the function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. determine the equation of the…

Explanation:

Step1: Determine the amplitude (a)

The amplitude is half the vertical distance between the maximum and minimum values. The maximum value is close to (0) and the minimum is (- 6), so (a=\frac{0 - (-6)}{1}=6).

Step2: Determine the period (T)

The period is the horizontal distance for one - complete cycle. From the graph, the function goes from (x = 0) to (x=2\pi) for one cycle, so (T = 2\pi).

Step3: Find the value of (b)

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).

Step4: Determine the function type

The graph starts at its minimum value when (x = 0). The cosine function (y=a\cos(bx)) starts at (y = a) when (x = 0) and the sine - shifted function (y=-a\sin(bx)) starts at its minimum when (x = 0). So the function is (y=-6\sin(x)).

Answer:

(y = - 6\sin(x))