Question

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 ).

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:

\(-1\)