question given the function h(x)=-x² - 4x + 7, 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 $y = h(x)$ over the interval $[a,b]$ is $\frac{h(b)-h(a)}{b - a}$. Here, $a=-7$ and $b = 0$.

Step2: Calculate $h(-7)$

Substitute $x=-7$ into $h(x)=-x^{2}-4x + 7$. $h(-7)=-(-7)^{2}-4\times(-7)+7=-49 + 28+7=-14$.

Step3: Calculate $h(0)$

Substitute $x = 0$ into $h(x)=-x^{2}-4x + 7$. $h(0)=-(0)^{2}-4\times(0)+7=7$.

Step4: Calculate the average rate of change

Use the formula $\frac{h(b)-h(a)}{b - a}=\frac{h(0)-h(-7)}{0-(-7)}$. $\frac{7-(-14)}{7}=\frac{7 + 14}{7}=\frac{21}{7}=3$.

Answer:

$3$