## Difference Quotients in Intermediate Algebra

In our intermediate algebra course at College of the Sequoias, we teach our students how to simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$. I do put a generic graph on the board and show my students that this represents the rate of change for a function, or the equivalent of slope for a non-linear function. However, the goal is simply to teach them how to work with functions and to simplify the difference quotient.

This is a great example of a problem that first appears to be overwhelming to a developmental math student, but when broken into smaller steps it is much more approachable.

I have my students start by evaluating $f(x+h)$. Suppose $f(x)=3x-4$.

$f(x+h)=3(x+h)-4=3x+3h-4$

Next they take this result, subtract the original function $f(x)$, and place the difference over h.

$\frac{f(x+h)-f(x)}{h}=\frac{3x+3h-4-(3x-4)}{h}$

To finish, we simplify the numerator and then simplify the fraction. The only terms that stay in the numerator will have an h in them, and we use that to simplify the fraction. (Note: If you a calculus student, beware! This only happens when $f(x)$ is a polynomial function.)

$\frac{f(x+h)-f(x)}{h}=\frac{3x+3h-4-(3x-4)}{h} \\ =\frac{3x+3h-4-3x+4}{h} \\ =\frac{3h}{h} \\ =3$

At this point I reinforce the connection between rate of change and slope, then we move on to a quadratic function. One essential tool that comes into play with quadratic functions is that:

$(x+h)^2=(x+h)(x+h) \\ =x^2+xh+xh+h^2 \\ =x^2+2xh+h^2$

I tell my students that they can use this fact $((x+h)^2=x^2+2xh+h^2)$ repeatedly, and there’s not much to be gained by squaring x+h each time they simplify one of these difference quotients.

Here’s an example for $f(x)=x^2-3x+8$.

$f(x+h)=(x+h)^2-3(x+h)+8 \\ =x^2+2xh+h^2-3x-3h+8$

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

I tell them again to notice that the only terms that “survive” in the numerator are the terms that originally contain h.

Summary

I hope that you find this helpful, whether you are an instructor teaching this topic to your students or if you are a student learning this for yourself. If you have any questions about simplifying difference quotients or working with functions, or if you have any other topics you are interested in, you can reach me through the contact page at my web site – georgewoodbury.com.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article related to General Teaching on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com