Answer
$y = -2~cos~(x+\frac{\pi}{4})$
We can see the graph below.
The period is $2\pi$
The amplitude is $2$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/cf1a21b6-c449-450f-b82f-432b5921d912/result_image/1531283156.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T021857Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=b4b802c2365acb249e77a5a93fd85c82562b4086f3f83277cce5f9c59922d7fb)
Work Step by Step
$y = -2~cos~(x+\frac{\pi}{4})$
When $x = 0$, then $y = -2~cos~(\frac{\pi}{4}) = -\sqrt{2}$
When $x = \frac{\pi}{4}$, then $y = -2~cos~(\frac{\pi}{2}) = 0$
When $x = \frac{3\pi}{4}$, then $y = -2~cos~(\pi)= 2$
When $x = \frac{5\pi}{4}$, then $y = -2~cos~(\frac{3\pi}{2}) = 0$
When $x = \frac{7\pi}{4}$, then $y = -2~cos~(2\pi) = -2$
When $x = 2\pi$, then $y = -2~cos~(\frac{9\pi}{4}) = -\sqrt{2}$
We can see the graph below.
The period is $2\pi$
The amplitude is $2$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/cf1a21b6-c449-450f-b82f-432b5921d912/steps_image/small_1531283156.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T021857Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=45ff263ab03f3b523769b2fe57320425c835a2bff98c45c0d6e94dbabf7b7f53)