Answer
This is a graph of $~~r = a~\theta~~$ for $a=-1$ and values of $\theta$ such that $0 \leq \theta \leq 12\pi$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/06f7f723-3cef-489d-85d4-6a0cb8bde6dd/result_image/1555830809.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T012436Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=51c3569285597a8b228cb56dd0d6f7a67169671a4718784276b264e046b099ae)
Work Step by Step
This is a graph of $~~r = a~\theta~~$ for $a=-1$ and values of $\theta$ such that $0 \leq \theta \leq 12\pi$
Note that $r = 0$ when $\theta = 0$, and the magnitude of $r$ continues to increase as the magnitude of $\theta$ increases. This leads to a spiral shape.
At the endpoint of the spiral, the magnitude of $r$ is $36\pi$ when $\theta = 12\pi$
We can see the graph in the window $[-40,40]$ by $[-40,40]$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/06f7f723-3cef-489d-85d4-6a0cb8bde6dd/steps_image/small_1555830809.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T012436Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=f9ef32d2211786b4a3662118a25c9a04d2bda224fafa2c498d9058a137a7e667)