find an equation for the graph.\ntype the equation of the given graph in the form y = a sin bx or y = a cos…

Explanation:

Step1: Determine the amplitude A

The amplitude is the distance from the mid - line to the maximum or minimum. The maximum value is 4 and the minimum is - 4, so the mid - line is (y = 0). The amplitude (A=\frac{4 - (-4)}{2}=4).

Step2: Determine the period T and find B

The period (T) is the distance between two consecutive peaks or troughs. From the graph, (T = 2\pi). The formula for the period of (y = A\sin Bx) or (y=A\cos Bx) is (T=\frac{2\pi}{B}). Since (T = 2\pi), then (2\pi=\frac{2\pi}{B}), solving for (B) gives (B = 1).

Step3: Choose the function type

The graph passes through the origin ((0,0)), so the function is of the form (y = A\sin Bx) (because (y=\sin x) passes through ((0,0)) and (y = \cos x) passes through ((0,1))).

Answer:

(y = 4\sin x)