the function graphed to the right is of the form y = c + cos x, y = c + sin x, y = cos (x - d), or y = sin…

Explanation:

Step1: Analyze the mid - line

The mid - line of the given graph is (y = 0). For the functions (y=c+\cos x) and (y = c+\sin x), if (c\neq0), the mid - line is (y = c). Since the mid - line is (y = 0), we can rule out (y=c+\cos x) and (y = c+\sin x) forms where (c\neq0).

Step2: Analyze the starting point

The standard cosine function (y=\cos x) has a maximum at (x = 0), and the standard sine function (y=\sin x) has a value of (0) at (x = 0). The given graph has a value of approximately (- 1) at (x = 0). The graph of (y=\sin(x - d)) or (y=\cos(x - d)) is a phase - shifted graph. The graph of (y=\cos x) is (y = 1) at (x = 0). The graph of (y=\sin x) is (y = 0) at (x = 0). We know that (\sin\left(x-\frac{3\pi}{2}\right)=\cos x). The graph of (y = \sin(x - d)): when (x = 0), (y=\sin(-d)). We want to find (d) such that the graph matches. The graph of (y=\sin(x - d)) has a value of (-1) at (x = 0). So (\sin(-d)=-1), then (-d=-\frac{\pi}{2}+2k\pi,k\in\mathbb{Z}), and the least positive value of (d) is (\frac{\pi}{2}).

Answer:

(y=\sin\left(x-\frac{\pi}{2}\right))