Question

find the difference quotient of f; that is, find $\frac{f(x + h)-f(x)}{h}$, h ≠ 0, for the following function. be sure to simplify.

$f(x)=x^{2}-4x + 6$

$\frac{f(x + h)-f(x)}{h}=square$ (simplify your answer.)

Explanation:

Step1: Find f(x + h)

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

$$\begin{align*} f(x + h)&=(x + h)^2-4(x + h)+6\\ &=x^{2}+2xh+h^{2}-4x-4h + 6 \end{align*}$$

\]

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

\[

$$\begin{align*} f(x + h)-f(x)&=(x^{2}+2xh+h^{2}-4x-4h + 6)-(x^{2}-4x + 6)\\ &=x^{2}+2xh+h^{2}-4x-4h + 6-x^{2}+4x - 6\\ &=2xh+h^{2}-4h \end{align*}$$

\]

Step3: Find the difference quotient

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}-4h}{h}\\ &=\frac{h(2x + h-4)}{h}\\ &=2x+h - 4 \end{align*}$$

\]

Answer:

$2x+h - 4$