given $f(x) = x^2 + 7x$, find the average rate of change of $f(x)$ on the interval $-4, -4 + h$. your answer will be an expression involving $h$.

Question

Accepted Answer

# Explanation:
## Step1: Recall the formula for average rate of change
The average rate of change of a function $ f(x) $ on the interval $[a, b]$ is given by $\frac{f(b) - f(a)}{b - a}$. Here, $ a=-4 $ and $ b = -4 + h $.

## Step2: Find $ f(-4) $
Substitute $ x=-4 $ into $ f(x)=x^{2}+7x $:
$ f(-4)=(-4)^{2}+7\times(-4)=16 - 28=-12 $

## Step3: Find $ f(-4 + h) $
Substitute $ x = -4+h $ into $ f(x)=x^{2}+7x $:
$$
\begin{align*}
f(-4 + h)&=(-4 + h)^{2}+7\times(-4 + h)\
&=16-8h+h^{2}-28 + 7h\
&=h^{2}-h - 12
\end{align*}
$$

## Step4: Calculate the average rate of change
Using the formula $\frac{f(b)-f(a)}{b - a}$, we have:
$$
\begin{align*}
\frac{f(-4 + h)-f(-4)}{(-4 + h)-(-4)}&=\frac{(h^{2}-h - 12)-(-12)}{h}\
&=\frac{h^{2}-h - 12 + 12}{h}\
&=\frac{h^{2}-h}{h}\
&=\frac{h(h - 1)}{h}\
&=h - 1\quad (h
eq0)
\end{align*}
$$

# Answer:
$ h - 1 $

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for average rate of change

Step2: Find \( f(-4) \)

Step3: Find \( f(-4 + h) \)

Step4: Calculate the average rate of change

Answer: