given the function ( f(x) = x^2 - 9x + 16 ), determine the average rate of change of the function over the interval ( -1 leq x leq 9 ).

Question

Accepted Answer

# Explanation:
## Step1: Recall the formula for average rate of change
The average rate of change of a function $ f(x) $ over the interval $[a, b]$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $ a=-1 $ and $ b = 9 $.

## Step2: Calculate $ f(-1) $
Substitute $ x=-1 $ into $ f(x)=x^{2}-9x + 16 $:
$$
\begin{align*}
f(-1)&=(-1)^{2}-9(-1)+16\
&=1 + 9+16\
&=26
\end{align*}
$$

## Step3: Calculate $ f(9) $
Substitute $ x = 9 $ into $ f(x)=x^{2}-9x + 16 $:
$$
\begin{align*}
f(9)&=9^{2}-9\times9+16\
&=81-81 + 16\
&=16
\end{align*}
$$

## Step4: Calculate the average rate of change
Using the formula $\frac{f(b)-f(a)}{b - a}$ with $ a=-1 $, $ b = 9 $, $ f(-1)=26 $ and $ f(9)=16 $:
$$
\begin{align*}
\text{Average rate of change}&=\frac{f(9)-f(-1)}{9-(-1)}\
&=\frac{16 - 26}{9 + 1}\
&=\frac{-10}{10}\
&=- 1
\end{align*}
$$

# Answer:
$-1$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for average rate of change

Step2: Calculate \( f(-1) \)

Step3: Calculate \( f(9) \)

Step4: Calculate the average rate of change

Answer: