question given the function h(x)=-x² + 5x + 12, determine the average rate of change of the function over…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = h(x)$ over the interval $[a,b]$ is $\frac{h(b)-h(a)}{b - a}$, where $a = 2$ and $b = 8$.

Step2: Calculate $h(2)$

Substitute $x = 2$ into $h(x)=-x^{2}+5x + 12$. $h(2)=-(2)^{2}+5\times2 + 12=-4 + 10+12=18$.

Step3: Calculate $h(8)$

Substitute $x = 8$ into $h(x)=-x^{2}+5x + 12$. $h(8)=-(8)^{2}+5\times8 + 12=-64+40 + 12=-12$.

Step4: Calculate the average rate of change

$\frac{h(8)-h(2)}{8 - 2}=\frac{-12-18}{6}=\frac{-30}{6}=-5$.

Answer:

$-5$