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

Question

Accepted Answer

# 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:
$2$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall average - rate - of - change formula

Step2: Calculate $f(-7)$

Step3: Calculate $f(3)$

Step4: Calculate average rate of change

Answer: