SAT II Math I : Solving Exponential Functions

Example Questions

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Example Question #1 : Solving Exponential Functions

Solve the following function:

and

and

Explanation:

You must get  by itself so you must add  to both side which results in

.

You must get the square root of both side to undue the exponent.

This leaves you with .

But since you square the  in the equation, the original value you plug can also be its negative value since squaring it will make it positive anyway.

Example Question #1 : Solving Exponential Functions

What is the horizontal asymptote of the graph of the equation  ?

Explanation:

The asymptote of this equation can be found by observing that  regardless of . We are thus solving for the value of as approaches zero.

So the value that  cannot exceed is , and the line  is the asymptote.

Example Question #1 : Solving And Graphing Exponential Equations

What is/are the asymptote(s) of the graph of the function

?

Explanation:

An exponential equation of the form  has only one asymptote - a horizontal one at . In the given function, , so its one and only asymptote is .

Example Question #1 : Solving Exponential Functions

Find the vertical asymptote of the equation.

There are no vertical asymptotes.

Explanation:

To find the vertical asymptotes, we set the denominator of the function equal to zero and solve.

Example Question #1 : Solving Exponential Functions

Consider the exponential function . Determine if there are any asymptotes and where they lie on the graph.

There are no asymptotes.  goes to positive infinity in both the  and  directions.

There is one vertical asymptote at .

There is one vertical asymptote at .

There is one horizontal asymptote at .

There is one horizontal asymptote at .

Explanation:

For positive  values,  increases exponentially in the  direction and goes to positive infinity, so there is no asymptote on the positive -axis. For negative  values, as  decreases, the term  becomes closer and closer to zero so  approaches  as we move along the negative  axis. As the graph below shows, this is forms a horizontal asymptote.

Example Question #1 : Solving Exponential Equations

Solve the equation for .

Explanation:

Begin by recognizing that both sides of the equation have a root term of .

Using the power rule, we can set the exponents equal to each other.

Example Question #1 : Solving Exponential Functions

Solve the equation for .

Explanation:

Begin by recognizing that both sides of the equation have the same root term, .

We can use the power rule to combine exponents.

Set the exponents equal to each other.

Example Question #2 : Solving And Graphing Exponential Equations

In 2009, the population of fish in a pond was 1,034. In 2013, it was 1,711.

Write an exponential growth function of the form  that could be used to model , the population of fish, in terms of , the number of years since 2009.

Explanation:

Solve for the values of and b:

In 2009,  and  (zero years since 2009). Plug this into the exponential equation form:

. Solve for  to get  .

In 2013,  and . Therefore,

or  .   Solve for  to get

.

Then the exponential growth function is

.

Example Question #8 : Solving Exponential Functions

Solve for .

Explanation:

8 and 4 are both powers of 2.

Solve for :

No solution
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