the Law of Cosines
Just as the Law of Sines provided the appropriate equations to solve a number of applications, the Law of Cosines is applicable to situations in which the given data fits the cosine models. We may see these in the fields of navigation, surveying, astronomy, and geometry, just to name a few.
### Using Heron’s Formula to Find the Area of a Triangle
We already learned how to find the area of an oblique triangle when we know two sides and an angle. We also know the formula to find the area of a triangle using the base and the height. When we know the three sides, however, we can use Heron’s formula instead of finding the height. Heron of Alexandria was a geometer who lived during the first century A.D. He discovered a formula for finding the area of oblique triangles when three sides are known.
### Key Equations
### Key Concepts
1. The Law of Cosines defines the relationship among angle measurements and lengths of sides in oblique triangles.
2. The Generalized Pythagorean Theorem is the Law of Cosines for two cases of oblique triangles: SAS and SSS. Dropping an imaginary perpendicular splits the oblique triangle into two right triangles or forms one right triangle, which allows sides to be related and measurements to be calculated. See and .
3. The Law of Cosines is useful for many types of applied problems. The first step in solving such problems is generally to draw a sketch of the problem presented. If the information given fits one of the three models (the three equations), then apply the Law of Cosines to find a solution. See and .
4. Heron’s formula allows the calculation of area in oblique triangles. All three sides must be known to apply Heron’s formula. See and See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, assume
is opposite side
is opposite side
and
is opposite side
If possible, solve each triangle for the unknown side. Round to the nearest tenth.
For the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.
For the following exercises, solve the triangle. Round to the nearest tenth.
For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.
### Graphical
For the following exercises, find the length of side
Round to the nearest tenth.
For the following exercises, find the measurement of angle
For the following exercises, solve for the unknown side. Round to the nearest tenth.
For the following exercises, find the area of the triangle. Round to the nearest hundredth.
### Extensions
For the following exercises, suppose that
represents the relationship of three sides of a triangle and the cosine of an angle.
For the following exercises, find the area of the triangle.
### Real-World Applications
|
# Further Applications of Trigonometry
## Polar Coordinates
Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see ). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.
### Plotting Points Using Polar Coordinates
When we think about plotting points in the plane, we usually think of rectangular coordinates
in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled
and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
The polar grid is scaled as the unit circle with the positive x-axis now viewed as the polar axis and the origin as the pole. The first coordinate
is the radius or length of the directed line segment from the pole. The angle
measured in radians, indicates the direction of
We move counterclockwise from the polar axis by an angle of
and measure a directed line segment the length of
in the direction of
Even though we measure
first and then
the polar point is written with the r-coordinate first. For example, to plot the point
we would move
units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in .
### Converting from Polar Coordinates to Rectangular Coordinates
When given a set of polar coordinates, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variables
and
Dropping a perpendicular from the point in the plane to the x-axis forms a right triangle, as illustrated in . An easy way to remember the equations above is to think of
as the adjacent side over the hypotenuse and
as the opposite side over the hypotenuse.
### Converting from Rectangular Coordinates to Polar Coordinates
To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.
### Transforming Equations between Polar and Rectangular Forms
We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.
### Identify and Graph Polar Equations by Converting to Rectangular Equations
We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are indeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will demonstrate that their graphs, while drawn on different grids, are identical.
### Key Equations
### Key Concepts
1. The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
2. To plot a point in the form
move in a counterclockwise direction from the polar axis by an angle of
and then extend a directed line segment from the pole the length of
in the direction of
If
is negative, move in a clockwise direction, and extend a directed line segment the length of
in the direction of
See .
3. If
is negative, extend the directed line segment in the opposite direction of
See .
4. To convert from polar coordinates to rectangular coordinates, use the formulas
and
See and .
5. To convert from rectangular coordinates to polar coordinates, use one or more of the formulas:
and
See .
6. Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See , , and .
7. Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane. See , , and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, convert the given polar coordinates to Cartesian coordinates. Remember to consider the quadrant in which the given point is located when determining
for the point.
For the following exercises, convert the given Cartesian coordinates to polar coordinates with
Remember to consider the quadrant in which the given point is located.
For the following exercises, convert the given Cartesian equation to a polar equation.
For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.
### Graphical
For the following exercises, find the polar coordinates of the point.
For the following exercises, plot the points.
For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.
### Technology
### Extensions
For the following exercise, graph the polar inequality. |
# Further Applications of Trigonometry
## Polar Coordinates: Graphs
The planets move through space in elliptical, periodic orbits about the sun, as shown in . They are in constant motion, so fixing an exact position of any planet is valid only for a moment. In other words, we can fix only a planet’s instantaneous position. This is one application of polar coordinates, represented as
We interpret
as the distance from the center of the sun and
as the planet’s angular bearing, or its direction from the center of the sun. In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates.
### Testing Polar Equations for Symmetry
Just as a rectangular equation such as
describes the relationship between
and
on a Cartesian grid, a polar equation describes a relationship between
and
on a polar grid. Recall that the coordinate pair
indicates that we move counterclockwise from the polar axis (positive x-axis) by an angle of
and extend a ray from the pole (origin)
units in the direction of
All points that satisfy the polar equation are on the graph.
Symmetry is a property that helps us recognize and plot the graph of any equation. If an equation has a graph that is symmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on one side would coincide with the portion on the other side. By performing three tests, we will see how to apply the properties of symmetry to polar equations. Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of
to determine the graph of a polar equation.
In the first test, we consider symmetry with respect to the line
(y-axis). We replace
with
to determine if the new equation is equivalent to the original equation. For example, suppose we are given the equation
This equation exhibits symmetry with respect to the line
In the second test, we consider symmetry with respect to the polar axis (
-axis). We replace
with
or
to determine equivalency between the tested equation and the original. For example, suppose we are given the equation
The graph of this equation exhibits symmetry with respect to the polar axis.
In the third test, we consider symmetry with respect to the pole (origin). We replace
with
to determine if the tested equation is equivalent to the original equation. For example, suppose we are given the equation
The equation has failed the symmetry test, but that does not mean that it is not symmetric with respect to the pole. Passing one or more of the symmetry tests verifies that symmetry will be exhibited in a graph. However, failing the symmetry tests does not necessarily indicate that a graph will not be symmetric about the line
the polar axis, or the pole. In these instances, we can confirm that symmetry exists by plotting reflecting points across the apparent axis of symmetry or the pole. Testing for symmetry is a technique that simplifies the graphing of polar equations, but its application is not perfect.
### Graphing Polar Equations by Plotting Points
To graph in the rectangular coordinate system we construct a table of
and
values. To graph in the polar coordinate system we construct a table of
and
values. We enter values of
into a polar equation and calculate
However, using the properties of symmetry and finding key values of
and
means fewer calculations will be needed.
### Finding Zeros and Maxima
To find the zeros of a polar equation, we solve for the values of
that result in
Recall that, to find the zeros of polynomial functions, we set the equation equal to zero and then solve for
We use the same process for polar equations. Set
and solve for
For many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values of
into the equation that result in the maximum value of the trigonometric functions. Consider
the maximum distance between the curve and the pole is 5 units. The maximum value of the cosine function is 1 when
so our polar equation is
and the value
will yield the maximum
Similarly, the maximum value of the sine function is 1 when
and if our polar equation is
the value
will yield the maximum
We may find additional information by calculating values of
when
These points would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar equation.
### Investigating Circles
Now we have seen the equation of a circle in the polar coordinate system. In the last two examples, the same equation was used to illustrate the properties of symmetry and demonstrate how to find the zeros, maximum values, and plotted points that produced the graphs. However, the circle is only one of many shapes in the set of polar curves.
There are five classic polar curves: cardioids, limaҫons, lemniscates, rose curves, and Archimedes’ spirals. We will briefly touch on the polar formulas for the circle before moving on to the classic curves and their variations.
### Investigating Cardioids
While translating from polar coordinates to Cartesian coordinates may seem simpler in some instances, graphing the classic curves is actually less complicated in the polar system. The next curve is called a cardioid, as it resembles a heart. This shape is often included with the family of curves called limaçons, but here we will discuss the cardioid on its own.
### Investigating Limaçons
The word limaçon is Old French for “snail,” a name that describes the shape of the graph. As mentioned earlier, the cardioid is a member of the limaçon family, and we can see the similarities in the graphs. The other images in this category include the one-loop limaçon and the two-loop (or inner-loop) limaçon. One-loop limaçons are sometimes referred to as dimpled limaçons when
and convex limaçons when
Another type of limaçon, the inner-loop limaçon, is named for the loop formed inside the general limaçon shape. It was discovered by the German artist Albrecht Dürer(1471-1528), who revealed a method for drawing the inner-loop limaçon in his 1525 book Underweysung der Messing. A century later, the father of mathematician Blaise Pascal, Étienne Pascal(1588-1651), rediscovered it.
### Investigating Lemniscates
The lemniscate is a polar curve resembling the infinity symbol
or a figure 8. Centered at the pole, a lemniscate is symmetrical by definition.
### Investigating Rose Curves
The next type of polar equation produces a petal-like shape called a rose curve. Although the graphs look complex, a simple polar equation generates the pattern.
### Investigating the Archimedes’ Spiral
The final polar equation we will discuss is the Archimedes’ spiral, named for its discoverer, the Greek mathematician Archimedes (c. 287 BCE-c. 212 BCE), who is credited with numerous discoveries in the fields of geometry and mechanics.
### Summary of Curves
We have explored a number of seemingly complex polar curves in this section. and summarize the graphs and equations for each of these curves.
### Key Concepts
1. It is easier to graph polar equations if we can test the equations for symmetry with respect to the line
the polar axis, or the pole.
2. There are three symmetry tests that indicate whether the graph of a polar equation will exhibit symmetry. If an equation fails a symmetry test, the graph may or may not exhibit symmetry. See .
3. Polar equations may be graphed by making a table of values for
and
4. The maximum value of a polar equation is found by substituting the value
that leads to the maximum value of the trigonometric expression.
5. The zeros of a polar equation are found by setting
and solving for
See .
6. Some formulas that produce the graph of a circle in polar coordinates are given by
and
See .
7. The formulas that produce the graphs of a cardioid are given by
and
for
and
See .
8. The formulas that produce the graphs of a one-loop limaçon are given by
and
for
See .
9. The formulas that produce the graphs of an inner-loop limaçon are given by
and
for
and
See .
10. The formulas that produce the graphs of a lemniscates are given by
and
where
See .
11. The formulas that produce the graphs of rose curves are given by
and
where
if
is even, there are
petals, and if
is odd, there are
petals. See and .
12. The formula that produces the graph of an Archimedes’ spiral is given by
See .
### Section Exercises
### Verbal
### Graphical
For the following exercises, test the equation for symmetry.
For the following exercises, graph the polar equation. Identify the name of the shape.
### Technology
For the following exercises, use a graphing calculator to sketch the graph of the polar equation.
For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of
and then explain the differences shown in the graphs.
### Extensions
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection. |
# Further Applications of Trigonometry
## Polar Form of Complex Numbers
“God made the integers; all else is the work of man.” This rather famous quote by nineteenth-century German mathematician Leopold Kronecker sets the stage for this section on the polar form of a complex number. Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as Pythagoras, Descartes, De Moivre, Euler, Gauss, and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.
We first encountered complex numbers in Complex Numbers. In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre’s Theorem.
### Plotting Complex Numbers in the Complex Plane
Plotting a complex number
is similar to plotting a real number, except that the horizontal axis represents the real part of the number,
and the vertical axis represents the imaginary part of the number,
### Finding the Absolute Value of a Complex Number
The first step toward working with a complex number in polar form is to find the absolute value. The absolute value of a complex number is the same as its magnitude, or
It measures the distance from the origin to a point in the plane. For example, the graph of
in , shows
### Writing Complex Numbers in Polar Form
The polar form of a complex number expresses a number in terms of an angle
and its distance from the origin
Given a complex number in rectangular form expressed as
we use the same conversion formulas as we do to write the number in trigonometric form:
We review these relationships in .
We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point
The modulus, then, is the same as
the radius in polar form. We use
to indicate the angle of direction (just as with polar coordinates). Substituting, we have
### Converting a Complex Number from Polar to Rectangular Form
Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given
first evaluate the trigonometric functions
and
Then, multiply through by
### Finding Products of Complex Numbers in Polar Form
Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. For the rest of this section, we will work with formulas developed by French mathematician Abraham De Moivre (1667-1754). These formulas have made working with products, quotients, powers, and roots of complex numbers much simpler than they appear. The rules are based on multiplying the moduli and adding the arguments.
### Finding Quotients of Complex Numbers in Polar Form
The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments.
### Finding Powers of Complex Numbers in Polar Form
Finding powers of complex numbers is greatly simplified using De Moivre’s Theorem. It states that, for a positive integer
is found by raising the modulus to the
power and multiplying the argument by
It is the standard method used in modern mathematics.
### Finding Roots of Complex Numbers in Polar Form
To find the in polar form, we use the
Root Theorem or De Moivre’s Theorem and raise the complex number to a power with a rational exponent. There are several ways to represent a formula for finding roots of complex numbers in polar form.
### Key Concepts
1. Complex numbers in the form
are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the x-axis as the real axis and the y-axis as the imaginary axis. See .
2. The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point:
See and .
3. To write complex numbers in polar form, we use the formulas
and
Then,
See and .
4. To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through by
See and .
5. To find the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the trigonometric functions, and multiply using the distributive property. See .
6. To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. See .
7. To find the power of a complex number
raise
to the power
and multiply
by
See .
8. Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the absolute value of the given complex number.
For the following exercises, write the complex number in polar form.
For the following exercises, convert the complex number from polar to rectangular form.
For the following exercises, find
in polar form.
For the following exercises, find
in polar form.
For the following exercises, find the powers of each complex number in polar form.
For the following exercises, evaluate each root.
### Graphical
For the following exercises, plot the complex number in the complex plane.
### Technology
For the following exercises, find all answers rounded to the nearest hundredth. |
# Further Applications of Trigonometry
## Parametric Equations
Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in . At any moment, the moon is located at a particular spot relative to the planet. But how do we write and solve the equation for the position of the moon when the distance from the planet, the speed of the moon’s orbit around the planet, and the speed of rotation around the sun are all unknowns? We can solve only for one variable at a time.
In this section, we will consider sets of equations given by
and
where
is the independent variable of time. We can use these parametric equations in a number of applications when we are looking for not only a particular position but also the direction of the movement. As we trace out successive values of
the orientation of the curve becomes clear. This is one of the primary advantages of using parametric equations: we are able to trace the movement of an object along a path according to time. We begin this section with a look at the basic components of parametric equations and what it means to parameterize a curve. Then we will learn how to eliminate the parameter, translate the equations of a curve defined parametrically into rectangular equations, and find the parametric equations for curves defined by rectangular equations.
### Parameterizing a Curve
When an object moves along a curve—or curvilinear path—in a given direction and in a given amount of time, the position of the object in the plane is given by the x-coordinate and the y-coordinate. However, both
and
vary over time and so are functions of time. For this reason, we add another variable, the parameter, upon which both
and
are dependent functions. In the example in the section opener, the parameter is time,
The
position of the moon at time,
is represented as the function
and the
position of the moon at time,
is represented as the function
Together,
and
are called parametric equations, and generate an ordered pair
Parametric equations primarily describe motion and direction.
When we parameterize a curve, we are translating a single equation in two variables, such as
and
into an equivalent pair of equations in three variables,
and
One of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object’s motion over time.
