Question

find the difference quotient of f; that is, find $\frac{f(x + h)-f(x)}{h}$, h $
eq$ 0, for the following function. f(x)=$x^{2}-7x + 3$. $\frac{f(x + h)-f(x)}{h}=square$ (simplify your answer.)

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$:
$f(x + h)=(x + h)^2-7(x + h)+3=x^{2}+2xh+h^{2}-7x - 7h+3$

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

$f(x + h)-f(x)=(x^{2}+2xh+h^{2}-7x - 7h+3)-(x^{2}-7x + 3)$
$=x^{2}+2xh+h^{2}-7x - 7h+3 - x^{2}+7x - 3=2xh+h^{2}-7h$

Step3: Find the difference quotient

$\frac{f(x + h)-f(x)}{h}=\frac{2xh+h^{2}-7h}{h}$
Factor out $h$ from the numerator: $\frac{h(2x + h-7)}{h}$
Cancel out $h$ (since $h
eq0$): $2x+h - 7$

Answer:

$2x+h - 7$