question given the function f(x)=x² + 4x + 1, determine the average rate of change of the function over the…

Explanation:

Step1: Recall average - rate - of - change formula

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, $a=-8$, $b = 4$, and $f(x)=x^{2}+4x + 1$.

Step2: Calculate $f(a)$

Substitute $x=-8$ into $f(x)$: $f(-8)=(-8)^{2}+4\times(-8)+1=64-32 + 1=33$.

Step3: Calculate $f(b)$

Substitute $x = 4$ into $f(x)$: $f(4)=4^{2}+4\times4+1=16 + 16+1=33$.

Step4: Calculate the average rate of change

$\frac{f(4)-f(-8)}{4-(-8)}=\frac{33 - 33}{4 + 8}=\frac{0}{12}=0$.

Answer:

$0$