Answer
Symmetrical about the polar axis
Work Step by Step
The rules for symmetry:
1. About the Polar Axis if the graph is the same when replacing θ with -θ
2. About the pole if the graph is the same when replacing r with -r or θ by $θ + \pi$
3. About the vertical line $θ = \frac{\pi}{2}$ if the graph is the same when replacing θ by $\pi - θ$
Given $r = 4 + 8 \cos θ$
1. $r = 4 + 8 \cos (-θ) = 4 + 8 \cos θ$
Yes, the same!
2. $-r = 4 + 8 \cos θ = -4 - 8 \cos θ$
Not the same, so not about the pole
3. $r = 4 + 8 \cos (\pi - θ) = 4 - 8 \cos θ$
Not the same, so not about the vertical line $θ = \frac{\pi}{2}$
Symmetrical about the polar axis
