given the function $h(x)=-x^{2}+4x + 12$, determine the average rate of change of the function over the…
given the function $h(x)=-x^{2}+4x + 12$, determine the average rate of change of the function over the interval $-1leq xleq8$.
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, $a=-1$ and $b = 8$.
Step2: Calculate $h(-1)$
Substitute $x=-1$ into $h(x)=-x^{2}+4x + 12$: $h(-1)=-(-1)^{2}+4\times(-1)+12=-1 - 4+12=7$.
Step3: Calculate $h(8)$
Substitute $x = 8$ into $h(x)=-x^{2}+4x + 12$: $h(8)=-8^{2}+4\times8+12=-64 + 32+12=-20$.
Step4: Calculate average rate of change
Use the formula $\frac{h(b)-h(a)}{b - a}=\frac{h(8)-h(-1)}{8-(-1)}$. Substitute $h(8)=-20$ and $h(-1)=7$: $\frac{-20 - 7}{8+1}=\frac{-27}{9}=-3$.
Answer:
$-3$