Question 1202196
<font color=black size=3>
Answer: <font color=red size=4>y = 2.5*cos(2x)-1</font>
This is when you are in degree mode.



=============================================================================



Explanation:


The image from the question is this
<img src = "https://i.gyazo.com/80871eaf3fe6163e135f3fbce5538d74.png">


The template for cosine is 
y = A*cos(B(x-C))+D


The variables are:
|A| = amplitude
B = helps determine the period, more on that later
C = phase shift, which is the side to side shift
D = up and down shift, useful to determine midline


The goal is to determine the values of A, B, C, and D.


-----------------------------


Let's start with A.


The amplitude is half of the height of the sine wave.
The peak occurs when y = 1.5 and the valley is when y = -3.5; these represent the largest and smallest possible y values respectively.


Subtract and use absolute value to find the vertical distance between peak and valley.
|peak - valley| = |1.5 - (-3.5)|
|peak - valley| = |1.5 + 3.5|
|peak - valley| = |5|
|peak - valley| = 5


Or you could say
|valley - peak| = |-3.5 - 1.5|
|valley - peak| = |-5|
|valley - peak| = 5


The vertical distance from peak to valley is 5 units.
Half of this is 5/2 = 2.5 and it is the amplitude.


|A| = 2.5 leads to either A = 2.5 or A = -2.5
We use A = 2.5 because the cosine graph is above the x axis when x = 0.


-----------------------------


Now onto the variable B.


This variable is tied to the period.
The period is the length of each cycle, aka wavelength.


One peak occurs at x = 0
The next neighboring peak to the right is when x = 180
This is a gap of 180-0 = 180 units
The period is 180 because the curve repeats itself every 180 units.


T = 180 = period
B = 2pi/T
B = 2pi/180
B = pi/90
That would be the value of B if we are in radian mode.


However, the next set of steps show a much nicer result for B when in degree mode.
B = 360/T
B = 360/180
B = 2


Because the x axis shows 90,180,270,etc it appears your teacher might want degree mode.


-----------------------------


The phase shift is C = 0 because cosine has x = 0 lead to a peak. We haven't shifted the peak left or right off the y axis. 


I recommend using a graphing tool like Desmos to look at y = cos(x) to see what I mean. 
GeoGebra is another handy graphing tool I use all the time.


-----------------------------


The midline is y = D to represent the horizontal line cutting through the middle of this cosine curve.


Apply the midpoint of the peak and valley
(peak+valley)/2 = (1.5+(-3.5))/2 = -2/2 = -1


The midline is y = -1 and we get D = -1


-----------------------------


Summary 
A = 2.5
B = 2 (when in degree mode)
C = 0
D = -1


Then
y = A*cos(B(x-C))+D
y = 2.5*cos(2(x-0))+(-1)
<font color=red>y = 2.5*cos(2x)-1</font>
This is when your calculator is in degree mode.


Graph using Desmos
<a href="https://www.desmos.com/calculator/ju2jkpfbkq">https://www.desmos.com/calculator/ju2jkpfbkq</a>
</font>