what is the average rate of change of f(x) on the interval -3, -2. f(x)=2x² + 12x + 16 -2 2 1/2 -1/2

Answer

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-3$, $b = - 2$, and $f(x)=2x^{2}+12x + 16$.

Step2: Calculate $f(-3)$

Substitute $x=-3$ into $f(x)$: [ \begin{align*} f(-3)&=2(-3)^{2}+12(-3)+16\ &=2\times9-36 + 16\ &=18-36 + 16\ &=-2 \end{align*} ]

Step3: Calculate $f(-2)$

Substitute $x=-2$ into $f(x)$: [ \begin{align*} f(-2)&=2(-2)^{2}+12(-2)+16\ &=2\times4-24 + 16\ &=8-24 + 16\ &=0 \end{align*} ]

Step4: Calculate average rate of change

[ \begin{align*} \frac{f(-2)-f(-3)}{-2-(-3)}&=\frac{0-(-2)}{-2 + 3}\ &=\frac{2}{1}\ &=2 \end{align*} ]