3.1 Functions, 3.1 Relations and Functions, Difference Quotient

Difference Quotient: Quadratic Function

Example:  Find the difference quotient for f(x)=x^2-9x

The Difference Quotient:{f(x+h)-f(x)}/h

Solution:

The Difference Quotient Formula
{f(x+h)-f(x)}/h
Write the difference quotient for the given function
={(x+h)^2-9(x+h)-(x^2-9x)}/h
Apply the exponent and use the distributive property
={(x+h)(x+h)-9x-9h-x^2+9x}/h
Multiply
={x^2+xh+xh+h^2-9x-9h-x^2+9x}/h
Combine the like terms
={x^2+2xh+h^2-9x-9h-x^2+9x}/h
Combine the like terms.  Only terms with h should remain
={2xh+h^2-9h}/h
Divide h into each term
={2xh}/h+{h^2}/h-{9h}/h
Cancel the common h from each term
=2x+h-9