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

$r^2=\sin \left(2\theta \right)$

The given equation is

$r^2=\sin \left(2\theta \right)$

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

role="math" localid="1646690348938" $$\begin{gathered} r^2=\sin \left(2\left(-\theta \right)\right) \\ {r}^2=-\sin \left(2\theta \right) \end{gathered}$$

The equation does not match with the original equation so the test fails

the graph may or may not be symmetric with respect to the polar axis

Substitute$\mathrm{\pi }-\mathrm{\theta }$ for$\theta$ in the equation

$$\begin{gathered} r^2=\sin \left(2\left(\mathrm{\pi }-\theta \right)\right) \\ {r}^2=\sin \left(2\mathrm{\pi }-2\theta \right) \\ {r}^2=-\sin \left(2\theta \right) \end{gathered}$$

The equation does not match the original equation so the test fails.

the graph may be or may not be symmetric with respect to the line$\theta =\frac{\mathrm{\pi }}{2}$

Substitute$-r$ for r in the equation

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

The equation match with the original equation so the test satisfied

the graph is symmetric with respect to the pole

Consider different values for $\theta$and determine coordinates of several points for the 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$r^2=\sin \left(2\theta \right)$using a graphing utility

Graph matches so our graph is correct.