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.
b. is the polar equation symmetrical with respect to the line \theta=\frac{\pi}{2}?
a. yes.
b. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the line \theta=\frac{\pi}{2}.
c. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the line \theta=\frac{\pi}{2}.

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 $r = 2-3\sin\theta$.
So, the polar equation failed the test for symmetry about the polar axis. But a failed test does not necessarily mean the graph is not symmetric.

Step2: Test for symmetry about the line $\theta=\frac{\pi}{2}$

Replace $(r,\theta)$ with $(-r,-\theta)$ in the equation $r = 2-3\sin\theta$.
We get $-r=2 - 3\sin(-\theta)$.
Since $\sin(-\theta)=-\sin\theta$, we have $-r=2 + 3\sin\theta$ or $r=-2 - 3\sin\theta$, which is not the same as the original equation.
Another way: Replace $\theta$ with $\pi-\theta$ in the original equation.
$r = 2-3\sin(\pi - \theta)$.
Since $\sin(\pi-\theta)=\sin\theta$, the equation remains $r = 2-3\sin\theta$.

Answer:

a. 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.
b. A. Yes.