Question

given the function $f(x)=-x^{2}-2x + 6$, determine the average rate of change of the function over the interval $-7leq xleq3$.

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-7$, $b = 3$, and $f(x)=-x^{2}-2x + 6$.

Step2: Calculate $f(-7)$

Substitute $x=-7$ into $f(x)$:
\[

$$\begin{align*} f(-7)&=-(-7)^{2}-2\times(-7)+6\\ &=-49 + 14+6\\ &=-29 \end{align*}$$

\]

Step3: Calculate $f(3)$

Substitute $x = 3$ into $f(x)$:
\[

$$\begin{align*} f(3)&=-3^{2}-2\times3+6\\ &=-9-6 + 6\\ &=-9 \end{align*}$$

\]

Step4: Calculate average rate of change

\[

$$\begin{align*} \frac{f(3)-f(-7)}{3-(-7)}&=\frac{-9-(-29)}{3 + 7}\\ &=\frac{-9 + 29}{10}\\ &=\frac{20}{10}\\ &=2 \end{align*}$$

\]

Answer:

$2$