given the function $f(x)=x^{2}-7x + 8$, determine the average rate of change of the function over the…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$, $b = 7$, and $f(x)=x^{2}-7x + 8$.

Step2: Calculate $f(a)$

Substitute $x=-1$ into $f(x)$: $f(-1)=(-1)^{2}-7\times(-1)+8=1 + 7+8=16$.

Step3: Calculate $f(b)$

Substitute $x = 7$ into $f(x)$: $f(7)=7^{2}-7\times7+8=49-49 + 8=8$.

Step4: Calculate the average rate of change

Using the formula $\frac{f(b)-f(a)}{b - a}$, we have $\frac{f(7)-f(-1)}{7-(-1)}=\frac{8 - 16}{7 + 1}=\frac{-8}{8}=-1$.

Answer:

$-1$