When we graph parametric equations, we can observe the individual behaviors of
and of
There are a number of shapes that cannot be represented in the form
meaning that they are not functions. For example, consider the graph of a circle, given as
Solving for
gives
or two equations:
and
If we graph
and
together, the graph will not pass the vertical line test, as shown in . Thus, the equation for the graph of a circle is not a function.
However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.
### Eliminating the Parameter
In many cases, we may have a pair of parametric equations but find that it is simpler to draw a curve if the equation involves only two variables, such as
and
Eliminating the parameter is a method that may make graphing some curves easier. However, if we are concerned with the mapping of the equation according to time, then it will be necessary to indicate the orientation of the curve as well. There are various methods for eliminating the parameter
from a set of parametric equations; not every method works for every type of equation. Here we will review the methods for the most common types of equations.
### Eliminating the Parameter from Polynomial, Exponential, and Logarithmic Equations
For polynomial, exponential, or logarithmic equations expressed as two parametric equations, we choose the equation that is most easily manipulated and solve for
We substitute the resulting expression for
into the second equation. This gives one equation in
and
### Eliminating the Parameter from Trigonometric Equations
Eliminating the parameter from trigonometric equations is a straightforward substitution. We can use a few of the familiar trigonometric identities and the Pythagorean Theorem.
First, we use the identities:
Solving for
and
we have
Then, use the Pythagorean Theorem:
Substituting gives
### Finding Cartesian Equations from Curves Defined Parametrically
When we are given a set of parametric equations and need to find an equivalent Cartesian equation, we are essentially “eliminating the parameter.” However, there are various methods we can use to rewrite a set of parametric equations as a Cartesian equation. The simplest method is to set one equation equal to the parameter, such as
In this case,
can be any expression. For example, consider the following pair of equations.
Rewriting this set of parametric equations is a matter of substituting
for
Thus, the Cartesian equation is
### Finding Parametric Equations for Curves Defined by Rectangular Equations
Although we have just shown that there is only one way to interpret a set of parametric equations as a rectangular equation, there are multiple ways to interpret a rectangular equation as a set of parametric equations. Any strategy we may use to find the parametric equations is valid if it produces equivalency. In other words, if we choose an expression to represent
and then substitute it into the
equation, and it produces the same graph over the same domain as the rectangular equation, then the set of parametric equations is valid. If the domain becomes restricted in the set of parametric equations, and the function does not allow the same values for
as the domain of the rectangular equation, then the graphs will be different.
### Key Concepts
1. Parameterizing a curve involves translating a rectangular equation in two variables,
and
into two equations in three variables, x, y, and t. Often, more information is obtained from a set of parametric equations. See , , and .
2. Sometimes equations are simpler to graph when written in rectangular form. By eliminating
an equation in
and
is the result.
3. To eliminate
solve one of the equations for
and substitute the expression into the second equation. See , , , and .
4. Finding the rectangular equation for a curve defined parametrically is basically the same as eliminating the parameter. Solve for
in one of the equations, and substitute the expression into the second equation. See .
5. There are an infinite number of ways to choose a set of parametric equations for a curve defined as a rectangular equation.
6. Find an expression for
such that the domain of the set of parametric equations remains the same as the original rectangular equation. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, eliminate the parameter
to rewrite the parametric equation as a Cartesian equation.
For the following exercises, rewrite the parametric equation as a Cartesian equation by building an
table.
For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting
or by setting
For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using
and
Identify the curve.
### Technology
For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.
For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.
### Extensions
|
# Further Applications of Trigonometry
## Parametric Equations: Graphs
While not every fan (or team manager) appreciates it, baseball and many other sports have become dependent on analytics, which involve complex data recording and quantitative evaluation used to understand and predict behavior. The earliest influence of analytics was mostly statistical; more recently, physics and other sciences have come into play. Foremost among these is the focus on launch angle and exit velocity, which when at certain values can almost guarantee a home run. On the other hand, emphasis on launch angle and focusing on home runs rather than overall hitting results in far more outs. Consider the following situation: it is the bottom of the ninth inning, with two outs and two players on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately
to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using parametric equations. In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.
### Graphing Parametric Equations by Plotting Points
In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.
### Applications of Parametric Equations
Many of the advantages of parametric equations become obvious when applied to solving real-world problems. Although rectangular equations in x and y give an overall picture of an object's path, they do not reveal the position of an object at a specific time. Parametric equations, however, illustrate how the values of x and y change depending on t, as the location of a moving object at a particular time.
A common application of parametric equations is solving problems involving projectile motion. In this type of motion, an object is propelled forward in an upward direction forming an angle of
to the horizontal, with an initial speed of
and at a height
above the horizontal.
The path of an object propelled at an inclination of
to the horizontal, with initial speed
and at a height
above the horizontal, is given by
where
accounts for the effects of gravity and
is the initial height of the object. Depending on the units involved in the problem, use
or
The equation for
gives horizontal distance, and the equation for
gives the vertical distance.
### Key Concepts
1. When there is a third variable, a third parameter on which
and
depend, parametric equations can be used.
2. To graph parametric equations by plotting points, make a table with three columns labeled
and
Choose values for
in increasing order. Plot the last two columns for
and
See and .
3. When graphing a parametric curve by plotting points, note the associated t-values and show arrows on the graph indicating the orientation of the curve. See and .
4. Parametric equations allow the direction or the orientation of the curve to be shown on the graph. Equations that are not functions can be graphed and used in many applications involving motion. See .
5. Projectile motion depends on two parametric equations:
and
Initial velocity is symbolized as
represents the initial angle of the object when thrown, and
represents the height at which the object is propelled.
### Section Exercises
### Verbal
### Graphical
For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.
For the following exercises, sketch the curve and include the orientation.
For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.
For the following exercises, graph the equation and include the orientation.
For the following exercises, use the parametric equations for integers a and b:
For the following exercises, describe the graph of the set of parametric equations.
For the following exercises, use a graphing utility to graph on the window
by
on the domain
for the following values of
and
, and include the orientation.
### Technology
For the following exercises, look at the graphs that were created by parametric equations of the form
Use the parametric mode on the graphing calculator to find the values of
and
to achieve each graph.
For the following exercises, use a graphing utility to graph the given parametric equations.
### Extensions
For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 64 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time
Neglect air resistance.
For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain. |
# Further Applications of Trigonometry
## Vectors
An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour, as shown in . What are the ground speed and actual bearing of the plane?
Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane’s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let’s examine the basics of vectors.
### A Geometric View of Vectors
A vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:
1. Lower case, boldfaced type, with or without an arrow on top such as
2. Given initial point
and terminal point
a vector can be represented as
The arrowhead on top is what indicates that it is not just a line, but a directed line segment.
3. Given an initial point of
and terminal point
a vector may be represented as
This last symbol
has special significance. It is called the standard position. The position vector has an initial point
and a terminal point
To change any vector into the position vector, we think about the change in the x-coordinates and the change in the y-coordinates. Thus, if the initial point of a vector
is
and the terminal point is
then the position vector is found by calculating
In , we see the original vector
and the position vector
### Finding Magnitude and Direction
To work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.
### Performing Vector Addition and Scalar Multiplication
Now that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector
as an arrow or directed line segment from the origin to the point
vectors can be situated anywhere in the plane. The sum of two vectors and , or vector addition, produces a third vector + , the resultant vector.
To find + , we first draw the vector , and from the terminal end of , we drawn the vector . In other words, we have the initial point of meet the terminal end of . This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum + is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of to the end of in a straight path, as shown in .
Vector subtraction is similar to vector addition. To find − , view it as + (−). Adding − is reversing direction of and adding it to the end of . The new vector begins at the start of and stops at the end point of −. See for a visual that compares vector addition and vector subtraction using parallelograms.
### Multiplying By a Scalar
While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.
### Finding Component Form
In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the
direction, and the vertical component is the
direction. For example, we can see in the graph in that the position vector
comes from adding the vectors 1 and 2. We have 1 with initial point
and terminal point
We also have 2 with initial point
and terminal point
Therefore, the position vector is
Using the Pythagorean Theorem, the magnitude of 1 is 2, and the magnitude of 2 is 3. To find the magnitude of , use the formula with the position vector.
The magnitude of is
To find the direction, we use the tangent function
Thus, the magnitude of is
and the direction is
off the horizontal.
### Finding the Unit Vector in the Direction of v
In addition to finding a vector’s components, it is also useful in solving problems to find a vector in the same direction as the given vector, but of magnitude 1. We call a vector with a magnitude of 1 a unit vector. We can then preserve the direction of the original vector while simplifying calculations.
Unit vectors are defined in terms of components. The horizontal unit vector is written as
and is directed along the positive horizontal axis. The vertical unit vector is written as
and is directed along the positive vertical axis. See .
### Performing Operations with Vectors in Terms of i and j
So far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar multiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar with the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of and .
### Performing Operations on Vectors in Terms of i and j
When vectors are written in terms of and we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.
### Calculating the Component Form of a Vector: Direction
We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using
For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.
Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with replacing
### Finding the Dot Product of Two Vectors
As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.
The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.
### Key Concepts
1. The position vector has its initial point at the origin. See .
2. If the position vector is the same for two vectors, they are equal. See .
3. Vectors are defined by their magnitude and direction. See .
4. If two vectors have the same magnitude and direction, they are equal. See .
5. Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements. See .
6. Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same. See and .
7. Vectors are comprised of two components: the horizontal component along the positive x-axis, and the vertical component along the positive y-axis. See .
8. The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.
9. The magnitude of a vector in the rectangular coordinate system is
See .
10. In the rectangular coordinate system, unit vectors may be represented in terms of and where represents the horizontal component and represents the vertical component. Then, = a + b is a scalar multiple of by real numbers
See and .
11. Adding and subtracting vectors in terms of i and j consists of adding or subtracting corresponding coefficients of i and corresponding coefficients of j. See .
12. A vector v = a + b is written in terms of magnitude and direction as
See .
13. The dot product of two vectors is the product of the terms plus the product of the terms. See .
14. We can use the dot product to find the angle between two vectors. and .
15. Dot products are useful for many types of physics applications. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the two vectors and are equal, where has an initial point
and a terminal point
and has an initial point
and a terminal point
.
For the following exercises, use the vectors = + 5, = −2− 3, and = 4 − .
For the following exercises, use the given vectors to compute + , − , and 2 − 3.
For the following exercises, find a unit vector in the same direction as the given vector.
For the following exercises, find the magnitude and direction of the vector,
### Graphical
For the following exercises, given draw 3 and
For the following exercises, use the vectors shown to sketch + , − , and 2.
For the following exercises, use the vectors shown to sketch 2 + .
For the following exercises, use the vectors shown to sketch − 3.
For the following exercises, write the vector shown in component form.
### Extensions
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.
### Real-World Applications
### Chapter Review Exercises
### Non-right Triangles: Law of Sines
For the following exercises, assume
is opposite side
is opposite side
and
is opposite side
Solve each triangle, if possible. Round each answer to the nearest tenth.
### Non-right Triangles: Law of Cosines
### Polar Coordinates
For the following exercises, convert the given Cartesian equation to a polar equation.
For the following exercises, convert the given polar equation to a Cartesian equation.
For the following exercises, convert to rectangular form and graph.
### Polar Coordinates: Graphs
For the following exercises, test each equation for symmetry.
### Polar Form of Complex Numbers
For the following exercises, find the absolute value of each complex number.
Write the complex number in polar form.
For the following exercises, convert the complex number from polar to rectangular form.
For the following exercises, find the product
in polar form.
For the following exercises, find the quotient
in polar form.
For the following exercises, find the powers of each complex number in polar form.
For the following exercises, evaluate each root.
For the following exercises, plot the complex number in the complex plane.
### Parametric Equations
For the following exercises, eliminate the parameter
to rewrite the parametric equation as a Cartesian equation.
### Parametric Equations: Graphs
For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.
### Vectors
For the following exercises, determine whether the two vectors, and are equal, where has an initial point
and a terminal point
and has an initial point
and a terminal point
For the following exercises, use the vectors and to evaluate the expression.
For the following exercises, find a unit vector in the same direction as the given vector.
For the following exercises, find the magnitude and direction of the vector.
For the following exercises, calculate
### Practice Test
Given
and
evaluate each expression.
For the following exercises, use the vectors = − 3 and = 2 + 3. |
# Systems of Equations and Inequalities
## Introduction to Systems of Equations and Inequalities
At the start of the Second World War, British military and intelligence officers recognized that defeating Nazi Germany would require the Allies to know what the enemy was planning. This task was complicated by the fact that the German military transmitted all of its communications through a presumably uncrackable code created by a machine called Enigma. The Germans had been encoding their messages with this machine since the early 1930s, and were so confident in its security that they used it for everyday military communications as well as highly important strategic messages. Concerned about the increasing military threat, other European nations began working to decipher the Enigma codes. Poland was the first country to make significant advances when it trained and recruited a new group of codebreakers: math students from Poznań University. With the help of intelligence obtained by French spies, Polish mathematicians, led by Marian Rejewski, were able to decipher initial codes and later to understand the wiring of the machines; eventually they create replicas. However, the German military eventually increased the complexity of the machines by adding additional rotors, requiring a new method of decryption.
The machine attached letters on a keyboard to three, four, or five rotors (depending on the version), each with 26 starting positions that could be set prior to encoding; a decryption code (called a cipher key) essentially conveyed these settings to the message recipient, and allowed people to interpret the message using another Enigma machine. Even with the simpler three-rotor scrambler, there were 17,576 different combinations of starting positions (26 x 26 x 26); plus the machine had numerous other methods of introducing variation. Not long after the war started, the British recruited a team of brilliant codebreakers to crack the Enigma code. The codebreakers, led by Alan Turing, used what they knew about the Enigma machine to build a mechanical computer that could crack the code. And that knowledge of what the Germans were planning proved to be a key part of the ultimate Allied victory of Nazi Germany in 1945.
The Enigma is perhaps the most famous cryptographic device ever known. It stands as an example of the pivotal role cryptography has played in society. Now, technology has moved cryptanalysis to the digital world.
Many ciphers are designed using invertible matrices as the method of message transference, as finding the inverse of a matrix is generally part of the process of decoding. In addition to knowing the matrix and its inverse, the receiver must also know the key that, when used with the matrix inverse, will allow the message to be read.
In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses. |
# Systems of Equations and Inequalities
## Systems of Linear Equations: Two Variables
### Learning Objectives
1. Determine whether an ordered pair is a solution of a system of equations (IA 4.1.1)
2. Solve a system of linear equations by graphing (IA 4.1.2)
### Objective: Determine whether an ordered pair is a solution of a system of equations (IA 4.1.1)
A system of linear equations is a group of two or more linear equations. For example,
is a system of linear equations
A solution to a system of linear equations is an ordered pair x,y that is a solution to every equation in the system.
### Practice Makes Perfect
Determine whether the ordered pairs are solutions to the given system.
at and
### Solve a system of linear equations by graphing (IA 4.1.2)
### Practice Makes Perfect
A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions.
### Introduction to Systems of Equations
In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.
In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.
The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.
In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. A consistent system of equations has at least one solution. A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored. The two lines have different slopes and intersect at one point in the plane. A consistent system is considered to be a dependent system if the equations have the same slope and the same y-intercepts. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions.
Another type of system of linear equations is an inconsistent system, which is one in which the equations represent two parallel lines. The lines have the same slope and different y-intercepts. There are no points common to both lines; hence, there is no solution to the system.
