given the function $f(x)=x^{2}+3x - 2$, 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, $a=-2$, $b = 2$, and $f(x)=x^{2}+3x - 2$.

Step2: Calculate $f(a)$ and $f(b)$

First, find $f(-2)$: [ \begin{align*} f(-2)&=(-2)^{2}+3\times(-2)-2\ &=4-6 - 2\ &=-4 \end{align*} ] Then, find $f(2)$: [ \begin{align*} f(2)&=2^{2}+3\times2-2\ &=4 + 6-2\ &=8 \end{align*} ]

Step3: Calculate the average rate of change

Substitute $f(-2)=-4$, $f(2)=8$, $a=-2$, and $b = 2$ into the average - rate - of change formula: [ \begin{align*} \frac{f(2)-f(-2)}{2-(-2)}&=\frac{8-(-4)}{2 + 2}\ &=\frac{8 + 4}{4}\ &=\frac{12}{4}\ &=3 \end{align*} ]

Answer:

$3$