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.

Explanation:

Step1: Recall the symmetry test for polar axis

To test for symmetry about the polar axis ($\theta = 0$), replace $\theta$ with $-\theta$ in the polar equation \(r = 1+4\cos\theta\).

Step2: Substitute \(\theta\) with \(-\theta\)

When we substitute \(\theta\) with \(-\theta\), we get \(r = 1 + 4\cos(-\theta)\). Since \(\cos(-\theta)=\cos\theta\) (cosine is an even - function, \(\cos(-x)=\cos x\) for all \(x\)), the equation becomes \(r = 1+4\cos\theta\), which is the same as the original equation.

Answer:

B. Yes.