Answer
Symmetric about the y-axis.
No symmetry about the x-axis.
No symmetry 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.
- about the x-axis:
$1+2\sin(-\theta)=1-2\sin\theta\neq r,\quad(r, -\theta)$ is not on the graph.
$1+2\sin(\pi-\theta)=1+2\sin\theta=r\neq-r,\quad(-r, \pi-\theta)$ is not on the graph.
No symmetry about the x-axis.
- about the y-axis:
$1+2\sin(\pi-\theta)=1+2\sin\theta=r,\quad(r, \pi-\theta)$ is on the graph.
Symmetric about the y-axis.
So, no symmetry about the origin.
To graph, select a few values for $\theta$ and calculate $r.$
Use the symmetry.
See the resulting image.
