Answer
Symmetric about the y-axis.
Symmetric about the x-axis.
Symmetric about the origin.
Work Step by Step
Testing for symmetry:
- about the x-axis: $(r,\theta)$ on the graph $\Rightarrow (r, -\theta)$ or $(-r, \pi-\theta)$ on the graph.
- about the y-axis: $(r,\theta)$ on the graph $\Rightarrow (r, \pi-\theta)$ or $(-r, -\theta)$ on the graph.
- about the origin: $(r,\theta)$ on the graph $\Rightarrow (-r, \theta)$ or $(r, \theta+\pi)$ on the graph.
If $(r,\theta)$ is on the graph, because cosine is an even function,
$\cos[2(-\theta)]=\cos(-2\theta)=\cos(2\theta)=(\pm r)^{2}$
Both $(r,-\theta)$ and $(-r,-\theta)$ are on the graph, so all three symmetries apply.
To graph, select a few values for $\theta \in[0,\pi/4]$ and calculate $r=2\sqrt{\cos 2\theta}.$
Use the symmetries.
See the resulting image.