### Solving Systems of Equations by Graphing
There are multiple methods of solving systems of linear equations. For a system of linear equations in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.
### Solving Systems of Equations by Substitution
Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving a system of linear equations that are more precise than graphing. One such method is solving a system of equations by the substitution method, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.
### Solving Systems of Equations in Two Variables by the Addition Method
A third method of solving systems of linear equations is the addition method. In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition.
### Identifying Inconsistent Systems of Equations Containing Two Variables
Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system consists of parallel lines that have the same slope but different
-intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as
### Expressing the Solution of a System of Dependent Equations Containing Two Variables
Recall that a dependent system of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as
### Using Systems of Equations to Investigate Profits
Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer’s revenue function is the function used to calculate the amount of money that comes into the business. It can be represented by the equation
where
quantity and
price. The revenue function is shown in orange in .
The cost function is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue in . The
-axis represents quantity in hundreds of units. The y-axis represents either cost or revenue in hundreds of dollars.
The point at which the two lines intersect is called the break-even point. We can see from the graph that if 700 units are produced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce and sell 700 units. They neither make money nor lose money.
The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The profit function is the revenue function minus the cost function, written as
Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.
### Key Concepts
1. A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously.
2. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. See .
3. Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution.
4. One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the equations on the same set of axes. See .
5. Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable in one equation and substitute the result into the second equation. See .
6. A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding opposite coefficients of corresponding variables. See .
7. It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when adding the two equations together. See , , and .
8. Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never intersect. See .
9. The solution to a system of dependent equations will always be true because both equations describe the same line. See .
10. Systems of equations can be used to solve real-world problems that involve more than one variable, such as those relating to revenue, cost, and profit. See and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the given ordered pair is a solution to the system of equations.
For the following exercises, solve each system by substitution.
For the following exercises, solve each system by addition.
For the following exercises, solve each system by any method.
### Graphical
For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.
### Technology
For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.
### Extensions
For the following exercises, solve each system in terms of
and
where
are nonzero numbers. Note that
and
### Real-World Applications
For the following exercises, solve for the desired quantity.
For the following exercises, use a system of linear equations with two variables and two equations to solve. |
# Systems of Equations and Inequalities
## Systems of Linear Equations: Three Variables
### Learning Objectives
1. Determine whether an ordered triple is a solution of a system of three linear equations with three variables (IA 4.4.1)
2. Solve a system of three linear equations with three variables (IA 4.4.2)
### Objective 1: Determine whether an ordered triple is a solution of a system of three linear equations with three variables (IA 4.4.1)
A linear equation with three variables where a, b, c, and d are real numbers and a, b, and c are not all 0, is of the form . The graph of a linear equation with three variables is a plane.
A system of linear equations with three variables is a set of linear equations with three variables. For example,
is a system of linear equations with three variables.
Solutions of a system of equations are the values of the variables that make all the equations true. A solution is represented by an ordered triple (x,y,z).
### Practice Makes Perfect
Determine whether the ordered pairs are solutions to the given system.
### Objective 2: Solve a system of three linear equations with three variables (IA 4.4.2)
When we solve a system of linear equations with three variables, we have many possible solutions.
The solutions are summarized in the table below.
### Practice Makes Perfect
Jordi received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. Jordi invested $4,000 more in mutual funds than in municipal bonds. He earned $670 in interest the first year. How much did Jordi invest in each type of fund?
Understanding the correct approach to setting up problems such as this one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visualization.
### Solving Systems of Three Equations in Three Variables
In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination, named after the prolific German mathematician Karl Friedrich Gauss. While there is no definitive order in which operations are to be performed, there are specific guidelines as to what type of moves can be made. We may number the equations to keep track of the steps we apply. The goal is to eliminate one variable at a time to achieve upper triangular form, the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution
which we call an ordered triple. A system in upper triangular form looks like the following:
The third equation can be solved for
and then we back-substitute to find
and
To write the system in upper triangular form, we can perform the following operations:
1. Interchange the order of any two equations.
2. Multiply both sides of an equation by a nonzero constant.
3. Add a nonzero multiple of one equation to another equation.
The solution set to a three-by-three system is an ordered triple
Graphically, the ordered triple defines the point that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a rectangular room. A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.
### Identifying Inconsistent Systems of Equations Containing Three Variables
Just as with systems of equations in two variables, we may come across an inconsistent system of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as
or some other contradiction.
### Expressing the Solution of a System of Dependent Equations Containing Three Variables
We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.
### Key Concepts
1. A solution set is an ordered triple
that represents the intersection of three planes in space. See .
2. A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. See .
3. Systems of three equations in three variables are useful for solving many different types of real-world problems. See .
4. A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. See .
5. Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
6. A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. See .
7. Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the ordered triple given is the solution to the system of equations.
For the following exercises, solve each system by elimination.
For the following exercises, solve each system by Gaussian elimination.
### Extensions
For the following exercises, solve the system for
and
### Real-World Applications
|
# Systems of Equations and Inequalities
## Systems of Nonlinear Equations and Inequalities: Two Variables
### Learning Objectives
1. Graph a parabola (IA 11.2.1)
2. Graph a circle (IA 11.1.4)
### Objective 1: Graph a parabola (IA 11.2.1)
A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
### Practice Makes Perfect
### Objective 2: Graph a circle (IA 11.1.4)
Any equation of the form is the standard form of the equation of a circle with center, (h,k) and radius. We can then graph the circle on a rectangular coordinate system using the center and radius.
### Practice Makes Perfect
Graph a circle.
Halley’s Comet () orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.
In this section, we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.
### Solving a System of Nonlinear Equations Using Substitution
A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form
Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.
### Intersection of a Parabola and a Line
There are three possible types of solutions for a system of nonlinear equations involving a parabola and a line.
### Intersection of a Circle and a Line
Just as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.
### Solving a System of Nonlinear Equations Using Elimination
We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of equations representing a circle and an ellipse.
### Graphing a Nonlinear Inequality
All of the equations in the systems that we have encountered so far have involved equalities, but we may also encounter systems that involve inequalities. We have already learned to graph linear inequalities by graphing the corresponding equation, and then shading the region represented by the inequality symbol. Now, we will follow similar steps to graph a nonlinear inequality so that we can learn to solve systems of nonlinear inequalities. A nonlinear inequality is an inequality containing a nonlinear expression. Graphing a nonlinear inequality is much like graphing a linear inequality.
Recall that when the inequality is greater than,
or less than,
the graph is drawn with a dashed line. When the inequality is greater than or equal to,
or less than or equal to,
the graph is drawn with a solid line. The graphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, the whole region works. That is the region we shade. See .
### Graphing a System of Nonlinear Inequalities
Now that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A system of nonlinear inequalities is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear. Graphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the feasible region.
### Key Concepts
1. There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no solution, the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points. See .
2. There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solution, the line is tangent to the circle; (3) two solutions, the line intersects the circle in two points. See .
3. There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle: (1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to each other; (3) two solutions, the circle and the ellipse intersect in two points; (4) three solutions, the circle and ellipse intersect in three places; (5) four solutions, the circle and the ellipse intersect in four points. See .
4. An inequality is graphed in much the same way as an equation, except for > or <, we draw a dashed line and shade the region containing the solution set. See .
5. Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, solve the system of nonlinear equations using substitution.
For the following exercises, solve the system of nonlinear equations using elimination.
For the following exercises, use any method to solve the system of nonlinear equations.
For the following exercises, use any method to solve the nonlinear system.
### Graphical
For the following exercises, graph the inequality.
For the following exercises, graph the system of inequalities. Label all points of intersection.
### Extensions
For the following exercises, graph the inequality.
For the following exercises, find the solutions to the nonlinear equations with two variables.
### Technology
For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.
### Real-World Applications
For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions. |
# Systems of Equations and Inequalities
## Partial Fractions
### Learning Objectives
1. Find the least common denominator of rational expressions (IA 7.2.3)
2. Solve a system of equations by elimination (IA 4.1.4)
### Objective 1: Find the least common denominator of rational expressions (IA 7.2.3)
A rational expression is an expression of the form where p and q are polynomials and .
are examples of rational expressions.
### Practice Makes Perfect
Find the least common denominator of the following rationals:
### Objective 2: Solve a system of equations by elimination (IA 4.1.4)
### Partial Fraction Decomposition
When we add rational expressions with unlike denominators such as and , we first need to find the LCD, then rewrite each fraction with the common denominator, and finally add the two numerators.
We want to do the opposite now.
Given a rational expression like, we would like to rewrite it as an addition of two simpler rational expressions and . Our goal is to find the values of A and B such that
Earlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three variables, and nonlinear systems. Here we introduce another way that systems of equations can be utilized—the decomposition of rational expressions.
Fractions can be complicated; adding a variable in the denominator makes them even more so. The methods studied in this section will help simplify the concept of a rational expression.
### Decomposing Where Q(x) Has Only Nonrepeated Linear Factors
Recall the algebra regarding adding and subtracting rational expressions. These operations depend on finding a common denominator so that we can write the sum or difference as a single, simplified rational expression. In this section, we will look at partial fraction decomposition, which is the undoing of the procedure to add or subtract rational expressions. In other words, it is a return from the single simplified rational expression to the original expressions, called the partial fraction.
For example, suppose we add the following fractions:
We would first need to find a common denominator,
Next, we would write each expression with this common denominator and find the sum of the terms.
Partial fraction decomposition is the reverse of this procedure. We would start with the solution and rewrite (decompose) it as the sum of two fractions.
We will investigate rational expressions with linear factors and quadratic factors in the denominator where the degree of the numerator is less than the degree of the denominator. Regardless of the type of expression we are decomposing, the first and most important thing to do is factor the denominator.
When the denominator of the simplified expression contains distinct linear factors, it is likely that each of the original rational expressions, which were added or subtracted, had one of the linear factors as the denominator. In other words, using the example above, the factors of
are
the denominators of the decomposed rational expression. So we will rewrite the simplified form as the sum of individual fractions and use a variable for each numerator. Then, we will solve for each numerator using one of several methods available for partial fraction decomposition.
### Decomposing Where Q(x) Has Repeated Linear Factors
Some fractions we may come across are special cases that we can decompose into partial fractions with repeated linear factors. We must remember that we account for repeated factors by writing each factor in increasing powers.
### Decomposing Where Q(x) Has a Nonrepeated Irreducible Quadratic Factor
So far, we have performed partial fraction decomposition with expressions that have had linear factors in the denominator, and we applied numerators
or
representing constants. Now we will look at an example where one of the factors in the denominator is a quadratic expression that does not factor. This is referred to as an irreducible quadratic factor. In cases like this, we use a linear numerator such as
etc.
### Decomposing When Q(x) Has a Repeated Irreducible Quadratic Factor
Now that we can decompose a simplified rational expression with an irreducible quadratic factor, we will learn how to do partial fraction decomposition when the simplified rational expression has repeated irreducible quadratic factors. The decomposition will consist of partial fractions with linear numerators over each irreducible quadratic factor represented in increasing powers.
### Key Concepts
1. Decompose
by writing the partial fractions as
Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations. See .
2. The decomposition of
with repeated linear factors must account for the factors of the denominator in increasing powers. See .
3. The decomposition of
with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in
See .
4. In the decomposition of
where
has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as
See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.
For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.
For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.
For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.
### Extensions
For the following exercises, find the partial fraction expansion.
For the following exercises, perform the operation and then find the partial fraction decomposition. |
# Systems of Equations and Inequalities
## Matrices and Matrix Operations
### Learning Objectives
1. Write the augmented matrix for a system of equations (IA 4.5.1)
2. Add, subtract matrices and multiply a matrix by a scalar
### Objective 1: Write the augmented matrix for a system of equations (IA 4.5.1)
A matrix is a rectangular array of numbers arranged in rows and columns.
A matrix with m rows and n columns has dimension m×n.
Each number in the matrix is called an element or entry in the matrix.
The matrix on the left below has 2 rows and 3 columns and so it has order 2×3. We say it is a 2 by 3 matrix.
We will use a matrix to represent systems of equations.
Each column then would be the coefficients of one of the variables in the system or the constants.
A vertical line replaces the equal signs.
We call the resulting matrix the augmented matrix for the system of equations.
### Practice Makes Perfect
Write each system of linear equations as an augmented matrix
### Objective 2: Add, subtract matrices and multiply a matrix by a scalar
We add or subtract matrices by adding or subtracting corresponding entries.
In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a 3 × 3 matrix and another 3 × 3 matrix, but we cannot add or subtract a 2 × 3 matrix and a 3 × 3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix.
The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.
### Practice Makes Perfect
Perform the indicated operations
Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. shows the needs of both teams.
A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.
### Finding the Sum and Difference of Two Matrices
To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named
and
are shown below.
### Describing Matrices
A matrix is often referred to by its size or dimensions:
indicating
rows and
columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix
identified as
we look for the entry in row
column
In matrix
shown below, the entry in row 2, column 3 is
A square matrix is a matrix with dimensions
meaning that it has the same number of rows as columns. The
matrix above is an example of a square matrix.
A row matrix is a matrix consisting of one row with dimensions
A column matrix is a matrix consisting of one column with dimensions
A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations.
### Adding and Subtracting Matrices
We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.
In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a
matrix and another
matrix, but we cannot add or subtract a
matrix and a
matrix because some entries in one matrix will not have a corresponding entry in the other matrix.
### Finding Scalar Multiples of a Matrix
Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.
Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in .
Converting the data to a matrix, we have
To calculate how much computer equipment will be needed, we multiply all entries in matrix
by 0.15.
We must round up to the next integer, so the amount of new equipment needed is
Adding the two matrices as shown below, we see the new inventory amounts.
This means
Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.
### Finding the Product of Two Matrices
In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If
is an
matrix and
is an
matrix, then the product matrix
is an
matrix. For example, the product
is possible because the number of columns in
is the same as the number of rows in
If the inner dimensions do not match, the product is not defined.
We multiply entries of
with entries of
according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.
To obtain the entries in row
of
we multiply the entries in row
of
by column
in
and add. For example, given matrices
and
where the dimensions of
are
and the dimensions of
are
the product of
will be a
matrix.
Multiply and add as follows to obtain the first entry of the product matrix
1. To obtain the entry in row 1, column 1 of
multiply the first row in
by the first column in
and add.
2. To obtain the entry in row 1, column 2 of
multiply the first row of
by the second column in
and add.
3. To obtain the entry in row 1, column 3 of
multiply the first row of
by the third column in
and add.
We proceed the same way to obtain the second row of
In other words, row 2 of
times column 1 of
row 2 of
times column 2 of
row 2 of
times column 3 of
When complete, the product matrix will be
### Key Concepts
1. A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
2. The dimensions of a matrix refer to the number of rows and the number of columns. A
matrix has three rows and two columns. See .
3. We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix. See , , , and .
4. Scalar multiplication involves multiplying each entry in a matrix by a constant. See .
5. Scalar multiplication is often required before addition or subtraction can occur. See .
6. Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
7. The product of two matrices,
and
is obtained by multiplying each entry in row 1 of
by each entry in column 1 of
then multiply each entry of row 1 of
by each entry in columns 2 of
and so on. See and .
8. Many real-world problems can often be solved using matrices. See .
9. We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.
For the following exercises, use the matrices below to perform scalar multiplication.
For the following exercises, use the matrices below to perform matrix multiplication.
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint:
)
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint:
)
### Technology
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.
