given the function h(x)= -x^2 - 4x + 7, determine the average rate of change of the function over the…
given the function h(x)= -x^2 - 4x + 7, determine the average rate of change of the function over the interval -7 ≤ x ≤ 0.
Answer
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 $h(a)$ and $h(b)$
First, find $h(-7)$: [ \begin{align*} h(-7)&=-(-7)^{2}-4\times(-7)+7\ &=-49 + 28+7\ &=-14 \end{align*} ] Then, find $h(0)$: [ \begin{align*} h(0)&=-(0)^{2}-4\times(0)+7\ &=7 \end{align*} ]
Step3: Calculate the average rate of change
Substitute $h(-7)=-14$ and $h(0) = 7$ into the average - rate - of change formula: [ \begin{align*} \frac{h(0)-h(-7)}{0-(-7)}&=\frac{7-(-14)}{7}\ &=\frac{7 + 14}{7}\ &=\frac{21}{7}\ &=3 \end{align*} ]
Answer:
3