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

$r=2^{\theta }$

The given equation is

$r=2^{\theta }$

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

$$\begin{gathered} r=2^{\left(-\theta \right)} \\ r=\frac{1}{2^{\theta }} \end{gathered}$$

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

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

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 line$\theta =\frac{\mathrm{\pi }}{2}$

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

$$\begin{gathered} -r=2^{\theta } \\ r=-2^{\theta } \end{gathered}$$

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 $\theta$in interval$\left(-2\mathrm{\pi },2\mathrm{\pi }\right)$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 spiral

Plot the graph of the equation using a graphing utility

Graph matches so our graph is correct.