### Extensions
For the following exercises, use the matrix below to perform the indicated operation on the given matrix. |
# Systems of Equations and Inequalities
## Solving Systems with Gaussian Elimination
### Learning Objectives
1. Use row operations on a matrix (IA 4.5.2)
2. Solve systems of equations using matrices (IA 4.5.3)
### Objective 1: Use row operations on a matrix (IA 4.5.2)
In the last section, we learned how to write the augmented matrix for a system of equations.
Once a system of equations is in its augmented matrix form, we will solve by elimination by performing operations on the rows that will lead us to the solution. Our goal will be to get 1 on the diagonal of the matrix and all entries below the diagonal must be zeros.
### Row Operations
In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.
1. Interchange any two rows.
2. Multiply a row by any real number except 0.
3. Add a nonzero multiple of one row to another row.
These actions are called row operations and will help us use the matrix to solve a system of equations.
### Practice Makes Perfect
### Objective 2: Solve systems of equations using matrices (IA 4.5.3)
To solve a system of equations using matrices, we transform the augmented matrix into a matrix in row-echelon form using row operations. For a consistent and independent system of equations, the augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.
Once we get the augmented matrix into row-echelon form, we can write the equivalent system of equations and solve for at least one variable. We then substitute this value in another equation to continue to solve for the other variables.
### Practice Makes Perfect
Carl Friedrich Gauss lived during the late 18th century and early 19th century, but he is still considered one of the most prolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.
We first encountered Gaussian elimination in Systems of Linear Equations: Two Variables. In this section, we will revisit this technique for solving systems, this time using matrices.
### Writing the Augmented Matrix of a System of Equations
A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
For example, consider the following
system of equations.
We can write this system as an augmented matrix:
We can also write a matrix containing just the coefficients. This is called the coefficient matrix.
A three-by-three system of equations such as
has a coefficient matrix
and is represented by the augmented matrix
Notice that the matrix is written so that the variables line up in their own columns: x-terms go in the first column, y-terms in the second column, and z-terms in the third column. It is very important that each equation is written in standard form
so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.
### Writing a System of Equations from an Augmented Matrix
We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.
### Performing Row Operations on a Matrix
Now that we can write systems of equations in augmented matrix form, we will examine the various row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows.
Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown.
We use row operations corresponding to equation operations to obtain a new matrix that is row-equivalent in a simpler form. Here are the guidelines to obtaining row-echelon form.
1. In any nonzero row, the first nonzero number is a 1. It is called a leading 1.
2. Any all-zero rows are placed at the bottom on the matrix.
3. Any leading 1 is below and to the right of a previous leading 1.
4. Any column containing a leading 1 has zeros in all other positions in the column.
To solve a system of equations we can perform the following row operations to convert the coefficient matrix to row-echelon form and do back-substitution to find the solution.
1. Interchange rows. (Notation:
)
2. Multiply a row by a constant. (Notation:
)
3. Add the product of a row multiplied by a constant to another row. (Notation:
Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in row-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method that uses row operations to obtain a 1 as the first entry so that row 1 can be used to convert the remaining rows.
### Solving a System of Linear Equations Using Matrices
We have seen how to write a system of equations with an augmented matrix, and then how to use row operations and back-substitution to obtain row-echelon form. Now, we will take row-echelon form a step farther to solve a 3 by 3 system of linear equations. The general idea is to eliminate all but one variable using row operations and then back-substitute to solve for the other variables.
### Key Concepts
1. An augmented matrix is one that contains the coefficients and constants of a system of equations. See .
2. A matrix augmented with the constant column can be represented as the original system of equations. See .
3. Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.
4. We can use Gaussian elimination to solve a system of equations. See , , and .
5. Row operations are performed on matrices to obtain row-echelon form. See .
6. To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions. See and .
7. A calculator can be used to solve systems of equations using matrices. See .
8. Many real-world problems can be solved using augmented matrices. See and .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, write the augmented matrix for the linear system.
For the following exercises, write the linear system from the augmented matrix.
For the following exercises, solve the system by Gaussian elimination.
### Extensions
For the following exercises, use Gaussian elimination to solve the system.
### Real-World Applications
For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. |
# Systems of Equations and Inequalities
## Solving Systems with Inverses
### Learning Objectives
1. Evaluate the determinant of a 2×2 matrix (IA 4.6.1)
2. Evaluate the determinant of a 3x3 matrix (IA 4.6.2)
### Objective 1: Evaluate the determinant of a 2×2 matrix (IA 4.6.1)
If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant.
### Practice Makes Perfect
Find the determinant of the 2x2 matrices.
### Objective 2: Evaluate the determinant of a 3×3 matrix (IA 4.6.2)
To evaluate the determinant of a 3×3 matrix, we must be able to evaluate the minor of an entry in the determinant.
The minor of an entry is the 2×2 determinant found by eliminating the row and column in the 3×3 determinant that contains the entry.
For example, to find the minor of entry a1, we eliminate the row and column which contain it. So, we eliminate the first row and first column. Then we write the 2×2 determinant that remains.
To find the minor of entry b2, we eliminate the row and column that contain it. So, we eliminate the second row and second column. Then we write the 2×2 determinant that remains.
### Strategy for evaluating the determinant of a 3x3 matrix
To evaluate a 3×3 determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier.
When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.
### Expanding by minors along the first row to evaluate a 3x3 determinant.
To evaluate a 3×3 determinant by expanding by minors along the first row, we use the following pattern:
NOTE: We can evaluate the determinant of a matrix by expanding minors along any row or column. When a row or a column has a zero entry, expanding by that row or column results in less calculations.
### Practice Makes Perfect
Soriya plans to invest $10,500 into two different bonds to spread out her risk. The first bond has an annual return of 10%, and the second bond has an annual return of 6%. In order to receive an 8.5% return from the two bonds, how much should Soriya invest in each bond? What is the best method to solve this problem?
There are several ways we can solve this problem. As we have seen in previous sections, systems of equations and matrices are useful in solving real-world problems involving finance. After studying this section, we will have the tools to solve the bond problem using the inverse of a matrix.
### Finding the Inverse of a Matrix
We know that the multiplicative inverse of a real number
is
and
For example,
and
The multiplicative inverse of a matrix is similar in concept, except that the product of matrix
and its inverse
equals the identity matrix. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. We identify identity matrices by
where
represents the dimension of the matrix. Observe the following equations.
The identity matrix acts as a 1 in matrix algebra. For example,
A matrix that has a multiplicative inverse has the properties
A matrix that has a multiplicative inverse is called an invertible matrix. Only a square matrix may have a multiplicative inverse, as the reversibility,
is a requirement. Not all square matrices have an inverse, but if
is invertible, then
is unique. We will look at two methods for finding the inverse of a
matrix and a third method that can be used on both
and
matrices.
### Finding the Multiplicative Inverse Using Matrix Multiplication
We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication.
### Finding the Multiplicative Inverse by Augmenting with the Identity
Another way to find the multiplicative inverse is by augmenting with the identity. When matrix
is transformed into
the augmented matrix
transforms into
For example, given
augment
with the identity
Perform row operations with the goal of turning
into the identity.
1. Switch row 1 and row 2.
2. Multiply row 2 by
and add to row 1.
3. Multiply row 1 by
and add to row 2.
4. Add row 2 to row 1.
5. Multiply row 2 by
The matrix we have found is
### Finding the Multiplicative Inverse of 2×2 Matrices Using a Formula
When we need to find the multiplicative inverse of a
matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.
If
is a
matrix, such as
the multiplicative inverse of
is given by the formula
where
If
then
has no inverse.
### Finding the Multiplicative Inverse of 3×3 Matrices
Unfortunately, we do not have a formula similar to the one for a
matrix to find the inverse of a
matrix. Instead, we will augment the original matrix with the identity matrix and use row operations to obtain the inverse.
Given a
matrix
augment
with the identity matrix
To begin, we write the augmented matrix with the identity on the right and
on the left. Performing elementary row operations so that the identity matrix appears on the left, we will obtain the inverse matrix on the right. We will find the inverse of this matrix in the next example.
### Solving a System of Linear Equations Using the Inverse of a Matrix
Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices:
is the matrix representing the variables of the system, and
is the matrix representing the constants. Using matrix multiplication, we may define a system of equations with the same number of equations as variables as
To solve a system of linear equations using an inverse matrix, let
be the coefficient matrix, let
be the variable matrix, and let
be the constant matrix. Thus, we want to solve a system
For example, look at the following system of equations.
From this system, the coefficient matrix is
The variable matrix is
And the constant matrix is
Then
looks like
Recall the discussion earlier in this section regarding multiplying a real number by its inverse,
To solve a single linear equation
for
we would simply multiply both sides of the equation by the multiplicative inverse (reciprocal) of
Thus,
The only difference between a solving a linear equation and a system of equations written in matrix form is that finding the inverse of a matrix is more complicated, and matrix multiplication is a longer process. However, the goal is the same—to isolate the variable.
We will investigate this idea in detail, but it is helpful to begin with a
system and then move on to a
system.
### Key Equations
### Key Concepts
1. An identity matrix has the property
See .
2. An invertible matrix has the property
See .
3. Use matrix multiplication and the identity to find the inverse of a
matrix. See .
4. The multiplicative inverse can be found using a formula. See .
5. Another method of finding the inverse is by augmenting with the identity. See .
6. We can augment a
matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse. See .
7. Write the system of equations as
and multiply both sides by the inverse of
See and .
8. We can also use a calculator to solve a system of equations with matrix inverses. See .
### Section Exercises
### Verbal
### Algebraic
In the following exercises, show that matrix
is the inverse of matrix
For the following exercises, find the multiplicative inverse of each matrix, if it exists.
For the following exercises, solve the system using the inverse of a
matrix.
For the following exercises, solve a system using the inverse of a
matrix.
### Technology
For the following exercises, use a calculator to solve the system of equations with matrix inverses.
### Extensions
For the following exercises, find the inverse of the given matrix.
### Real-World Applications
For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. |
# Systems of Equations and Inequalities
## Solving Systems with Cramer's Rule
### Learning Objectives
1. Use Cramer’s Rule to solve systems of equations (IA 4.6.3)
### Objective 1: Use Cramer’s Rule to solve systems of equations (IA 4.6.3)
Cramer’s Rule uses determinants to solve systems of equations.
### Practice Makes Perfect
### Practice Makes Perfect
We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.
### Evaluating the Determinant of a 2×2 Matrix
A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an invertible matrix and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.
### Using Cramer’s Rule to Solve a System of Two Equations in Two Variables
We will now introduce a final method for solving systems of equations that uses determinants. Known as Cramer’s Rule, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750 in Introduction à l'Analyse des lignes Courbes algébriques. Cramer’s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.
Cramer’s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.
To understand Cramer’s Rule, let’s look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables.
We eliminate one variable using row operations and solve for the other. Say that we wish to solve for
If equation (2) is multiplied by the opposite of the coefficient of
in equation (1), equation (1) is multiplied by the coefficient of
in equation (2), and we add the two equations, the variable
will be eliminated.
Now, solve for
Similarly, to solve for
we will eliminate
Solving for
gives
Notice that the denominator for both
and
is the determinant of the coefficient matrix.
We can use these formulas to solve for
and
but Cramer’s Rule also introduces new notation:
1.
determinant of the coefficient matrix
2.
determinant of the numerator in the solution of
3.
determinant of the numerator in the solution of
The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express
and
as a quotient of two determinants.
### Evaluating the Determinant of a 3 × 3 Matrix
Finding the determinant of a 2×2 matrix is straightforward, but finding the determinant of a 3×3 matrix is more complicated. One method is to augment the 3×3 matrix with a repetition of the first two columns, giving a 3×5 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.
Find the determinant of the 3×3 matrix.
1. Augment
with the first two columns.
2. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.
3. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.
The algebra is as follows:
### Using Cramer’s Rule to Solve a System of Three Equations in Three Variables
Now that we can find the determinant of a 3 × 3 matrix, we can apply Cramer’s Rule to solve a system of three equations in three variables. Cramer’s Rule is straightforward, following a pattern consistent with Cramer’s Rule for 2 × 2 matrices. As the order of the matrix increases to 3 × 3, however, there are many more calculations required.
When we calculate the determinant to be zero, Cramer’s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.
Consider a 3 × 3 system of equations.
where
If we are writing the determinant
we replace the
column with the constant column. If we are writing the determinant
we replace the
column with the constant column. If we are writing the determinant
we replace the
column with the constant column. Always check the answer.
### Understanding Properties of Determinants
There are many properties of determinants. Listed here are some properties that may be helpful in calculating the determinant of a matrix.
### Key Concepts
1. The determinant for
is
See .
2. Cramer’s Rule replaces a variable column with the constant column. Solutions are
See .
3. To find the determinant of a 3×3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). See .
4. To solve a system of three equations in three variables using Cramer’s Rule, replace a variable column with the constant column for each desired solution:
See .
5. Cramer’s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. See and .
6. Certain properties of determinants are useful for solving problems. For example:
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the determinant.
For the following exercises, solve the system of linear equations using Cramer’s Rule.
For the following exercises, solve the system of linear equations using Cramer’s Rule.
### Technology
For the following exercises, use the determinant function on a graphing utility.
### Real-World Applications
For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.
For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.
For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown in .
### Review Exercises
### Systems of Linear Equations: Two Variables
For the following exercises, determine whether the ordered pair is a solution to the system of equations.
For the following exercises, use substitution to solve the system of equations.
For the following exercises, use addition to solve the system of equations.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.
### Systems of Linear Equations: Three Variables
For the following exercises, solve the system of three equations using substitution or addition.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.
### Systems of Nonlinear Equations and Inequalities: Two Variables
For the following exercises, solve the system of nonlinear equations.
For the following exercises, graph the inequality.
For the following exercises, graph the system of inequalities.
### Partial Fractions
For the following exercises, decompose into partial fractions.
### Matrices and Matrix Operations
For the following exercises, perform the requested operations on the given matrices.
### Solving Systems with Gaussian Elimination
For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.
For the following exercises, write the augmented matrix from the system of linear equations.
For the following exercises, solve the system of linear equations using Gaussian elimination.
### Solving Systems with Inverses
For the following exercises, find the inverse of the matrix.
For the following exercises, find the solutions by computing the inverse of the matrix.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.
### Solving Systems with Cramer's Rule
For the following exercises, find the determinant.
For the following exercises, use Cramer’s Rule to solve the linear systems of equations.
### Practice Test
Is the following ordered pair a solution to the system of equations?
For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.
For the following exercises, graph the following inequalities.
For the following exercises, write the partial fraction decomposition.
For the following exercises, perform the given matrix operations.
For the following exercises, use Gaussian elimination to solve the systems of equations.
For the following exercises, use the inverse of a matrix to solve the systems of equations.
For the following exercises, use Cramer’s Rule to solve the systems of equations.
For the following exercises, solve using a system of linear equations. |
# Analytic Geometry
## Introduction to Analytic Geometry
The Greek mathematician Menaechmus (c. 380–c. 320 BCE) is generally credited with discovering the shapes formed by the intersection of a plane and a right circular cone. Depending on how he tilted the plane when it intersected the cone, he formed different shapes at the intersection–beautiful shapes with near-perfect symmetry.
It was also said that Aristotle may have had an intuitive understanding of these shapes, as he observed the orbit of the planet to be circular. He presumed that the planets moved in circular orbits around Earth, and for nearly 2000 years this was the commonly held belief.
It was not until the Renaissance movement that Johannes Kepler noticed that the orbits of the planet were not circular in nature. His published law of planetary motion in the 1600s changed our view of the solar system forever. He claimed that the sun was at one end of the orbits, and the planets revolved around the sun in an oval-shaped path.
