find the average rate of change of f(x)=-x^2 + 2x + 2 from x = 2 to x = 6. simplify your answer as much as possible.

Question

Accepted Answer

# 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 given by $\frac{f(b)-f(a)}{b - a}$. Here, $a = 2$, $b = 6$, and $f(x)=-x^{2}+2x + 2$.
## Step2: Calculate $f(2)$
Substitute $x = 2$ into $f(x)$:
$$
\begin{align*}
f(2)&=-(2)^{2}+2\times2 + 2\
&=-4 + 4+2\
&=2
\end{align*}
$$
## Step3: Calculate $f(6)$
Substitute $x = 6$ into $f(x)$:
$$
\begin{align*}
f(6)&=-(6)^{2}+2\times6 + 2\
&=-36+12 + 2\
&=-22
\end{align*}
$$
## Step4: Calculate the average rate of change
$$
\begin{align*}
\frac{f(6)-f(2)}{6 - 2}&=\frac{-22 - 2}{4}\
&=\frac{-24}{4}\
&=-6
\end{align*}
$$
# Answer:
$-6$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall average rate - of - change formula

Step2: Calculate $f(2)$

Step3: Calculate $f(6)$

Step4: Calculate the average rate of change

Answer: