Question

find the difference quotient of f; that is, find $\frac{f(x + h)-f(x)}{h}$, h $
eq$ 0, for the following function. be sure to simplify.
f(x) = $x^{2}-9x + 6$
$\frac{f(x + h)-f(x)}{h}=square$ (simplify your answer.)

Explanation:

Step1: Find f(x + h)

Substitute \(x+h\) into \(f(x)\):
\[

$$\begin{align*} f(x + h)&=(x + h)^2-9(x + h)+6\\ &=x^{2}+2xh+h^{2}-9x-9h + 6 \end{align*}$$

\]

Step2: Calculate f(x + h)-f(x)

\[

$$\begin{align*} f(x + h)-f(x)&=(x^{2}+2xh+h^{2}-9x-9h + 6)-(x^{2}-9x + 6)\\ &=x^{2}+2xh+h^{2}-9x-9h + 6-x^{2}+9x - 6\\ &=2xh+h^{2}-9h \end{align*}$$

\]

Step3: Find the difference - quotient

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}-9h}{h}\\ &=\frac{h(2x + h-9)}{h}\\ &=2x+h - 9 \end{align*}$$

\]

Answer:

\(2x+h - 9\)