Other objects in the solar system (and perhaps other systems) follow a similar elliptical path, including the spectacular rings of Saturn. Using this understanding as a basis, 19th century mathematicians like James Clerk Maxwell and Sofya Kovalevskaya showed that despite their appearance through the telescopes of the day (and even in current telescopes), the rings are not solid and continuous, but are rather composed of small particles. Even after the Voyager and Cassini missions have provided close-up and detailed data regarding the ring structures, full understanding of their construction relies heavily on mathematical analysis. Of particular interest are the influences of Saturn's moons and moonlets, and the ways they both disrupt and preserve the ring structure.
In this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected by a plane. We will begin by studying each of three figures created in this manner. We will develop defining equations for each figure and then learn how to use these equations to solve a variety of problems. |
# Analytic Geometry
## The Ellipse
### Learning Objectives
1. Complete the square of a binomial expression. (IA 9.2.1)
2. Graph a circle. (IA 11.1.4)
### Objective 1: Complete the square of a binomial expression. (IA 9.2.1)
But what happens if we have to solve an equation where the trinomial is not a perfect square?
For example, ? For these types of equations, we can use a process called completing the square.
Recall .
We can use the Binomial Squares Pattern to make a perfect square.
### Practice Makes Perfect
Determine what number would have to be added to the given terms to create a perfect square trinomial. Then rewrite as a binomial squared.
### Objective 2: Graph a circle. (IA 11.1.4)
A circle is all points in a plane that are a fixed distance from a given point in the plane. The given point is called the center, (h, k) and the fixed distance is called the radius, r, of the circle.
The standard or graphing form of the equation of a circle with center, (h, k) and radius, r, is .
The general form of the equation of a circle is . If we are given an equation in general form, we can change it to standard, also called the graphing form, by completing the squares in both x and y. Then we can graph the circle using its center and radius.
### Practice Makes Perfect
Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in , is such a room.Architect of the Capitol. http://www.aoc.gov. Accessed April 15, 2014. It is an semi-circular room called a whispering chamber because the shape makes it possible for sound to travel along the walls and dome. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.
### Writing Equations of Ellipses in Standard Form
A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in .
Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points
in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See .
Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See .
In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the x- and y-axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.
To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.
### Deriving the Equation of an Ellipse Centered at the Origin
To derive the equation of an ellipse centered at the origin, we begin with the foci
and
The ellipse is the set of all points
such that the sum of the distances from
to the foci is constant, as shown in .
If
is a vertex of the ellipse, the distance from
to
is
The distance from
to
is
. The sum of the distances from the foci to the vertex is
If
is a point on the ellipse, then we can define the following variables:
By the definition of an ellipse,
is constant for any point
on the ellipse. We know that the sum of these distances is
for the vertex
It follows that
for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.
Thus, the standard equation of an ellipse is
This equation defines an ellipse centered at the origin. If
the ellipse is stretched further in the horizontal direction, and if
the ellipse is stretched further in the vertical direction.
### Writing Equations of Ellipses Centered at the Origin in Standard Form
Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we’ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.
The key features of the ellipse are its center, vertices, co-vertices, foci, and lengths and positions of the major and minor axes. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.
### Writing Equations of Ellipses Not Centered at the Origin
Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translated
units horizontally and
units vertically, the center of the ellipse will be
This translation results in the standard form of the equation we saw previously, with
replaced by
and y replaced by
### Graphing Ellipses Centered at the Origin
Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form
for horizontal ellipses and
for vertical ellipses.
### Graphing Ellipses Not Centered at the Origin
When an ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point,
we use the standard forms
for horizontal ellipses and
for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.
### Solving Applied Problems Involving Ellipses
Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See . In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper. When these chambers are placed in unexpected places, such as the ones inside Bush International Airport in Houston and Grand Central Terminal in New York City, they can induce surprised reactions among travelers.
### Key Equations
### Key Concepts
1. An ellipse is the set of all points
in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
2. When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See and .
3. When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See and .
4. When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See and .
5. Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.
For the following exercises, find the foci for the given ellipses.
### Graphical
For the following exercises, graph the given ellipses, noting center, vertices, and foci.
For the following exercises, use the given information about the graph of each ellipse to determine its equation.
For the following exercises, given the graph of the ellipse, determine its equation.
### Extensions
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula
### Real-World Applications
|
# Analytic Geometry
## The Hyperbola
### Learning Objectives
1. Use the Distance Formula. (IA 11.1.1)
2. Graph a hyperbola with center at (0,0). (IA 11.4.1)
### Objective 1: Use the Distance Formula. (IA 11.1.1)
### Practice Makes Perfect
Use the Distance Formula.
### Objective 2: Graph a hyperbola with center at (0,0). (IA 11.4.1)
A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola.
The line through the foci is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola.
Notice that, unlike the equation of an ellipse, the denominator of is not always and the denominator of is not always .
Notice that when the term is positive, the transverse axis is on the x-axis. When the term is positive, the transverse axis is on the y-axis.
### Practice Makes Perfect
Graph a hyperbola with center at (0,0).
What do paths of comets, supersonic booms, ancient Grecian pillars, and natural draft cooling towers have in common? They can all be modeled by the same type of conic. For instance, when something moves faster than the speed of sound, a shock wave in the form of a cone is created. A portion of a conic is formed when the wave intersects the ground, resulting in a sonic boom. See .
Most people are familiar with the sonic boom created by supersonic aircraft, but humans were breaking the sound barrier long before the first supersonic flight. The crack of a whip occurs because the tip is exceeding the speed of sound. The bullets shot from many firearms also break the sound barrier, although the bang of the gun usually supersedes the sound of the sonic boom.
### Locating the Vertices and Foci of a Hyperbola
In analytic geometry, a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other. See .
Like the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all points
in a plane such that the difference of the distances between
and the foci is a positive constant.
Notice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the difference of two distances, whereas the ellipse is defined in terms of the sum of two distances.
As with the ellipse, every hyperbola has two axes of symmetry. The transverse axis is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The conjugate axis is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two asymptotes that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle. See .
In this section, we will limit our discussion to hyperbolas that are positioned vertically or horizontally in the coordinate plane; the axes will either lie on or be parallel to the x- and y-axes. We will consider two cases: those that are centered at the origin, and those that are centered at a point other than the origin.
### Deriving the Equation of a Hyperbola Centered at the Origin
Let
and
be the foci of a hyperbola centered at the origin. The hyperbola is the set of all points
such that the difference of the distances from
to the foci is constant. See .
If
is a vertex of the hyperbola, the distance from
to
is
The distance from
to
is
The difference of the distances from the foci to the vertex is
If
is a point on the hyperbola, we can define the following variables:
By definition of a hyperbola,
is constant for any point
on the hyperbola. We know that the difference of these distances is
for the vertex
It follows that
for any point on the hyperbola. As with the derivation of the equation of an ellipse, we will begin by applying the distance formula. The rest of the derivation is algebraic. Compare this derivation with the one from the previous section for ellipses.
This equation defines a hyperbola centered at the origin with vertices
and co-vertices
### Writing Equations of Hyperbolas in Standard Form
Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin.
### Hyperbolas Centered at the Origin
Reviewing the standard forms given for hyperbolas centered at
we see that the vertices, co-vertices, and foci are related by the equation
Note that this equation can also be rewritten as
This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices.
### Hyperbolas Not Centered at the Origin
Like the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translated
units horizontally and
units vertically, the center of the hyperbola will be
This translation results in the standard form of the equation we saw previously, with
replaced by
and
replaced by
Like hyperbolas centered at the origin, hyperbolas centered at a point
have vertices, co-vertices, and foci that are related by the equation
We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given.
### Graphing Hyperbolas Centered at the Origin
When we have an equation in standard form for a hyperbola centered at the origin, we can interpret its parts to identify the key features of its graph: the center, vertices, co-vertices, asymptotes, foci, and lengths and positions of the transverse and conjugate axes. To graph hyperbolas centered at the origin, we use the standard form
for horizontal hyperbolas and the standard form
for vertical hyperbolas.
### Graphing Hyperbolas Not Centered at the Origin
Graphing hyperbolas centered at a point
other than the origin is similar to graphing ellipses centered at a point other than the origin. We use the standard forms
for horizontal hyperbolas, and
for vertical hyperbolas. From these standard form equations we can easily calculate and plot key features of the graph: the coordinates of its center, vertices, co-vertices, and foci; the equations of its asymptotes; and the positions of the transverse and conjugate axes.
### Solving Applied Problems Involving Hyperbolas
As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy, physics, engineering, and architecture. The design efficiency of hyperbolic cooling towers is particularly interesting. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to generate power efficiently. Because of their hyperbolic form, these structures are able to withstand extreme winds while requiring less material than any other forms of their size and strength. See . For example, a 500-foot tower can be made of a reinforced concrete shell only 6 or 8 inches wide!
The first hyperbolic towers were designed in 1914 and were 35 meters high. Today, the tallest cooling towers are in France, standing a remarkable 170 meters tall. In we will use the design layout of a cooling tower to find a hyperbolic equation that models its sides.
### Key Equations
### Key Concepts
1. A hyperbola is the set of all points
in a plane such that the difference of the distances between
and the foci is a positive constant.
2. The standard form of a hyperbola can be used to locate its vertices and foci. See .
3. When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. See and .
4. When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. See and .
5. Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.
For the following exercises, find the equations of the asymptotes for each hyperbola.
### Graphical
For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.
For the following exercises, given information about the graph of the hyperbola, find its equation.
For the following exercises, given the graph of the hyperbola, find its equation.
### Extensions
For the following exercises, express the equation for the hyperbola as two functions, with
as a function of
Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.
### Real-World Applications
For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.
For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information. |
# Analytic Geometry
## The Parabola
### Learning Objectives
1. Graph vertical parabolas. (IA 11.2.1)
2. Graph horizontal parabolas. (IA 11.2.2)
### Objective 1: Graph vertical parabolas. (IA 11.2.1)
A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here.
### Practice Makes Perfect
Graph vertical parabolas.
### Objective 2: Graph horizontal parabolas. (IA 11.2.2)
Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the x and y in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.
### Practice Makes Perfect
Katherine Johnson is the pioneering NASA mathematician who was integral to the successful and safe flight and return of many human missions as well as satellites. Prior to the work featured in the movie Hidden Figures, she had already made major contributions to the space program. She provided trajectory analysis for the Mercury mission, in which Alan Shepard became the first American to reach space, and she and engineer Ted Sopinski authored a monumental paper regarding placing an object in a precise orbital position and having it return safely to Earth. Many of the orbits she determined were made up of parabolas, and her ability to combine different types of math enabled an unprecedented level of precision. Johnson said, "You tell me when you want it and where you want it to land, and I'll do it backwards and tell you when to take off."
Johnson's work on parabolic orbits and other complex mathematics resulted in successful orbits, Moon landings, and the development of the Space Shuttle program. Applications of parabolas are also critical to other areas of science. Parabolic mirrors (or reflectors) are able to capture energy and focus it to a single point. The advantages of this property are evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes, microphones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little maintenance. In this section we will explore the parabola and its uses, including low-cost, energy-efficient solar designs.
### Graphing Parabolas with Vertices at the Origin
In The Ellipse, we saw that an ellipse is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a parabola. See .
Like the ellipse and hyperbola, the parabola can also be defined by a set of points in the coordinate plane. A parabola is the set of all points
in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
In Quadratic Functions, we learned about a parabola’s vertex and axis of symmetry. Now we extend the discussion to include other key features of the parabola. See . Notice that the axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus.
The line segment that passes through the focus and is parallel to the directrix is called the latus rectum. The endpoints of the latus rectum lie on the curve. By definition, the distance
from the focus to any point
on the parabola is equal to the distance from
to the directrix.
To work with parabolas in the coordinate plane, we consider two cases: those with a vertex at the origin and those with a vertex at a point other than the origin. We begin with the former.
Let
be a point on the parabola with vertex
focus
and directrix
as shown in . The distance
from point
to point
on the directrix is the difference of the y-values:
The distance from the focus
to the point
is also equal to
and can be expressed using the distance formula.
Set the two expressions for
equal to each other and solve for
to derive the equation of the parabola. We do this because the distance from
to
equals the distance from
to
We then square both sides of the equation, expand the squared terms, and simplify by combining like terms.
The equations of parabolas with vertex
are
when the x-axis is the axis of symmetry and
when the y-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.
The key features of a parabola are its vertex, axis of symmetry, focus, directrix, and latus rectum. See . When given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the parabola.
A line is said to be tangent to a curve if it intersects the curve at exactly one point. If we sketch lines tangent to the parabola at the endpoints of the latus rectum, these lines intersect on the axis of symmetry, as shown in .
### Writing Equations of Parabolas in Standard Form
In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.
### Graphing Parabolas with Vertices Not at the Origin
Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated
units horizontally and
units vertically, the vertex will be
This translation results in the standard form of the equation we saw previously with
replaced by
and
replaced by
To graph parabolas with a vertex
other than the origin, we use the standard form
for parabolas that have an axis of symmetry parallel to the x-axis, and
for parabolas that have an axis of symmetry parallel to the y-axis. These standard forms are given below, along with their general graphs and key features.
### Solving Applied Problems Involving Parabolas
As we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the Olympic torch, have a very unique reflecting property. When rays of light parallel to the parabola’s axis of symmetry are directed toward any surface of the mirror, the light is reflected directly to the focus. See . This is why the Olympic torch is ignited when it is held at the focus of the parabolic mirror.
Parabolic mirrors have the ability to focus the sun’s energy to a single point, raising the temperature hundreds of degrees in a matter of seconds. Thus, parabolic mirrors are featured in many low-cost, energy efficient solar products, such as solar cookers, solar heaters, and even travel-sized fire starters.
### Key Equations
### Key Concepts
1. A parabola is the set of all points
in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
2. The standard form of a parabola with vertex
and the x-axis as its axis of symmetry can be used to graph the parabola. If
the parabola opens right. If
the parabola opens left. See .
3. The standard form of a parabola with vertex
and the y-axis as its axis of symmetry can be used to graph the parabola. If
the parabola opens up. If
the parabola opens down. See .
4. When given the focus and directrix of a parabola, we can write its equation in standard form. See .
5. The standard form of a parabola with vertex
and axis of symmetry parallel to the x-axis can be used to graph the parabola. If
the parabola opens right. If
the parabola opens left. See .
6. The standard form of a parabola with vertex
and axis of symmetry parallel to the y-axis can be used to graph the parabola. If
the parabola opens up. If
the parabola opens down. See .
7. Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.
For the following exercises, rewrite the given equation in standard form, and then determine the vertex
focus
and directrix
of the parabola.
### Graphical
For the following exercises, graph the parabola, labeling the focus and the directrix.
For the following exercises, find the equation of the parabola given information about its graph.
For the following exercises, determine the equation for the parabola from its graph.
### Extensions
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.
### Real-World Applications
|
# Analytic Geometry
## Rotation of Axes
### Learning Objectives
1. Using rotation of axes formulas.
2. Identify conic sections by their equations. (IA 11.4.3)
### Objective 1: Using rotation of axes formulas.
If a point
on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle
from the positive
x
-axis, then the coordinates of the point with respect to the new axes are
The following rotations of axes formulas define the relationship between (x,y) and (x’,y’):
### Practice Makes Perfect
Using rotation of axes formulas:
### Objective 2: Identify conic sections by their equations. (IA 11.4.3)
We can identify a conic from its equations by looking at the signs and coefficients of the variables that are squared.
