Question

the graph of a cosine function is shown. which two points on the midline of the function are separated by a distance of one period?

Explanation:

Step1: Identify the mid - line

The mid - line of the cosine function is the horizontal line that the graph oscillates around. For this graph, the mid - line is $y = 1.5$.

Step2: Recall the definition of period

The period of a periodic function is the horizontal distance between two consecutive points with the same phase. For a cosine function, we can look for two consecutive points on the mid - line with the same slope.

Step3: Find the points on the mid - line

The points on the mid - line $y = 1.5$ are: $( \frac{\pi}{8},1.5)$, $(\frac{\pi}{2},1.5)$. The horizontal distance between two consecutive such points with the same behavior (going up or going down) gives the period. The period of this cosine function is $\frac{\pi}{2}-\frac{\pi}{8}=\frac{4\pi - \pi}{8}=\frac{3\pi}{8}$. The two points on the mid - line separated by one period are $(\frac{\pi}{8},1.5)$ and $(\frac{5\pi}{8},1.5)$.

Answer:

The points $(\frac{\pi}{8},1.5)$ and $(\frac{5\pi}{8},1.5)$