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

Question

Quizplus · Accepted Answer

To test for symmetry in polar coordinates, we use specific tests for each type of symmetry. For symmetry with respect to the pole, we replace $r$ with $-r$ and see if the equation remains unchanged or equivalent. Here, replacing $r$ with $-r$ gives $-r = 2 - 2 \sin(4\theta)$, which can be rewritten as $r = 2\sin(4\theta) - 2$, showing that the graph is symmetric with respect to the pole. Testing for symmetry with respect to $\theta = \frac{\pi}{2}$ and the polar axis does not yield equivalent equations, indicating that the graph does not have those symmetries.

Test the Graph of the Following Equation for Symmetry with Respect