question given the function f(x)=-x² + x + 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 $\frac{f(b)-f(a)}{b - a}$. Here, $a = 0$, $b = 7$, and $f(x)=-x^{2}+x + 8$.

Step2: Calculate $f(7)$

Substitute $x = 7$ into $f(x)$: $f(7)=-(7)^{2}+7 + 8=-49+7 + 8=-34$.

Step3: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$: $f(0)=-(0)^{2}+0 + 8=8$.

Step4: Calculate the average rate of change

$\frac{f(7)-f(0)}{7 - 0}=\frac{-34 - 8}{7}=\frac{-42}{7}=-6$.

Answer:

$-6$