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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
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# Rotating a Parabola

Sketch the graph of

Solution

Because A = 1, B = -4, and C = 4, you have

The trigonometric identity 2θ = (cot2 θ - 1)/(2 cot θ) produces

from which you can obtain the equation

Considering 0 < θ < π/2, it follows that 2 cot θ = 4. Thus,

From the triangle in the figure below, you can obtain and

Consequently, you can write the following.

Substituting these expressions into the original equation produces

which simplifies to

5(y')2 + 5x' + 10y' + 1 = 0

By completing the square, you can obtain the standard form

The graph of the equation is a parabola with its vertex at and its axis parallel to the x'-axis in the y'-system, as shown in the figure below.
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