91Ó°ÊÓ

The given equation is

$r^2=\sin \left(2\mathrm{θ}\right)$

Substitute $-\mathrm{θ}$for $\mathrm{θ}$in the equation

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

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$\mathrm{π}-\mathrm{θ}$ for$\mathrm{θ}$ in the equation

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

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$\mathrm{θ}=\frac{\mathrm{π}}{2}$

Substitute$-r$ for r in the equation

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

The equation match with the original equation so the test satisfied

the graph is symmetric with respect to the pole

Consider different values for $\mathrm{θ}$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$r^2=\sin \left(2\mathrm{θ}\right)$using a graphing utility

Graph matches so our graph is correct.