select all the correct locations on the graph. the graph of a cosine function is shown. which two points on…

Explanation:

Step1: Recall period - definition

The period of a periodic function is the horizontal distance between two consecutive points with the same phase. For a cosine - type function, two points on the mid - line separated by one period have an $x$ - value difference equal to the period of the function.

Step2: Identify mid - line points

The mid - line of the given cosine function is $y = 1.5$. We need to find two points on $y = 1.5$ such that the difference in their $x$ - coordinates is the period of the function.

Step3: Determine period from graph

By observing the graph, we can see that the function repeats itself. The distance between two consecutive peaks (or troughs) gives the period. The period of the function is $\frac{\pi}{2}$. Let's consider the $x$ - coordinates of the points on the mid - line $y = 1.5$. The points on the mid - line have $x$ values: $x_1=\frac{\pi}{8},x_2 = \frac{5\pi}{8}$. The difference $\Delta x=\frac{5\pi}{8}-\frac{\pi}{8}=\frac{4\pi}{8}=\frac{\pi}{2}$.

Answer:

The two points on the mid - line ($y = 1.5$) that are separated by one period are the points with $x$ - coordinates $\frac{\pi}{8}$ and $\frac{5\pi}{8}$.