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

The graph is symmetric with respect to the polar axis because replacing $\theta$ with $-\theta$ in the equation $r = 5 + 4\cos(\theta)$ does not change the equation, indicating symmetry with respect to the polar axis. There is no symmetry with respect to $\theta = \frac{\pi}{2}$ or the pole, as the corresponding tests for symmetry (replacing $\theta$ with $\pi - \theta$ and $r$ with $-r$, respectively) would alter the equation in ways that do not preserve its original form.

Test the Graph of the Following Equation for Symmetry with Respect