what is the average rate of change of the function $g(x) = -x^2 + 3x + 1$ on the interval from $x = 2$ to $x = 6$?

Question

Accepted Answer

# Explanation:
## Step1: Recall the average rate of change formula
The average rate of change of a function $ g(x) $ on the interval $[a, b]$ is given by $\frac{g(b) - g(a)}{b - a}$. Here, $ a = 2 $, $ b = 8 $, and $ g(x)=-x^{2}+3x + 1 $.

## Step2: Calculate $ g(8) $
Substitute $ x = 8 $ into $ g(x) $:
$$
\begin{align*}
g(8)&=-(8)^{2}+3(8)+1\
&=-64 + 24+1\
&=-39
\end{align*}
$$

## Step3: Calculate $ g(2) $
Substitute $ x = 2 $ into $ g(x) $:
$$
\begin{align*}
g(2)&=-(2)^{2}+3(2)+1\
&=-4 + 6+1\
&=3
\end{align*}
$$

## Step4: Apply the average rate of change formula
Now, use $\frac{g(8)-g(2)}{8 - 2}$. Substitute $ g(8)=-39 $ and $ g(2)=3 $:
$$
\begin{align*}
\frac{g(8)-g(2)}{8 - 2}&=\frac{-39 - 3}{6}\
&=\frac{-42}{6}\
&=-7
\end{align*}
$$

# Answer:
$-7$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the average rate of change formula

Step2: Calculate \( g(8) \)

Step3: Calculate \( g(2) \)

Step4: Apply the average rate of change formula

Answer: