The graph of a trigonometric function oscillates between y=1 and y=-7. It reaches its
maximum at x=pi and its minimum at x=3pi. Which of the following could be the equation of the function?

y = 4 sin(½ x) − 3

Step-by-step explanation:

The function is either sine or cosine:

y = A sin(2π/T x) + C

y = A cos(2π/T x) + C

where A is the amplitude, T is the period, and C is the midline.

The midline is the average of the min and max:

C = (1 + -7) / 2

C = -3

The amplitude is half the difference between the min and max:

A = (1 − -7) / 2

A = 4

The maximum is at x = π, and the minimum is at x = 3π.  The difference, 2π, is half the period.  So T = 4π.

Plugging in, the options are:

y = 4 sin(½ x) − 3

y = 4 cos(½ x) − 3

Since the maximum is at x = π, this must be a sine wave.

y = 4 sin(½ x) − 3