Difference Quotients

A Difference Quotient is the formula used to find the slope of the secant line of two points on a graph, or the tangent line of a single point on a graph.

Simply, for any function f(x) the difference quotient for that function is

With h being any nonzero value in the domain of the original function.

How does one form this formula?

First take a simple, curvy graph and pick any point, lets call this point (a, f(a)) with "a" being the x value of the point, and "f(a)" being the output, or y value.
Then add any value "h" to "a" and plot a point on the graph there, creating the point (a+h, f(a+h)). Your graph will look something like this.

Since you now have 2 points, and therefore a line secant to this graph, let us find this lines slope by using the formula of the slope of a line.

And there it is, proven right above, in simplest terms the difference quotient is merely the formula of the slope of a line. But you can do so much more with the difference quotient than a plain slope formula, you can manipulate it, as long as you have a knowledge of limits, to find the slope of the tangent line of any point on a graph. Although that is for next year in calculus, for now you only need the basic knowledge of how it is formed and how it could possibly be applied.

Also here are some good tips.

• If you get 1, you probably did something very wrong.
• Stuff is going to cancel each other out in the numerator, so if it doesn't happen you distributed wrong in the f(x+h)
• It's going to divide so that there is no denominator, if it doesn't you distributed wrong again
• And as always, if it just looks funky double check your work!

By Jacob D on Nick C's account because mine was being weird.