Question

test for symmetry and then graph the polar equation.
r = 1 + 4\cos\theta
a. is the graph of the polar equation symmetric 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 graph of the polar equation symmetric with respect to the line \theta=\frac{\pi}{2}?
a. 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}.
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. yes.

Explanation:

Step1: Test for symmetry about the polar axis ($\theta = 0$)

Replace $\theta$ with $-\theta$ in the equation $r = 1+4\cos\theta$.
Since $\cos(-\theta)=\cos\theta$, we get $r = 1 + 4\cos(-\theta)=1 + 4\cos\theta$. The equation remains the same.

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

Replace $\theta$ with $\pi-\theta$ in the equation $r = 1+4\cos\theta$.
We have $r=1 + 4\cos(\pi-\theta)$. Using the identity $\cos(\pi-\theta)=-\cos\theta$, the equation becomes $r = 1-4\cos\theta$, which is not the same as the original equation $r = 1+4\cos\theta$.

Answer:

a. B. Yes.
b. 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 $\theta=\frac{\pi}{2}$.