Question

given the function $h(x) = -x^2 - 4x + 7$, determine the average rate of change of the function over the interval $-7 \leq x \leq 0$.

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, \( a=-7 \) and \( b = 0 \).

Step2: Calculate \( h(-7) \)

Substitute \( x=-7 \) into \( h(x)=-x^{2}-4x + 7 \):
\[

$$\begin{align*} h(-7)&=-(-7)^{2}-4(-7)+7\\ &=-49 + 28+7\\ &=-14 \end{align*}$$

\]

Step3: Calculate \( h(0) \)

Substitute \( x = 0 \) into \( h(x)=-x^{2}-4x + 7 \):
\[
h(0)=-(0)^{2}-4(0)+7=7
\]

Step4: Calculate the average rate of change

Using the formula \(\frac{h(b)-h(a)}{b - a}\) with \( a=-7 \), \( b = 0 \), \( h(-7)=-14 \) and \( h(0)=7 \):
\[

$$\begin{align*} \text{Average rate of change}&=\frac{h(0)-h(-7)}{0-(-7)}\\ &=\frac{7-(-14)}{7}\\ &=\frac{21}{7}\\ &=3 \end{align*}$$

\]

Answer:

The average rate of change of the function \( h(x) \) over the interval \(-7\leq x\leq0\) is \( 3 \).