Identify and graph each polar equation. Verify your graph using a graphing utility.

$r^2=9\cos \left(2\theta \right)$

The given equation is

$r^2=9\cos \left(2\theta \right)$

Substitute $-\theta$for $\theta$in the equation

$$\begin{gathered} r^2=9\cos \left(2\left(-\theta \right)\right) \\ {r}^2=9\cos \left(2\theta \right) \end{gathered}$$

The equation matches with the original equation so the test is satisfied.

the graph is symmetric with respect to the polar axis

Substitute $\pi -\theta$for $\theta$in the equation

role="math" localid="1646689488157" $$\begin{gathered} r^2=9\cos \left(2\left(\mathrm{\pi }-\theta \right)\right) \\ {r}^2=9\cos \left(2\mathrm{\pi }-2\theta \right) \\ {r}^2=9\cos \left(2\theta \right) \end{gathered}$$

The equation matches the original equation so the test is satisfied.

the graph is symmetric with respect to the line$\theta =\frac{\mathrm{\pi }}{2}$

The equation is symmetric with respect to the polar axis and line$\theta =\frac{\mathrm{\pi }}{2}$

so the equation is symmetric with respect to the pole

Consider different values for $\theta$in interval$\left(0,\frac{\mathrm{\pi }}{2}\right)$and determine coordinates of several points for graph

Locates points and use them to plot the graph of the equation

The graph state that the equation is a lemniscate.

Plot the graph of the equation$r^2=9\cos \left(2\theta \right)$using a graphing utility

Graph matches so our graph is correct.