Question

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.

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:

$-6$