find the average rate of change of $f(x)=-x^{2}+2x + 2$ from $x = 2$ to $x = 6$. simplify your answer 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, $a = 2$, $b = 6$, and $f(x)=-x^{2}+2x + 2$.

Step2: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: $f(2)=-(2)^{2}+2\times2 + 2=-4 + 4+2=2$.

Step3: Calculate $f(6)$

Substitute $x = 6$ into $f(x)$: $f(6)=-(6)^{2}+2\times6 + 2=-36+12 + 2=-22$.

Step4: Calculate the average rate of change

$\frac{f(6)-f(2)}{6 - 2}=\frac{-22 - 2}{4}=\frac{-24}{4}=-6$.

Answer:

$-6$