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

$r=3^{\theta }$

The given equation is

$r=3^{\theta }$

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

$$\begin{gathered} r=3^{-\theta } \\ r=\frac{1}{3^{\theta }} \end{gathered}$$

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

role="math" localid="1646691976744" $$\begin{gathered} r=3^{\pi -\theta } \\ r=\frac{3^{\mathrm{\pi }}}{3^{\theta }} \end{gathered}$$

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

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

Substitute$-r$for $r$in the equation

$$\begin{gathered} -r=3^{\theta } \\ r=-3^{\theta } \end{gathered}$$

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

the graph may or may not be 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 spiral

Plot the graph of the equation$r=3^{\theta }$using a graphing utility

Graph matches so our graph is correct.