given the function $f(x)=x^{2}+4x + 1$, determine the average rate of change of the function over the interval $-8leq xleq4$.

Question

Accepted Answer

# Explanation:
## Step1: Recall the formula for average rate of change
The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here $a=-8$ and $b = 4$.
## Step2: Calculate $f(-8)$
Substitute $x=-8$ into $f(x)=x^{2}+4x + 1$.
$$
\begin{align*}
f(-8)&=(-8)^{2}+4\times(-8)+1\
&=64-32 + 1\
&=33
\end{align*}
$$
## Step3: Calculate $f(4)$
Substitute $x = 4$ into $f(x)=x^{2}+4x + 1$.
$$
\begin{align*}
f(4)&=4^{2}+4\times4+1\
&=16 + 16+1\
&=33
\end{align*}
$$
## Step4: Calculate the average rate of change
$$
\begin{align*}
\frac{f(4)-f(-8)}{4-(-8)}&=\frac{33 - 33}{4 + 8}\
&=\frac{0}{12}\
&=0
\end{align*}
$$

# Answer:
$0$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for average rate of change

Step2: Calculate \(f(-8)\)

Step3: Calculate \(f(4)\)

Step4: Calculate the average rate of change

Answer: