Determine if the graph is symmetric about the x-axis, the y-axis, or the origin.
-$$r = \frac { 3 } { 1 - 4 \sin \theta }$$
A) No symmetry
B) $x$-axis only
C) x-axis, $y$-axis, origin
D) $y$-axis only

Question

Quizplus · Accepted Answer

To determine the symmetry of the graph of the polar equation $r = \frac { 3 } { 1 - 4 \sin \theta }$, we check for symmetry about the x-axis, y-axis, and the origin using specific tests. - Symmetry about the y-axis (polar axis) is tested by replacing $\theta$ with $-\theta$ and seeing if the equation remains unchanged. In this case, replacing $\theta$ with $-\theta$ gives $r = \frac { 3 } { 1 - 4 \sin(-\theta) }$. Since $\sin(-\theta) = -\sin(\theta)$, the equation becomes $r = \frac { 3 } { 1 + 4 \sin(\theta) }$, which is not equivalent to the original equation, indicating it does not have symmetry about the y-axis.- Symmetry about the x-axis is not directly applicable because polar equations inherently represent points in a radial and angular manner, and symmetry about the x-axis would imply that for every point $r, \theta$, there is a point $r, -\theta$, which is not a direct implication of the given equation.- Symmetry about the origin is tested by replacing $r$ with $-r$ and $\theta$ with $\theta + \pi$. This test is not directly applicable here without further manipulation and does not directly lead to the conclusion of origin symmetry.However, my explanation above contains a mistake in the analysis of symmetry about the y-axis. The correct approach to determine the symmetry of the given polar equation involves understanding how the equation behaves under transformations that correspond to reflections over the axes or rotations (which would indicate origin symmetry).For the given equation $r = \frac { 3 } { 1 - 4 \sin \theta }$, the correct symmetry test involves:- For y-axis symmetry, replace $\theta$ with $\pi - \theta$. Since $\sin(\pi - \theta) = \sin(\theta)$, the equation remains unchanged, indicating y-axis symmetry.- For x-axis symmetry, you would look for an equation that remains unchanged when $\theta$ is replaced with $-\theta$, but this is more relevant for Cartesian coordinates. Polar coordinates inherently do not have this type of symmetry due to their nature.- For origin symmetry, replace $r$ with $-r$ and $\theta$ with $\theta + \pi$. This test is not straightforward for this equation without further algebraic manipulation.The correct analysis shows that the equation is symmetric about the y-axis because replacing $\theta$ with $\pi - \theta$ does not change the equation, indicating that for every point $r, \theta$, there is a point $r, \pi - \theta$ that lies on the graph, which corresponds to reflection across the y-axis. My initial explanation incorrectly analyzed these symmetries, leading to an incorrect conclusion. The correct answer is that the graph has symmetry about the y-axis only.

Determine If the Graph Is Symmetric About the X-Axis, the Y-Axis