Question

test for symmetry and then identify the graph of the polar equation.
r = 2 - 3\sin\theta
a. is the polar equation symmetrical with respect to the polar axis?
a. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the polar axis.
b. yes.
c. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the polar axis.

Explanation:

Step1: Test for symmetry about the polar axis ($\theta = 0$)

Replace $\theta$ with $-\theta$ in the equation $r = 2-3\sin\theta$.
We get $r = 2-3\sin(-\theta)$.
Since $\sin(-\theta)=-\sin\theta$, the equation becomes $r = 2 + 3\sin\theta$.
This is not the same as the original equation $r = 2-3\sin\theta$.

Step2: Analyze the result of the symmetry test

A failed symmetry test (when we replace $\theta$ with $-\theta$ and do not get the original equation) does not guarantee that the graph is not symmetric about the polar axis. Symmetry tests are sufficient conditions (if the test passes, the graph has the symmetry), but not necessary conditions (a failed test does not mean the graph does not have the symmetry). For example, $r=\sin(2\theta)$ fails the test for symmetry about the polar axis (when we replace $\theta$ with $-\theta$, we get $r =-\sin(2\theta)$), but it is symmetric about the line $\theta=\frac{\pi}{2}$ and other axes.

Answer:

C. The polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the polar axis.