find the average rate of change of f(x)=-2x^2 + 2 from x = 5 to x = 8. simplify your answer as much as…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 5$, $b = 8$, and $f(x)=-2x^{2}+2$.

Step2: Calculate $f(5)$

Substitute $x = 5$ into $f(x)$: $f(5)=-2\times5^{2}+2=-2\times25 + 2=-50 + 2=-48$.

Step3: Calculate $f(8)$

Substitute $x = 8$ into $f(x)$: $f(8)=-2\times8^{2}+2=-2\times64+2=-128 + 2=-126$.

Step4: Calculate the average rate of change

$\frac{f(8)-f(5)}{8 - 5}=\frac{-126-(-48)}{3}=\frac{-126 + 48}{3}=\frac{-78}{3}=-26$.

Answer:

$-26$