given the function ( f(x)=-x^{2}+5x + 14 ), determine the average rate of change of the function over the interval ( -2leq xleq9 ).

Question

Accepted Answer

# Explanation:
## Step1: Find $ f(-2) $
Substitute $ x = -2 $ into $ f(x)=-x^{2}+5x + 14 $.
$$
\begin{align*}
f(-2)&=-(-2)^{2}+5\times(-2)+14\
&=-4-10 + 14\
&=0
\end{align*}
$$
## Step2: Find $ f(9) $
Substitute $ x = 9 $ into $ f(x)=-x^{2}+5x + 14 $.
$$
\begin{align*}
f(9)&=-9^{2}+5\times9+14\
&=-81 + 45+14\
&=-22
\end{align*}
$$
## Step3: Calculate the average rate of change
The formula for 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 $a=-2$, $b = 9$, $f(a)=0$, $f(b)=-22$.
$$
\begin{align*}
\frac{f(9)-f(-2)}{9-(-2)}&=\frac{-22-0}{9 + 2}\
&=\frac{-22}{11}\
&=-2
\end{align*}
$$

# Answer:
$-2$

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QUESTION IMAGE

Question

Explanation:

Step1: Find \( f(-2) \)

Step2: Find \( f(9) \)

Step3: Calculate the average rate of change

Answer: