Question

given the function $h(x) = x^2 + 9x + 14$, determine the average rate of change of the function over the interval $-8 \leq x \leq 0$.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( h(x) \) over the interval \([a, b]\) is given by \(\frac{h(b) - h(a)}{b - a}\). Here, \( a=-8 \) and \( b = 0 \).

Step2: Calculate \( h(-8) \)

Substitute \( x=-8 \) into \( h(x)=x^{2}+9x + 14 \):
\[

$$\begin{align*} h(-8)&=(-8)^{2}+9\times(-8)+14\\ &=64-72 + 14\\ &=64 + 14-72\\ &=78-72\\ &=6 \end{align*}$$

\]

Step3: Calculate \( h(0) \)

Substitute \( x = 0 \) into \( h(x)=x^{2}+9x + 14 \):
\[
h(0)=0^{2}+9\times0+14=14
\]

Step4: Calculate the average rate of change

Using the formula \(\frac{h(b)-h(a)}{b - a}\) with \( a=-8 \), \( b = 0 \), \( h(-8)=6 \) and \( h(0)=14 \):
\[
\frac{h(0)-h(-8)}{0-(-8)}=\frac{14 - 6}{0 + 8}=\frac{8}{8}=1
\]

Answer:

1