Question

graphs and functions
finding a difference quotient for a linear or quadratic function
find the difference quotient $\frac{f(x + h)-f(x)}{h}$, where $h
eq0$, for the function below.
$f(x)=3x^{2}+8$
simplify your answer as much as possible.
$\frac{f(x + h)-f(x)}{h}=$

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$:
$f(x + h)=3(x + h)^2+8=3(x^{2}+2xh+h^{2})+8 = 3x^{2}+6xh + 3h^{2}+8$

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

$f(x + h)-f(x)=(3x^{2}+6xh + 3h^{2}+8)-(3x^{2}+8)=6xh+3h^{2}$

Step3: Find the difference - quotient

$\frac{f(x + h)-f(x)}{h}=\frac{6xh + 3h^{2}}{h}=\frac{h(6x + 3h)}{h}=6x+3h$

Answer:

$6x + 3h$