Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$, the polar axis, and the pole.
$$r = 5 + 4 \cos ( \theta )$$
A) The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only.
B) The graph is symmetric with respect to the polar axis only.
C) The graph is symmetric with respect to the pole only.
D) The graph is symmetric with respect to all three.
E) The graph has no symmetries.

Question

Quizplus · Accepted Answer

To check for symmetry with respect to $	heta=\frac{\pi}{2}$, we substitute $	heta=\frac{\pi}{2}-\phi$, which gives us $r=5+4\cos\left(\frac{\pi}{2}-\phiight)=5-4\sin\phi$. This is not equivalent to the original equation, so there is no symmetry with respect to $	heta=\frac{\pi}{2}$.

To check for symmetry with respect to the polar axis, we substitute $	heta=-	heta$ to get $r=5+4\cos(-	heta)=5+4\cos	heta$. This is equivalent to the original equation, so there is symmetry with respect to the polar axis.

To check for symmetry with respect to the pole, we substitute $	heta=	heta+\pi$, which gives us $r=5+4\cos(	heta+\pi)=5-4\cos	heta$. This is not equivalent to the original equation, so there is no symmetry with respect to the pole.

Therefore, the correct answer is B) The graph is symmetric with respect to the polar axis only.

Test the Graph of the Following Equation for Symmetry with Respect