Question

test for symmetry and graph the polar equation.
r = 6 + 6\cos\theta
a. is the polar equation symmetrical 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 (\(\theta = 0\)), we replace \(\theta\) with \(-\theta\) in the polar equation \(r = f(\theta)\). The cosine function has the property \(\cos(-\theta)=\cos\theta\).

Step2: Substitute \(\theta\) with \(-\theta\) in the given equation

Given \(r = 6+6\cos\theta\), when we substitute \(\theta\) with \(-\theta\), we get \(r = 6 + 6\cos(-\theta)\). Since \(\cos(-\theta)=\cos\theta\), the equation becomes \(r=6 + 6\cos\theta\), which is the same as the original equation.

Answer:

B. Yes.