15. determine the period of the following graph then write the equation of function

Explanation:

Step1: Identify period - definition

The period of a periodic function is the horizontal length of one complete cycle.

Step2: Observe graph

From the graph, we can see that one complete cycle occurs from (x = 0) to (x = 3\pi). So the period (T=3\pi).

Step3: General form of sine - cosine function

The general form of a sine - cosine function is (y = A\sin(Bx - C)+D) or (y = A\cos(Bx - C)+D). The period formula is (T=\frac{2\pi}{|B|}). Since (T = 3\pi), we have (3\pi=\frac{2\pi}{|B|}), solving for (B) gives (|B|=\frac{2}{3}).

Step4: Determine amplitude

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

Step5: Determine phase shift and vertical shift

There is no phase shift ((C = 0)) and no vertical shift ((D = 0)). Let's assume the function is a cosine function (y=A\cos(Bx)), substituting (A = 2) and (B=\frac{2}{3}), we get (y = 2\cos(\frac{2}{3}x)).

Answer:

Period: (3\pi), Equation of the function: (y = 2\cos(\frac{2}{3}x))