Which of the following could be the equation of the function below?

On a coordinate plane, a curve crosses the y-axis at y = negative 1.5. It has a maximum at negative 1.5 and a minimum at negative 2.5. It goes through 2 cycles at pi.
y = 0.5 cosine (2 (x minus StartFraction pi Over 2 EndFraction)) + 2
y = 0.5 cosine (4 (x minus StartFraction pi Over 2 EndFraction)) minus 2
y = 0.5 cosine (4 (x + pi)) + 2
y = 0.5 cosine (2 (x minus pi)) minus 2

Second option: y = 0.5cos(4*(x+pi/2)) - 2

Step-by-step explanation:

From the format of the figure, we can say that is a cosine function (starts in the maximum value and then goes down).

We can write a generic form of a cosine function:

y = a * cos(bx + c) + d

The period of this generic function is 2pi/b.

In this graph, as the cosine completes 2 periods when x = pi, we know that the period is pi/2, so we can find the value of b:

2pi/b = pi/2

b = 4

Then, the amplitude of the cosine function is 2 (goes from 1 to -1), and in this graph, the amplitude is 1 (goes from -1.5 to -2.5), so as we have half the amplitude, we have that a = 0.5

Now, to find d, we know that a cosine usually have a mean value of 0, and in this graph we see that the mean value is -2, so we have that c = -2.

The value of c represents a shift in the graph. In this case, we have a shift of zero from the original cosine, but as the period is pi/2, we can have any multiple value of pi/2 as a value of d (d = -pi/2, d = 0, d = pi/2, ...)

So the answer is the second option:

y = 0.5cos(4*(x+pi/2)) - 2

y/x=18

step-by-step explanation: