what is the rate of change between the interval of x = π/4 and x = π/2? a -8/π b -π/8 c -32/π d -π/32

Explanation:

Step1: Recall rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[x_1,x_2]$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. Here, $x_1=\frac{\pi}{4}$ and $x_2=\frac{\pi}{2}$.

Step2: Estimate function values from graph

From the graph, when $x = \frac{\pi}{4}$, $y=- 7$ and when $x=\frac{\pi}{2}$, $y = 1$.

Step3: Calculate rate of change

Substitute into the formula: $\frac{1-(-7)}{\frac{\pi}{2}-\frac{\pi}{4}}=\frac{1 + 7}{\frac{2\pi-\pi}{4}}=\frac{8}{\frac{\pi}{4}}=\frac{32}{\pi}$. But since the function is decreasing over the interval, the rate of change is $-\frac{32}{\pi}$.

Answer:

C. $-\frac{32}{\pi}$