Identify and graph each polar equation. Verify your graph using a graphing utility.
$r=1+2\sin \theta$

The given equation is

$r=1+2\sin \theta$

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

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

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="1646681410227" $$\begin{gathered} r=1+2\sin \left(\mathrm{\pi }-\theta \right) \\ r=1+2\sin \theta \end{gathered}$$

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

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

Substitute$-r$ for r in the equation

$$\begin{gathered} -r=1+2\sin \theta \\ r=-1-2\sin \theta \end{gathered}$$

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 pole

Consider different values for $\theta$in interval$\left(\frac{-\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$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 limacon with an inner loop

Plot the graph of the equation $r=1+2\sin \theta$using a graphing utility

Graph matches so our graph is correct.