the graph of a cosine function is shown. which two points on the mid - line of the function are separated by…

Answer:

We need to first identify the mid - line of the cosine function. The mid - line of a cosine function (y = A\cos(Bx - C)+D) is (y = D). From the graph, the mid - line is (y = 1.5). The period of a cosine function (y=\cos(Bx)) is (T=\frac{2\pi}{B}). For a standard cosine - like graph, we look for two points on the mid - line that are one full cycle apart. Let's assume the general form of the cosine function (y = A\cos(Bx - C)+D). The period (T) of the cosine function is the horizontal distance between two consecutive points with the same phase. Looking at the graph, if we consider the points on the mid - line (y = 1.5), we can see that the point at (x=\frac{\pi}{8}) and the point at (x=\frac{5\pi}{8}) are separated by a distance of (\frac{5\pi}{8}-\frac{\pi}{8}=\frac{4\pi}{8}=\frac{\pi}{2}). If we assume the function has a period (T = \frac{\pi}{2}), and we check the points on the mid - line. The two points on the mid - line (y = 1.5) that are separated by one period are the point with (x) - coordinate (\frac{\pi}{8}) and the point with (x) - coordinate (\frac{5\pi}{8}).

Explanation:

Step1: Identify mid - line

The mid - line of the function is (y = 1.5) by observing the graph.

Step2: Recall period concept

The period is the horizontal distance between two repeating points.

Step3: Locate points on mid - line

Find points on (y = 1.5) and measure the horizontal distance. Points at (x=\frac{\pi}{8}) and (x = \frac{5\pi}{8}) on (y = 1.5) have a distance of (\frac{5\pi}{8}-\frac{\pi}{8}=\frac{\pi}{2}) which is one period (assuming the period of the function is (\frac{\pi}{2}) from the graph).