question
given the function $g(x) = -x^2 - 9x + 24$, determine the average rate of change of the function over the interval $-9 \leq x \leq -2$.

Question

Accepted Answer

# Explanation:
## Step1: Recall the formula for average rate of change
The average rate of change of a function $ g(x) $ over the interval $[a, b]$ is given by $\frac{g(b)-g(a)}{b - a}$. Here, $ a=-9 $ and $ b = - 2$.

## Step2: Calculate $ g(-9) $
Substitute $ x=-9 $ into $ g(x)=-x^{2}-9x + 24 $:
$$
\begin{align*}
g(-9)&=-(-9)^{2}-9\times(-9)+24\
&=-81 + 81+24\
&=24
\end{align*}
$$

## Step3: Calculate $ g(-2) $
Substitute $ x = - 2$ into $ g(x)=-x^{2}-9x + 24 $:
$$
\begin{align*}
g(-2)&=-(-2)^{2}-9\times(-2)+24\
&=-4 + 18+24\
&=38
\end{align*}
$$

## Step4: Calculate the average rate of change
Using the formula $\frac{g(b)-g(a)}{b - a}$ with $ a=-9 $, $ b=-2 $, $ g(a) = 24$ and $ g(b)=38$:
$$
\begin{align*}
\text{Average rate of change}&=\frac{g(-2)-g(-9)}{-2-(-9)}\
&=\frac{38 - 24}{-2 + 9}\
&=\frac{14}{7}\
&=2
\end{align*}
$$

# Answer:
The average rate of change is $ 2 $.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for average rate of change

Step2: Calculate \( g(-9) \)

Step3: Calculate \( g(-2) \)

Step4: Calculate the average rate of change

Answer: