Question

Find an equation of the cosine function whose graph is shown below.

Answer 1

Answer:

$y=3\text{cos}(2x)+1$

Step-by-step explanation:

We have been given an image of trigonometric function. We are asked to find the equation of the given function.

We know that standard form of a cosine function is $y=A\cdot \text{cos}(Bx-C)+d$, where,

A = Amplitude of function

$\frac{2\pi}{B}$ = Period of function,

C = Horizontal shift or phase shift,

D =  Vertical shift.

Upon looking at our given function we can see that amplitude of our given function is 3 as average of maximum and minimum of our given function is $\frac{4--2}{2}=\frac{4+2}{2}=\frac{6}{2}=3$.

We know that mid-line of our given function is $y=1$, therefore, our function is shifted upwards by 1 unit.

We can see that period of our given function is $\pi$.

Let us find the value of B using formula:

$\pi=\frac{2\pi}{B}$

$B=\frac{2\pi}{\pi}$

$B=2$

Therefore, our required equation is $y=3\text{cos}(2x)+1$.

Answer 2

Short Answer: y = 3*cos(2x) + 1
Remark
This is not part of the answer, but it will help you to see what is going on. Begin by shifting the cos graph 1 unit down. That will that the cos has a minimum of -3 and a max of + 3

What that tells you is the part of the equation is
y = 3*cos(x)

Step One
Show that the graph is of something that resembles  y = 3*cos(x)
the test way to check this is to put in 0 for x
3*cos(0) = 3*1 = 3. But why is it 4 and -2 instead of 3 and -3.

Step Two 
Show how the graph is shifted up one space.
y movement is always recorded behind how the variable is determined.
So if the graph is shifting up one, you should do this.
y = 3cos(x) + 1

Step Three
The graph seems to be starting over at n*pi rather than n*2pi. How do we adjust for that?
There are 2 choices. Either there is a 4 in front of the x or there is a 1/2 in front of  the x. Before just telling you, consider the graph below.
Violet is 3*cos(1/2 x)
Blue is 3*cos (x)
orange is 3*cos(2*x)

You want the graph that starts over again at x = 3.14 rather than at x = 6.28 
The one that starts over at y = 3*cos(2x)
The rule is that if you want to compress a trigonometric function, use a constant such that a > 1 y = 3*cos(a*x). It is the a I'm trying to explain.

Step 4
Is there a phase shift? 
No. If there was cos(x) would not have a maximum at x = 0

Answer y = 3*cos(2x) + 1