find the average rate of change of the function f(x)= - 1x² + 2x + 8, from x = 0 to x = 3. note, 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 given by $\frac{f(b)-f(a)}{b - a}$. Here, $a = 0$, $b = 3$, and $f(x)=-x^{2}+2x + 8$.

Step2: Calculate $f(3)$

Substitute $x = 3$ into $f(x)$: $f(3)=-(3)^{2}+2\times3 + 8=-9 + 6+8=5$.

Step3: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$: $f(0)=-(0)^{2}+2\times0 + 8=8$.

Step4: Calculate the average rate of change

Using the formula $\frac{f(3)-f(0)}{3 - 0}$, we substitute $f(3)=5$ and $f(0)=8$: $\frac{5 - 8}{3}=\frac{-3}{3}=-1$.

Answer:

$-1$