find a function of the form $y = a \\sin(kx) + c$ or $y = a \\cos(kx) + c$ whose graph matches the function…

Explanation:

Step1: Determine the vertical shift (C)

The mid - line of the sinusoidal function is the average of the maximum and minimum values. The maximum value (y_{max}=-2) and the minimum value (y_{min}=-4). [C=\frac{y_{max} + y_{min}}{2}=\frac{-2+( - 4)}{2}=-3]

Step2: Determine the amplitude (A)

The amplitude (A) is the distance from the mid - line to the maximum (or minimum) value. [A=\vert y_{max}-C\vert=\vert-2-( - 3)\vert = 1]

Step3: Determine the period (T) and (k)

The period (T) is the horizontal length of one full cycle. From the graph, the period (T = 16). We know that the formula for the period of (y = A\sin(kx)+C) or (y=A\cos(kx)+C) is (T=\frac{2\pi}{k}). Since (T = 16), then (16=\frac{2\pi}{k}), and (k=\frac{\pi}{8})

Step4: Determine the function type

Since the graph passes through the point ((0,-3)) (the mid - line), and for (y = A\sin(kx)+C), when (x = 0), (y=C) (because (\sin(0)=0)). For (y = A\cos(kx)+C), when (x = 0), (y=A + C). So the function is of the form (y=\sin(kx)+C)

Answer:

(y=\sin\left(\frac{\pi}{8}x\right)-3)