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 graph oscillates between - 5 and 5. The amplitude (a) is the distance from the mid - line to the maximum or minimum value. So (a = 5).

Step2: Determine the period and (b)

The period (T) of a sine or cosine function (y=a\sin(bx)) or (y = a\cos(bx)) is given by (T=\frac{2\pi}{b}). The graph goes from (x = 0) to (x = 2\pi) and completes half of a cycle. So the period (T = 4\pi). Since (T=\frac{2\pi}{b}), then (4\pi=\frac{2\pi}{b}), solving for (b) gives (b=\frac{1}{2}).

Step3: Determine the function type

The graph passes through the point ((0, - 5)). For (y=a\sin(bx)), when (x = 0), (y=a\sin(0)=0). For (y=a\cos(bx)), when (x = 0), (y=a\cos(0)=a). Since the graph has (y=-5) at (x = 0) and (a = 5), the function is (y=-5\cos(\frac{1}{2}x)).

Answer:

(y=-5\cos(\frac{1}{2}x))