Question

given the function $h(x)=-x^{2}+6x + 11$, determine the average rate of change of the function over the interval $-3leq xleq6$.

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 $a=-3$ and $b = 6$.

Step2: Calculate $h(a)$

Substitute $x=-3$ into $h(x)=-x^{2}+6x + 11$.
$h(-3)=-(-3)^{2}+6\times(-3)+11=-9-18 + 11=-16$.

Step3: Calculate $h(b)$

Substitute $x = 6$ into $h(x)=-x^{2}+6x + 11$.
$h(6)=-6^{2}+6\times6+11=-36 + 36+11=11$.

Step4: Calculate the average rate of change

Substitute $h(-3)=-16$ and $h(6)=11$ into the average - rate - of - change formula.
$\frac{h(6)-h(-3)}{6-(-3)}=\frac{11-(-16)}{6 + 3}=\frac{11 + 16}{9}=\frac{27}{9}=3$.

Answer:

$3$