Answer
$y = csc~(x-\frac{\pi}{4})$
We can see the graph below.
![](https://gradesaver.s3.amazonaws.com/uploads/solution/8d85c423-4c9e-42fe-82b8-bbcb49546758/result_image/1531468480.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T021942Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=ae69ef3bcfcd6fd2cf5a590b89c9321b87aa1d26c93bc586a650626465a207d4)
Work Step by Step
$y = csc~(x-\frac{\pi}{4})$
When $x = -\frac{3\pi}{4}$, then $y = csc~(-\pi)$ (which is undefined)
When $x = -\frac{\pi}{4}$, then $y = csc~(-\frac{\pi}{2}) = -1$
When $x = \frac{\pi}{4}$, then $y = csc~0$ (which is undefined)
When $x = \frac{3\pi}{4}$, then $y = csc~(\frac{\pi}{2}) = 1$
When $x = \frac{5\pi}{4}$, then $y = csc~2\pi$ (which is undefined)
We can see the graph below.
![](https://gradesaver.s3.amazonaws.com/uploads/solution/8d85c423-4c9e-42fe-82b8-bbcb49546758/steps_image/small_1531468480.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T021942Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=59d6a337fbf015a4388121ae658757090fed279f9dbe6ffd67199929bf96d535)