find the average rate of change of the function f(x)= - 1x² + 6x + 2, from x=-3 to x=0. note, the directions…

Explanation:

Step1: Recall the average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $a=-3$, $b = 0$, and $f(x)=-x^{2}+6x + 2$.

Step2: Calculate $f(-3)$

Substitute $x=-3$ into $f(x)$: [ \begin{align*} f(-3)&=-(-3)^{2}+6\times(-3)+2\ &=-9-18 + 2\ &=-25 \end{align*} ]

Step3: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$: [ \begin{align*} f(0)&=-(0)^{2}+6\times(0)+2\ &=2 \end{align*} ]

Step4: Calculate the average rate of change

[ \begin{align*} \frac{f(0)-f(-3)}{0-(-3)}&=\frac{2-(-25)}{3}\ &=\frac{2 + 25}{3}\ &=\frac{27}{3}\ &=9 \end{align*} ]

Answer:

9