### Practice Makes Perfect
Identify conic sections by their equations.
As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See .
Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in . A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.
### Identifying Nondegenerate Conics in General Form
In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.
where
and
are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.
You may notice that the general form equation has an
term that we have not seen in any of the standard form equations. As we will discuss later, the
term rotates the conic whenever
is not equal to zero.
### Finding a New Representation of the Given Equation after Rotating through a Given Angle
Until now, we have looked at equations of conic sections without an
term, which aligns the graphs with the x- and y-axes. When we add an
term, we are rotating the conic about the origin. If the x- and y-axes are rotated through an angle, say
then every point on the plane may be thought of as having two representations:
on the Cartesian plane with the original x-axis and y-axis, and
on the new plane defined by the new, rotated axes, called the x'-axis and y'-axis. See .
We will find the relationships between
and
on the Cartesian plane with
and
on the new rotated plane. See .
The original coordinate x- and y-axes have unit vectors
and
The rotated coordinate axes have unit vectors
and
The angle
is known as the angle of rotation. See . We may write the new unit vectors in terms of the original ones.
Consider a vector in the new coordinate plane. It may be represented in terms of its coordinate axes.
Because
we have representations of
and
in terms of the new coordinate system.
### Writing Equations of Rotated Conics in Standard Form
Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to transform the equation of a conic given in the form
into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in the
and
coordinate system without the
term, by rotating the axes by a measure of
that satisfies
We have learned already that any conic may be represented by the second degree equation
where
and
are not all zero. However, if
then we have an
term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle
where
### Identifying Conics without Rotating Axes
Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is
If we apply the rotation formulas to this equation we get the form
It may be shown that
The expression does not vary after rotation, so we call the expression invariant. The discriminant,
is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.
### Key Equations
### Key Concepts
1. Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
2. A nondegenerate conic section has the general form
where
and
are not all zero. The values of
and
determine the type of conic. See .
3. Equations of conic sections with an
term have been rotated about the origin. See .
4. The general form can be transformed into an equation in the
and
coordinate system without the
term. See and .
5. An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, determine which conic section is represented based on the given equation.
For the following exercises, find a new representation of the given equation after rotating through the given angle.
For the following exercises, determine the angle
that will eliminate the
term and write the corresponding equation without the
term.
### Graphical
For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.
For the following exercises, graph the equation relative to the
system in which the equation has no
term.
For the following exercises, determine the angle of rotation in order to eliminate the
term. Then graph the new set of axes.
For the following exercises, determine the value of
based on the given equation. |
# Analytic Geometry
## Conic Sections in Polar Coordinates
Most of us are familiar with orbital motion, such as the motion of a planet around the sun or an electron around an atomic nucleus. Within the planetary system, orbits of planets, asteroids, and comets around a larger celestial body are often elliptical. Comets, however, may take on a parabolic or hyperbolic orbit instead. And, in reality, the characteristics of the planets’ orbits may vary over time. Each orbit is tied to the location of the celestial body being orbited and the distance and direction of the planet or other object from that body. As a result, we tend to use polar coordinates to represent these orbits.
In an elliptical orbit, the periapsis is the point at which the two objects are closest, and the apoapsis is the point at which they are farthest apart. Generally, the velocity of the orbiting body tends to increase as it approaches the periapsis and decrease as it approaches the apoapsis. Some objects reach an escape velocity, which results in an infinite orbit. These bodies exhibit either a parabolic or a hyperbolic orbit about a body; the orbiting body breaks free of the celestial body’s gravitational pull and fires off into space. Each of these orbits can be modeled by a conic section in the polar coordinate system.
### Identifying a Conic in Polar Form
Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph. Consider the parabola
shown in .
In The Parabola, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line). In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus
at the pole, and a line, the directrix, which is perpendicular to the polar axis.
If
is a fixed point, the focus, and
is a fixed line, the directrix, then we can let
be a fixed positive number, called the eccentricity, which we can define as the ratio of the distances from a point on the graph to the focus and the point on the graph to the directrix. Then the set of all points
such that
is a conic. In other words, we can define a conic as the set of all points
with the property that the ratio of the distance from
to
to the distance from
to
is equal to the constant
For a conic with eccentricity
1. if
the conic is an ellipse
2. if
the conic is a parabola
3. if
the conic is an hyperbola
With this definition, we may now define a conic in terms of the directrix,
the eccentricity
and the angle
Thus, each conic may be written as a polar equation, an equation written in terms of
and
### Graphing the Polar Equations of Conics
When graphing in Cartesian coordinates, each conic section has a unique equation. This is not the case when graphing in polar coordinates. We must use the eccentricity of a conic section to determine which type of curve to graph, and then determine its specific characteristics. The first step is to rewrite the conic in standard form as we have done in the previous example. In other words, we need to rewrite the equation so that the denominator begins with 1. This enables us to determine
and, therefore, the shape of the curve. The next step is to substitute values for
and solve for
to plot a few key points. Setting
equal to
and
provides the vertices so we can create a rough sketch of the graph.
### Defining Conics in Terms of a Focus and a Directrix
So far we have been using polar equations of conics to describe and graph the curve. Now we will work in reverse; we will use information about the origin, eccentricity, and directrix to determine the polar equation.
### Key Concepts
1. Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus
at the pole, and a line, the directrix, which is perpendicular to the polar axis.
2. A conic is the set of all points
where eccentricity
is a positive real number. Each conic may be written in terms of its polar equation. See .
3. The polar equations of conics can be graphed. See , , and .
4. Conics can be defined in terms of a focus, a directrix, and eccentricity. See and .
5. We can use the identities
and
to convert the equation for a conic from polar to rectangular form. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.
For the following exercises, convert the polar equation of a conic section to a rectangular equation.
For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.
### Extensions
Recall from Rotation of Axes that equations of conics with an
term have rotated graphs. For the following exercises, express each equation in polar form with
as a function of
### Chapter Review Exercises
### The Ellipse
For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.
For the following exercises, graph the ellipse, noting center, vertices, and foci.
For the following exercises, use the given information to find the equation for the ellipse.
### The Hyperbola
For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.
For the following exercises, graph the hyperbola, labeling vertices and foci.
For the following exercises, find the equation of the hyperbola.
### The Parabola
For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.
For the following exercises, graph the parabola, labeling vertex, focus, and directrix.
For the following exercises, write the equation of the parabola using the given information.
### Rotation of Axes
For the following exercises, determine which of the conic sections is represented.
For the following exercises, determine the angle
that will eliminate the
term, and write the corresponding equation without the
term.
For the following exercises, graph the equation relative to the
system in which the equation has no
term.
### Conic Sections in Polar Coordinates
For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.
For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.
For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.
### Practice Test
For the following exercises, write the equation in standard form and state the center, vertices, and foci.
For the following exercises, sketch the graph, identifying the center, vertices, and foci.
For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.
For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.
For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.
For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.
For the following exercises, determine which conic section is represented by the given equation, and then determine the angle
that will eliminate the
term.
For the following exercises, rewrite in the
system without the
term, and graph the rotated graph.
For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.
For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci. |
# Sequences, Probability and Counting Theory
## Introduction to Sequences, Probability and Counting Theory
A lottery winner has some big decisions to make regarding what to do with the winnings. Buy a new home? A luxury convertible? A cruise around the world?
The likelihood of winning the lottery is slim, but we all love to fantasize about what we could buy with the winnings. One of the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series of regular payments, called an annuity, over an extended period of time.
This decision is often based on many factors, such as tax implications, interest rates, and investment strategies. There are also personal reasons to consider when making the choice, and one can make many arguments for either decision. However, most lottery winners opt for the lump sum.
In this chapter, we will explore the mathematics behind situations such as these. We will take an in-depth look at annuities. We will also look at the branch of mathematics that would allow us to calculate the number of ways to choose lottery numbers and the probability of winning. |
# Sequences, Probability and Counting Theory
## Sequences and Their Notations
### Learning Objectives
1. Write the first few terms of a sequence (IA 12.1.1)
2. Find a formula for the general term (nth term) of a sequence (IA 12.1.2)
### Objective 1: Write the first few terms of a sequence (IA 12.1.1).
A patient takes a 30 mg antibiotic capsule. At the end of that hour, the amount of antibiotic remaining in her body is only 90% of the amount in the beginning of that hour. The 30mg dose is taken at time t = 1 hour. How much of this dose remains at the end of 1 hour? 2hours? 3 hours? 4 hours?
This ordered list of numbers 27, 24.3, 21.87, 19.68, … is a sequence. Each number in the list is a term.
A sequence is a function whose domain is the counting numbers. A sequence may have an infinite number of terms or a finite number of terms. Our sequence has three dots (ellipsis) at the end which indicates the list never ends. If the domain is the set of all counting numbers, then the sequence is an infinite sequence.
Often when working with sequences we do not want to write out all the terms. We want a more compact way to show how each term is defined. When we worked with functions, we wrote and we said the expression 2x was the rule that defined values in the range.
While a sequence is a function, we do not use the usual function notation. Instead of writing the function as , we would write it as . The is the , the term in the nth position where n is a value in the domain. The formula for writing the nth term of the sequence is called the general term or formula of the sequence.
General sequence terms are denoted as follows:
### Practice Makes Perfect
Write the first few terms of a sequence.
### Objective 2: Find a formula for the general term (nth term) of a sequence (IA 12.1.2)
Sometimes we have a few terms of a sequence and it would be helpful to know the general term or . To find the general term, we look for patterns in the terms. Often the patterns involve multiples or powers. We also look for a pattern in the signs of the terms.
### Practice Makes Perfect
A video game company launches an exciting new advertising campaign. They predict the number of online visits to their website, or hits, will double each day. The model they are using shows 2 hits the first day, 4 hits the second day, 8 hits the third day, and so on. See .
If their model continues, how many hits will there be at the end of the month? To answer this question, we’ll first need to know how to determine a list of numbers written in a specific order. In this section, we will explore these kinds of ordered lists.
### Writing the Terms of a Sequence Defined by an Explicit Formula
One way to describe an ordered list of numbers is as a sequence. A sequence is a function whose domain is a subset of the counting numbers. The sequence established by the number of hits on the website is
The ellipsis (…) indicates that the sequence continues indefinitely. Each number in the sequence is called a term. The first five terms of this sequence are 2, 4, 8, 16, and 32.
Listing all of the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at the end of the month would require listing out as many as 31 terms. A more efficient way to determine a specific term is by writing a formula to define the sequence.
One type of formula is an explicit formula, which defines the terms of a sequence using their position in the sequence. Explicit formulas are helpful if we want to find a specific term of a sequence without finding all of the previous terms. We can use the formula to find the nth term of the sequence, where
is any positive number. In our example, each number in the sequence is double the previous number, so we can use powers of 2 to write a formula for the
term.
The first term of the sequence is
the second term is
the third term is
and so on. The
term of the sequence can be found by raising 2 to the
power. An explicit formula for a sequence is named by a lower case letter
with the subscript
The explicit formula for this sequence is
Now that we have a formula for the
term of the sequence, we can answer the question posed at the beginning of this section. We were asked to find the number of hits at the end of the month, which we will take to be 31 days. To find the number of hits on the last day of the month, we need to find the 31st term of the sequence. We will substitute 31 for
in the formula.
If the doubling trend continues, the company will get
hits on the last day of the month. That is over 2.1 billion hits! The huge number is probably a little unrealistic because it does not take consumer interest and competition into account. It does, however, give the company a starting point from which to consider business decisions.
Another way to represent the sequence is by using a table. The first five terms of the sequence and the
term of the sequence are shown in .
Graphing provides a visual representation of the sequence as a set of distinct points. We can see from the graph in that the number of hits is rising at an exponential rate. This particular sequence forms an exponential function.
Lastly, we can write this particular sequence as
A sequence that continues indefinitely is called an infinite sequence. The domain of an infinite sequence is the set of counting numbers. If we consider only the first 10 terms of the sequence, we could write
This sequence is called a finite sequence because it does not continue indefinitely.
### Investigating Alternating Sequences
Sometimes sequences have terms that are alternate. In fact, the terms may actually alternate in sign. The steps to finding terms of the sequence are the same as if the signs did not alternate. However, the resulting terms will not show increase or decrease as
increases. Let’s take a look at the following sequence.
Notice the first term is greater than the second term, the second term is less than the third term, and the third term is greater than the fourth term. This trend continues forever. Do not rearrange the terms in numerical order to interpret the sequence.
### Investigating Piecewise Explicit Formulas
We’ve learned that sequences are functions whose domain is over the positive integers. This is true for other types of functions, including some piecewise functions. Recall that a piecewise function is a function defined by multiple subsections. A different formula might represent each individual subsection.
### Finding an Explicit Formula
Thus far, we have been given the explicit formula and asked to find a number of terms of the sequence. Sometimes, the explicit formula for the
term of a sequence is not given. Instead, we are given several terms from the sequence. When this happens, we can work in reverse to find an explicit formula from the first few terms of a sequence. The key to finding an explicit formula is to look for a pattern in the terms. Keep in mind that the pattern may involve alternating terms, formulas for numerators, formulas for denominators, exponents, or bases.
### Writing the Terms of a Sequence Defined by a Recursive Formula
Sequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,…. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.
Each term of the Fibonacci sequence depends on the terms that come before it. The Fibonacci sequence cannot easily be written using an explicit formula. Instead, we describe the sequence using a recursive formula, a formula that defines the terms of a sequence using previous terms.
A recursive formula always has two parts: the value of an initial term (or terms), and an equation defining
in terms of preceding terms. For example, suppose we know the following:
We can find the subsequent terms of the sequence using the first term.
So the first four terms of the sequence are
.
The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms.
To find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We were told previously that the eighth and ninth terms are 21 and 34, so
### Using Factorial Notation
The formulas for some sequences include products of consecutive positive integers. , written as
is the product of the positive integers from 1 to
For example,
An example of formula containing a factorial is
The sixth term of the sequence can be found by substituting 6 for
The factorial of any whole number
is
We can therefore also think of
as
### Key Equations
### Key Concepts
1. A sequence is a list of numbers, called terms, written in a specific order.
2. Explicit formulas define each term of a sequence using the position of the term. See , , and .
3. An explicit formula for the
term of a sequence can be written by analyzing the pattern of several terms. See .
4. Recursive formulas define each term of a sequence using previous terms.
5. Recursive formulas must state the initial term, or terms, of a sequence.
6. A set of terms can be written by using a recursive formula. See and .
7. A factorial is a mathematical operation that can be defined recursively.
8. The factorial of
is the product of all integers from 1 to
See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, write the first four terms of the sequence.
For the following exercises, write the first eight terms of the piecewise sequence.
For the following exercises, write an explicit formula for each sequence.
For the following exercises, write the first five terms of the sequence.
For the following exercises, write the first eight terms of the sequence.
For the following exercises, write a recursive formula for each sequence.
For the following exercises, evaluate the factorial.
For the following exercises, write the first four terms of the sequence.
### Graphical
For the following exercises, graph the first five terms of the indicated sequence
For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.
For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.
### Technology
Follow these steps to evaluate a sequence defined recursively using a graphing calculator:
For the following exercises, use the steps above to find the indicated term or terms for the sequence.
Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.
Using a TI-83, do the following.
For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.
