Answer
This is a graph of $~~r = a~\theta~~$ for $a=1.5$ and values of $\theta$ such that $-4\pi \leq \theta \leq 4\pi$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/9aff5c3c-aac6-4600-9e87-1f2c620d38fc/result_image/1555829503.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T020250Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=0b58493ceb04fd575066e2b567f5d9fb62f969b02c0c7e1524571a5ae1f66819)
Work Step by Step
This is a graph of $~~r = a~\theta~~$ for $a=1.5$ and values of $\theta$ such that $-4\pi \leq \theta \leq 4\pi$
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.
There is symmetry about the vertical axis because both positive and negative values of $\theta$ are included.
At the endpoints of the spiral, $r = 6\pi$ when $\theta = 4\pi$ and the magnitude of $r$ is $6\pi$ when $\theta = -4\pi$
We can see the graph in the window $[-20,20]$ by $[-20,20]$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/9aff5c3c-aac6-4600-9e87-1f2c620d38fc/steps_image/small_1555829503.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T020250Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=f7090786167ed5299c819b8d49c4b3aafa65ea5815ca418ff713270c204798a5)