Question

Find the difference quotient and simplify your answer.

Answer 1

We're given that

$f(x)=x^2-2x+6$

which means

$f(x+h)=(x+h)^2-2(x+h)+6=x^2+2xh+h^2-2x-2h+6$

When we subtract these, several terms cancel:

$(x^2+2xh+h^2-2x-2h+6)-(x^2-2x+6)=2xh+h^2-2h$

So the difference quotient is

$\dfrac{f(x+h)-f(x)}h=\dfrac{2xh+h^2-2h}h$

and since h ≠ 0, we can cancel it out to end up with

$\dfrac{f(x+h)-f(x)}h=2x+h-2$

Now plug in x = 8 (this could have been done at any prior point):

$\dfrac{f(8+h)-f(8)}h=14+h$