### Extensions
|
# Sequences, Probability and Counting Theory
## Arithmetic Sequences
### Learning Objectives
1. Determine if a sequence is arithmetic (IA 12.2.1)
2. Find the general term (nth term) of an arithmetic sequence (IA 12.2.2)
### Objective 1: Determine if a sequence is arithmetic (IA 12.2.1)
An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.
The difference between consecutive terms, d, and is called the common difference, for n greater than or equal to two.
### Practice Makes Perfect
Determine if each sequence is arithmetic. If so, indicate the common difference.
### Practice Makes Perfect
### Objective 2: Find the general term (nth term) of an arithmetic sequence (IA 12.2.2)
In the last section, we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.
Let’s write the first few terms of a sequence where the first term is and the common difference is d. We will then look for a pattern.
As we look for a pattern we see that each term starts with .
The first term adds 0d to the , the second term adds 1d, the third term adds 2d, the fourth term adds 3d, and the fifth term adds 4d. The number of ds that were added to is one less than the number of the term. We then have the formula for the general term of an arithmetic sequence.
### Practice Makes Perfect
Find the general term (nth term) of an arithmetic sequence.
Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.
As an example, consider a woman who starts a small contracting business. She purchases a new truck for $25,000. After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years. The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years. In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck’s value.
### Finding Common Differences
The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is –3,400.
The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any term of the sequence, and add 3 to find the subsequent term.
### Writing Terms of Arithmetic Sequences
Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of
and
into formula below.
### Using Recursive Formulas for Arithmetic Sequences
Some arithmetic sequences are defined in terms of the previous term using a recursive formula. The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.
### Using Explicit Formulas for Arithmetic Sequences
We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.
To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.
The common difference is
, so the sequence represents a linear function with a slope of
. To find the
-intercept, we subtract
from
. You can also find the
-intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in .
Recall the slope-intercept form of a line is
When dealing with sequences, we use
in place of
and
in place of
If we know the slope and vertical intercept of the function, we can substitute them for
and
in the slope-intercept form of a line. Substituting
for the slope and
for the vertical intercept, we get the following equation:
We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is
, which simplifies to
### Finding the Number of Terms in a Finite Arithmetic Sequence
Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.
### Solving Application Problems with Arithmetic Sequences
In many application problems, it often makes sense to use an initial term of
instead of
In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:
### Key Equations
### Key Concepts
1. An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
2. The constant between two consecutive terms is called the common difference.
3. The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See .
4. The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See and .
5. A recursive formula for an arithmetic sequence with common difference
is given by
See .
6. As with any recursive formula, the initial term of the sequence must be given.
7. An explicit formula for an arithmetic sequence with common difference
is given by
See .
8. An explicit formula can be used to find the number of terms in a sequence. See .
9. In application problems, we sometimes alter the explicit formula slightly to
See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the common difference for the arithmetic sequence provided.
For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.
For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.
For the following exercises, write the first five terms of the arithmetic series given two terms.
For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.
For the following exercises, find the first term given two terms from an arithmetic sequence.
For the following exercises, find the specified term given two terms from an arithmetic sequence.
For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence.
For the following exercises, write a recursive formula for each arithmetic sequence.
For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.
For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.
For the following exercises, write an explicit formula for each arithmetic sequence.
For the following exercises, find the number of terms in the given finite arithmetic sequence.
### Graphical
For the following exercises, determine whether the graph shown represents an arithmetic sequence.
For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence.
### Technology
For the following exercises, follow the steps to work with the arithmetic sequence
using a graphing calculator:
For the following exercises, follow the steps given above to work with the arithmetic sequence
using a graphing calculator.
### Extensions
|
# Sequences, Probability and Counting Theory
## Geometric Sequences
### Learning Objectives
1. Determine if a sequence is geometric (IA 12.3.1).
2. Find the general term (nth term) of a geometric sequence (IA 12.3.2).
### Objective 1: Determine if a sequence is geometric (IA 12.3.1)
A sequence is called a geometric sequence if the ratio between consecutive terms is always the same.
The ratio between consecutive terms in a geometric sequence is r, the common ratio, where n is greater than or equal to two.
### Practice Makes Perfect
Determine if each sequence is geometric. If so, indicate the common ratio.
### Practice Makes Perfect
### Objective 2: Find the general term (nth term) of a geometric sequence (IA 12.3.2)
Let’s find the formula for the general term of a geometric sequence.
Let’s write the first few terms of the sequence where the first term is and the common ratio is . We will then look for a pattern.
### Practice Makes Perfect
Find the general term (nth term) of a geometric sequence.
Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.
### Finding Common Ratios
The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.
### Writing Terms of Geometric Sequences
Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is
and the common ratio is
we can find subsequent terms by multiplying
to get
then multiplying the result
to get
and so on.
The first four terms are
### Using Recursive Formulas for Geometric Sequences
A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.
### Using Explicit Formulas for Geometric Sequences
Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.
Let’s take a look at the sequence
This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is
The graph of the sequence is shown in .
### Solving Application Problems with Geometric Sequences
In real-world scenarios involving geometric sequences, we may need to use an initial term of
instead of
In these problems, we can alter the explicit formula slightly by using the following formula:
### Key Equations
### Key Concepts
1. A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
2. The constant ratio between two consecutive terms is called the common ratio.
3. The common ratio can be found by dividing any term in the sequence by the previous term. See .
4. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See and .
5. A recursive formula for a geometric sequence with common ratio
is given by
for
.
6. As with any recursive formula, the initial term of the sequence must be given. See .
7. An explicit formula for a geometric sequence with common ratio
is given by
See .
8. In application problems, we sometimes alter the explicit formula slightly to
See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, find the common ratio for the geometric sequence.
For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.
For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.
For the following exercises, write the first five terms of the geometric sequence, given any two terms.
For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.
For the following exercises, find the specified term for the geometric sequence, given the first four terms.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write a recursive formula for each geometric sequence.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write an explicit formula for each geometric sequence.
For the following exercises, find the specified term for the geometric sequence given.
For the following exercises, find the number of terms in the given finite geometric sequence.
### Graphical
For the following exercises, determine whether the graph shown represents a geometric sequence.
For the following exercises, use the information provided to graph the first five terms of the geometric sequence.
### Extensions
|
# Sequences, Probability and Counting Theory
## Series and Their Notations
### Learning Objectives
1. Use summation notation to write a sum. (IA 12.1.5)
2. Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)
### Objective 1: Use summation notation to write a sum. (IA 12.1.5)
A series is the sum of the terms of a sequence. For example, 1 + 6 + 11+ 16 + 21 + 26 + 31 is the sum of the first seven terms arithmetic sequence with general term,
We write a series by using the summation notation. In order to write that summation, we will need to find the general term of our sequence and the summation will look like:
For the series, 1 + 6 + 11 + 16 + 21 + 26 + 31 + .... the summation notation is
### Practice Makes Perfect
Use summation notation to write the sum.
### Objective 2: Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)
### Practice Makes Perfect
A parent decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.
### Using Summation Notation
To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.
The of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation
represents the partial sum.
Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma,
to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series.
If we interpret the given notation, we see that it asks us to find the sum of the terms in the series
for
through
We can begin by substituting the terms for
and listing out the terms of this series.
We can find the sum of the series by adding the terms:
### Using the Formula for Arithmetic Series
Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference,
The sum of the terms of an arithmetic sequence is called an arithmetic series. We can write the sum of the first
terms of an arithmetic series as:
We can also reverse the order of the terms and write the sum as
If we add these two expressions for the sum of the first
terms of an arithmetic series, we can derive a formula for the sum of the first
terms of any arithmetic series.
Because there are
terms in the series, we can simplify this sum to
We divide by 2 to find the formula for the sum of the first
terms of an arithmetic series.
Use the formula to find the sum of each arithmetic series.
### Using the Formula for Geometric Series
Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio,
We can write the sum of the first
terms of a geometric series as
Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first
terms of a geometric series. We will begin by multiplying both sides of the equation by
Next, we subtract this equation from the original equation.
Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for
divide both sides by
Use the formula to find the indicated partial sum of each geometric series.
### Using the Formula for the Sum of an Infinite Geometric Series
Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first
terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is
This series can also be written in summation notation as
where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.
### Determining Whether the Sum of an Infinite Geometric Series is Defined
If the terms of an infinite geometric sequence approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:
The common ratio
As
gets very large, the values of
get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with
approach 0; the sum of a geometric series is defined when
Determine whether the sum of the infinite series is defined.
### Finding Sums of Infinite Series
When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first
terms of a geometric series.
We will examine an infinite series with
What happens to
as
increases?
The value of
decreases rapidly. What happens for greater values of
As
gets very large,
gets very small. We say that, as
increases without bound,
approaches 0. As
approaches 0,
approaches 1. When this happens, the numerator approaches
This give us a formula for the sum of an infinite geometric series.
Find the sum, if it exists.
### Solving Annuity Problems
At the beginning of the section, we looked at a problem in which a parent invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the parent invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.
We can find the value of the annuity right after the last deposit by using a geometric series with
and
After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.
We can find the value of the annuity after
deposits using the formula for the sum of the first
terms of a geometric series. In 6 years, there are 72 months, so
We can substitute
into the formula, and simplify to find the value of the annuity after 6 years.
After the last deposit, the parent will have a total of $4,320.44 in the account. Notice, the parent made 72 payments of $50 each for a total of
This means that because of the annuity, the parent earned $720.44 interest in their college fund.
### Key Equations
### Key Concepts
1. The sum of the terms in a sequence is called a series.
2. A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See .
3. The sum of the terms in an arithmetic sequence is called an arithmetic series.
4. The sum of the first
terms of an arithmetic series can be found using a formula. See and .
5. The sum of the terms in a geometric sequence is called a geometric series.
6. The sum of the first
terms of a geometric series can be found using a formula. See and .
7. The sum of an infinite series exists if the series is geometric with
8. If the sum of an infinite series exists, it can be found using a formula. See , , and .
9. An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, express each description of a sum using summation notation.
For the following exercises, express each arithmetic sum using summation notation.
For the following exercises, use the formula for the sum of the first
terms of each arithmetic sequence.
For the following exercises, express each geometric sum using summation notation.
For the following exercises, use the formula for the sum of the first
terms of each geometric sequence, and then state the indicated sum.
For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.
### Graphical
For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.
For the following exercises, use the geometric series
### Numeric
For the following exercises, find the indicated sum.
For the following exercises, use the formula for the sum of the first terms of an arithmetic series to find the sum.
For the following exercises, use the formula for the sum of the first
terms of a geometric series to find the partial sum.
For the following exercises, find the sum of the infinite geometric series.
For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.
### Extensions
### Real-World Applications
|
# Sequences, Probability and Counting Theory
## Counting Principles
### Learning Objectives
1. Solve counting problems using the addition principle.
2. Solve counting problems using the multiplication principle.
### Objective 1: Solve counting problems using the addition principle.
In probability theory, an outcome is a possible result of an experiment or trial.
In probability theory, an event is a set of outcomes of an experiment.
Disjoint events cannot happen at the same time. In other words, they are mutually exclusive.
The addition principle applies when we are making only one selection.
### Practice Makes Perfect
Solve counting problems using the addition principle.
### Objective 2: Solve counting problems using the multiplication principle.
The Multiplication Principle applies when we are making more than one selection.
### Practice Makes Perfect
Solve counting problems using the multiplication principle.
A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!
We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.
### Using the Addition Principle
The company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. The Addition Principle tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. By the Addition Principle, there are 8 total options, as we can see in .
### Using the Multiplication Principle
The Multiplication Principle applies when we are making more than one selection. Suppose we are choosing an appetizer, an entrée, and a dessert. If there are 2 appetizer options, 3 entrée options, and 2 dessert options on a fixed-price dinner menu, there are a total of 12 possible choices of one each as shown in the tree diagram in .
The possible choices are:
1. soup, chicken, cake
2. soup, chicken, pudding
3. soup, fish, cake
4. soup, fish, pudding
5. soup, steak, cake
6. soup, steak, pudding
7. salad, chicken, cake
8. salad, chicken, pudding
9. salad, fish, cake
10. salad, fish, pudding
11. salad, steak, cake
12. salad, steak, pudding
We can also find the total number of possible dinners by multiplying.
We could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.
### Finding the Number of Permutations of n Distinct Objects
The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation.
### Finding the Number of Permutations of n Distinct Objects Using the Multiplication Principle
To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.
There are four options for the first place, so we write a 4 on the first line.
After the first place has been filled, there are three options for the second place so we write a 3 on the second line.
After the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally, we find the product.
There are 24 possible permutations of the paintings.
A family of five is having portraits taken. Use the Multiplication Principle to find the following.
### Finding the Number of Permutations of n Distinct Objects Using a Formula
For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of
objects and we want to choose
objects from the set in order, we write
Another way to write this is
a notation commonly seen on computers and calculators. To calculate
we begin by finding
the number of ways to line up all
objects. We then divide by
to cancel out the
items that we do not wish to line up.
Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is
Using factorials, we get the same result.
There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.
Note that the formula stills works if we are choosing all
objects and placing them in order. In that case we would be dividing by
or
which we said earlier is equal to 1. So the number of permutations of
objects taken
at a time is
or just
A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.
### Find the Number of Combinations Using the Formula
So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with combinations. A selection of
objects from a set of
objects where the order does not matter can be written as
Just as with permutations,
can also be written as
In this case, the general formula is as follows.
An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are
ways to order 3 paintings. There are
or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are not selecting 1 painting. There are 4 paintings we could choose not to select, so there are 4 ways to select 3 of the 4 paintings.
### Finding the Number of Subsets of a Set
We have looked only at combination problems in which we chose exactly
objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can be ordered. How many different pizzas are possible?
To answer this question, we need to consider pizzas with any number of toppings. There is
way to order a pizza with no toppings. There are
ways to order a pizza with exactly one topping. If we continue this process, we get
There are 32 possible pizzas. This result is equal to
We are presented with a sequence of choices. For each of the
objects we have two choices: include it in the subset or not. So for the whole subset we have made
choices, each with two options. So there are a total of
possible resulting subsets, all the way from the empty subset, which we obtain when we say “no” each time, to the original set itself, which we obtain when we say “yes” each time.
### Finding the Number of Permutations of n Non-Distinct Objects
We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the
permutations we counted are duplicates. The general formula for this situation is as follows.
In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are
ways to order the stars and
ways to order the moon.
There are 3,326,400 ways to order the sheet of stickers.
### Key Equations
### Key Concepts
1. If one event can occur in
ways and a second event with no common outcomes can occur in
ways, then the first or second event can occur in
ways. See .
2. If one event can occur in
ways and a second event can occur in
ways after the first event has occurred, then the two events can occur in
ways. See .
3. A permutation is an ordering of
objects.
4. If we have a set of
objects and we want to choose
objects from the set in order, we write
5. Permutation problems can be solved using the Multiplication Principle or the formula for
See and .
6. A selection of objects where the order does not matter is a combination.
7. Given
distinct objects, the number of ways to select
objects from the set is
and can be found using a formula. See .
8. A set containing
distinct objects has
subsets. See .
9. For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See .
### Section Exercises
### Verbal
For the following exercises, assume that there are
ways an event
can happen,
ways an event
can happen, and that
are non-overlapping.
Answer the following questions.
### Numeric
For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.
For the following exercises, compute the value of the expression.
For the following exercises, find the number of subsets in each given set.
For the following exercises, find the distinct number of arrangements.
### Extensions
### Real-World Applications
|
# Sequences, Probability and Counting Theory
## Binomial Theorem
### Learning Objectives
1. Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)
### Objective 1: Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)
Pascal’s triangle helps us find the coefficients of the terms in the expansion of a binomial.
To find the coefficients of the terms, we write our expansion again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients. The array to the right is called Pascal’s Triangle.
Notice that in each expansion the powers of a in each term decrease from n to 0, and the powers of b increase from 0 to n.
Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.
To find the coefficients of the expansion of the binomial , go to the row that has the value n as a second entry.
### Practice Makes Perfect
Use Pascal’s Triangle to expand a binomial.
A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find
without multiplying the binomial by itself
times.
### Identifying Binomial Coefficients
In Counting Principles, we studied combinations. In the shortcut to finding
we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation
instead of
but it can be calculated in the same way. So
The combination
is called a binomial coefficient. An example of a binomial coefficient is
### Using the Binomial Theorem
When we expand
by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand
we might multiply
by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.
First, let’s examine the exponents. With each successive term, the exponent for
decreases and the exponent for
increases. The sum of the two exponents is
for each term.
Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:
These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.
Another way to see the coefficients is to examine the expansion of a binomial in general form,
to successive powers 1, 2, 3, and 4.
Can you guess the next expansion for the binomial
See , which illustrates the following:
1. There are
terms in the expansion of
2. The degree (or sum of the exponents) for each term is
3. The powers on
begin with
and decrease to 0.
4. The powers on
begin with 0 and increase to
5. The coefficients are symmetric.
To determine the expansion on
we see
thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of
the pattern is as follows:
1. Introduce
and then for each successive term reduce the exponent on
by 1 until
is reached.
2. Introduce
and then increase the exponent on
by 1 until
is reached.
The next expansion would be
But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as Pascal's Triangle, shown in . Pascal didn't invent the triangle. The underlying principles had been developed and written about for over 1500 years, first by the Indian mathematician (and poet) Pingala in the second century BCE. Others throughout Asia and Europe worked with the concepts throughout, and the triangle was first published in its graphical form by Omar Khayyam, an Iranian mathematician and astronomer, for whom the triangle is named in Iran. French mathematician Blaise Pascal repopularized it when he republished it and used it to solve a number of probability problems.
To generate Pascal’s Triangle, we start by writing a 1. In the row below, row 2, we write two 1’s. In the 3rd row, flank the ends of the rows with 1’s, and add
to find the middle number, 2. In the
row, flank the ends of the row with 1’s. Each element in the triangle is the sum of the two elements immediately above it.
To see the connection between Pascal’s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.
### Using the Binomial Theorem to Find a Single Term
Expanding a binomial with a high exponent such as
can be a lengthy process.
Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.
Note the pattern of coefficients in the expansion of
The second term is
The third term is
We can generalize this result.
### Key Equations
### Key Concepts
1.
is called a binomial coefficient and is equal to
See .
2. The Binomial Theorem allows us to expand binomials without multiplying. See .
3. We can find a given term of a binomial expansion without fully expanding the binomial. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, evaluate the binomial coefficient.
For the following exercises, use the Binomial Theorem to expand each binomial.
For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.
For the following exercises, find the indicated term of each binomial without fully expanding the binomial.
### Graphical
For the following exercises, use the Binomial Theorem to expand the binomial
Then find and graph each indicated sum on one set of axes.
### Extensions
|
# Sequences, Probability and Counting Theory
## Probability
### Learning Objectives
1. Introduction to Sample Spaces and Computing Basic Probabilities.
### Objective 1: Introduction to Sample Spaces and Computing Basic Probabilities.
Many events in life are inherently uncertain: will it snow tomorrow? Am I going to get an ‘A’ in this course? None of these questions can be answered with certainty, however, we might say that some are unlikely, and others are more likely.
The probability of an event is a description of how likely it is that an event will happen. A probability is a number between 0 and 1 (that is, between 0% and 100%), where probabilities closer to 100% are very likely to occur, and probabilities closer to 0% are very unlikely to occur. A probability of 0% means the event is impossible, and a probability of 100% means the event will certainly occur.
A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. It is defined by its sample space, events within the sample space, and probabilities associated with each event.
The sample space S for a probability model is the set of all possible outcomes. For example, the sample space for rolling a dice is the set 1,2,3,4,5,6.This notation is referred to as roster notation.
An event A is a subset of the sample space S. For example, the event “Rolling an even number” is the subset 2,4,6.
To calculate the probability of an event, we divide the number of possible outcomes of the event by the number of possible outcomes of the sample space.
It is important to note that in order to use this formula, all outcomes must be equally likely to happen.
For example, the probability of rolling an even number with a standard dice is:
### Practice Makes Perfect
Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in . They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions.
### Constructing Probability Models
Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is An event is any subset of a sample space.
The likelihood of an event is known as probability. The probability of an event
is a number that always satisfies
where 0 indicates an impossible event and 1 indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like .
The sum of the probabilities listed in a probability model must equal 1, or 100%.
### Computing Probabilities of Equally Likely Outcomes
Let
be a sample space for an experiment. When investigating probability, an event is any subset of
When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in
Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in
so the probability of the event is
### Computing the Probability of the Union of Two Events
We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events
is the event that occurs if either or both events occur.
Suppose the spinner in is spun. We want to find the probability of spinning orange or spinning a
There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is
There are a total of 6 sections, and 2 of them have a
So the probability of spinning a
is
If we added these two probabilities, we would be counting the sector that is both orange and a
twice. To find the probability of spinning an orange or a
we need to subtract the probability that the sector is both orange and has a
The probability of spinning orange or a
is
### Computing the Probability of Mutually Exclusive Events
Suppose the spinner in is spun again, but this time we are interested in the probability of spinning an orange or a
There are no sectors that are both orange and contain a
so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is
Notice that with mutually exclusive events, the intersection of
and
is the empty set. The probability of spinning an orange is
and the probability of spinning a
is
We can find the probability of spinning an orange or a
simply by adding the two probabilities.
The probability of spinning an orange or a
is
### Using the Complement Rule to Compute Probabilities
We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event
denoted
is the set of outcomes in the sample space that are not in
For example, suppose we are interested in the probability that a horse will lose a race. If event
is the horse winning the race, then the complement of event
is the horse losing the race.
To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.
The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is
the probability of the horse losing the race is simply
### Computing Probability Using Counting Theory
Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.
Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are
ways to select 2 phones that are not defective. There are 8 phones, so there are
ways to select 2 phones. The probability of selecting 2 phones that are not defective is:
### Key Equations
### Key Concepts
1. Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
2. The probabilities in a probability model must sum to 1. See .
3. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. See .
4. To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See .
5. To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See .
6. The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See .
7. In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See .
### Section Exercises
### Verbal
### Numeric
For the following exercises, use the spinner shown in to find the probabilities indicated.
For the following exercises, two coins are tossed.
For the following exercises, four coins are tossed.
For the following exercises, one card is drawn from a standard deck of
cards. Find the probability of drawing the following:
For the following exercises, two dice are rolled, and the results are summed.
For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:
For the following exercises, use this scenario: a bag of M&Ms contains
blue,
brown,
orange,
yellow,
red, and
green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.
### Extensions
Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting
numbers from the numbers
to
After the player makes his selections,
winning numbers are randomly selected from numbers
to
A win occurs if the player has correctly selected
or
of the
winning numbers. (Round all answers to the nearest hundredth of a percent.)
### Real-World Applications
Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).United States Census Bureau. http://www.census.gov
### Chapter Review Exercises
### Sequences and Their Notation
### Arithmetic Sequences
### Geometric Sequences
### Series and Their Notation
### Counting Principles
### Binomial Theorem
### Probability
For the following exercises, assume two die are rolled.
For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)
### Practice Test
For the following exercises, use the spinner in . |
# Introduction to Calculus
## Introduction to Calculus
Sifan Hassan, an Ethiopian-born Dutch runner, has dominated distance running for several years. She became the first runner to win both the 1500 and 10,000 meter races at a World Championship. During the Tokyo Olympics, she joined only one other runner in history when she medaled in the rarely attempted distance triple: Winning the and won the gold medal in both the 5000 and 10,000 meter races and winning he bronze in the 1500. Hassan's signature racing style is to stay at the back of the pack for much of the race, and then move up during the final laps.
Hassan does not run at her top speed at every instant. How then, do we approximate her speed at any given instant? We will find the answer to this and many related questions in this chapter. |
# Introduction to Calculus
## Finding Limits: Numerical and Graphical Approaches
Intuitively, we know what a limit is. A car can go only so fast and no faster. A trash can might hold 33 gallons and no more. It is natural for measured amounts to have limits. What, for instance, is the limit to the height of a woman? The tallest woman on record was Jinlian Zeng from China, who was 8 ft 1 in.https://en.wikipedia.org/wiki/Human_height and http://en.wikipedia.org/wiki/List_of_tallest_people Is this the limit of the height to which women can grow? Perhaps not, but there is likely a limit that we might describe in inches if we were able to determine what it was.
To put it mathematically, the function whose input is a woman and whose output is a measured height in inches has a limit. In this section, we will examine numerical and graphical approaches to identifying limits.
### Understanding Limit Notation
We have seen how a sequence can have a limit, a value that the sequence of terms moves toward as the nu mber of terms increases. For example, the terms of the sequence
gets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function
then as the input
gets closer and closer to
the output y-coordinate gets closer and closer to
We say that the output “approaches”
provides a visual representation of the mathematical concept of limit. As the input value
approaches
the output value
approaches
We write the equation of a limit as
This notation indicates that as
approaches
both from the left of
and the right of
the output value approaches
Consider the function
We can factor the function as shown.
Notice that
cannot be 7, or we would be dividing by 0, so 7 is not in the domain of the original function. In order to avoid changing the function when we simplify, we set the same condition,
for the simplified function. We can represent the function graphically as shown in .
What happens at
is completely different from what happens at points close to
on either side. The notation
indicates that as the input
approaches 7 from either the left or the right, the output approaches 8. The output can get as close to 8 as we like if the input is sufficiently near 7.
What happens at
When
there is no corresponding output. We write this as
This notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as
Notice that the limit of a function can exist even when
is not defined at
Much of our subsequent work will be determining limits of functions as
nears
even though the output at
does not exist.
### Understanding Left-Hand Limits and Right-Hand Limits
We can approach the input of a function from either side of a value—from the left or the right. shows the values of
as described earlier and depicted in .
Values described as “from the left” are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left in are
and
The corresponding outputs are
and
These values are getting closer to 8. The limit of values of
as
approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function
as
approaches 7.
Values described as “from the right” are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right in are
and
The corresponding outputs are
and
These values are getting closer to 8. The limit of values of
as
approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function
as
approaches 7.
shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input
within the interval
to produce an output value of
within the interval
We also see that we can get output values of
successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.
provides a visual representation of the left- and right-hand limits of the function. From the graph of
we observe the output can get infinitesimally close to
as
approaches 7 from the left and as
approaches 7 from the right.
To indicate the left-hand limit, we write
To indicate the right-hand limit, we write
### Understanding Two-Sided Limits
In the previous example, the left-hand limit and right-hand limit as
approaches
are equal. If the left- and right-hand limits are equal, we say that the function
has a two-sided limit as
approaches
More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.
### Finding a Limit Using a Graph
To visually determine if a limit exists as
approaches
we observe the graph of the function when
is very near to
In we observe the behavior of the graph on both sides of
To determine if a left-hand limit exists, we observe the branch of the graph to the left of
but near
This is where
We see that the outputs are getting close to some real number
so there is a left-hand limit.
To determine if a right-hand limit exists, observe the branch of the graph to the right of
but near
This is where
We see that the outputs are getting close to some real number
so there is a right-hand limit.
If the left-hand limit and the right-hand limit are the same, as they are in , then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit.
Finally, we can look for an output value for the function
when the input value
is equal to
The coordinate pair of the point would be
If such a point exists, then
has a value. If the point does not exist, as in , then we say that
does not exist.
### Finding a Limit Using a Table
Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of
approach
from both sides. Then we determine if the output values get closer and closer to some real value, the limit
Let’s consider an example using the following function:
To create the table, we evaluate the function at values close to
We use some input values less than 5 and some values greater than 5 as in . The table values show that when
but nearing 5, the corresponding output gets close to 75. When
but nearing 5, the corresponding output also gets close to 75.
Because
then
Remember that
does not exist.
### Key Concepts
1. A function has a limit if the output values approach some value
as the input values approach some quantity
See .
2. A shorthand notation is used to describe the limit of a function according to the form
which indicates that as
approaches
both from the left of
and the right of
the output value gets close to
3. A function has a left-hand limit if
approaches
as
approaches
where
A function has a right-hand limit if
approaches
as
approaches
where
4. A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.
5. A graph provides a visual method of determining the limit of a function.
6. If the function has a limit as
approaches
the branches of the graph will approach the same
coordinate near
from the left and the right. See .
7. A table can be used to determine if a function has a limit. The table should show input values that approach
from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit. See .
8. A graphing utility can also be used to find a limit. See .
### Section Exercises
### Verbal
### Graphical
For the following exercises, estimate the functional values and the limits from the graph of the function
provided in .
For the following exercises, draw the graph of a function from the functional values and limits provided.
For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as
approaches 0.
For the following exercises, use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as
approaches
If the function has a limit as
approaches
state it. If not, discuss why there is no limit.
### Numeric
For the following exercises, use numerical evidence to determine whether the limit exists at
If not, describe the behavior of the graph of the function near
Round answers to two decimal places.
For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as
approaches the given value.
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as
approaches
If the function has a limit as
approaches
state it. If not, discuss why there is no limit.
### Extensions
|
# Introduction to Calculus
## Finding Limits: Properties of Limits
Consider the rational function
The function can be factored as follows:
Does this mean the function
is the same as the function
The answer is no. Function
does not have
in its domain, but
does. Graphically, we observe there is a hole in the graph of
at
as shown in and no such hole in the graph of
as shown in .
So, do these two different functions also have different limits as
approaches 7?
Not necessarily. Remember, in determining a limit of a function as
approaches
what matters is whether the output approaches a real number as we get close to
The existence of a limit does not depend on what happens when
equals
Look again at and . Notice that in both graphs, as
approaches 7, the output values approach 8. This means
Remember that when determining a limit, the concern is what occurs near
not at
In this section, we will use a variety of methods, such as rewriting functions by factoring, to evaluate the limit. These methods will give us formal verification for what we formerly accomplished by intuition.
### Finding the Limit of a Sum, a Difference, and a Product
Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits.
Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.
### Finding the Limit of a Polynomial
Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit of a polynomial function as
approaches
is equivalent to simply evaluating the function for
.
### Finding the Limit of a Power or a Root
When a limit includes a power or a root, we need another property to help us evaluate it. The square of the limit of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.
### Finding the Limit of a Quotient
Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.
### Key Concepts
1. The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves. See .
2. The limit of a polynomial function can be found by finding the sum of the limits of the individual terms. See and .
3. The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution. See .
4. The limit of the root of a function equals the corresponding root of the limit of the function.
5. One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify. See .
6. Another method of finding the limit of a complex fraction is to find the LCD. See .
7. A limit containing a function containing a root may be evaluated using a conjugate. See .
8. The limits of some functions expressed as quotients can be found by factoring. See .
9. One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful. See .
### Section Exercises
### Verbal
### Algebraic
For the following exercises, evaluate the limits algebraically.
For the following exercise, use the given information to evaluate the limits:
.
For the following exercises, evaluate the following limits.
For the following exercises, find the average rate of change
### Graphical
For the following exercises, refer to .
### Real-World Applications
|
Subsets and Splits