The following is an idea in progress. It is inspired by Nate Burchell's post on the Sine of the Times blog titled "Create Parametric Curves Graphically and Kinesthetically", in which he talks about "tactile functions". With a tactile function students manipulate graphs directly instead of using equations. In the Geogebra applet below we can explore this option for learning about cubic functions.
To manipulate the cubic function above you can drag any of the four given points. This applet is embedded into the blogpost, so you can try it now.
My hope is that students can use this applet to focus on some of the conceptual ideas behind cubic functions instead of the procedural emphasis that is sometimes the focus of the textbook. Some questions/explorations that we might ask students to think about during this unit are:
1. Create a cubic function with roots of -4, -1, 2 and y-intercept of -2.
2. Create a cubic function with a factor of (x-2), two negative x-intercepts, and y-intercept of -4
3. Create a cubic function with a repeated root at x=2, a factor of (x+3) and also has y values that approach infinity as x approaches infinity (Note: I'd use notation with students, but not so easy on blogger.)
4. Create a cubic function with a turning point at (4,2) and x-intercept at 3. What else must be true about this cubic?
Some of these questions may seem too simple without the algebra involved, but the truth is that our textbooks often rush straight to the algebra without giving time for students to develop their understanding of the concept. Playing with the graphs first might be a helpful way to frontload the unit on cubic functions. You can access this applet along with similar applets for quadratic and quartic functions here. Use the arrows at the top of the page next to the numbers 1,2,3 to navigate between applets.
A goal of mine when using Geogebra is to make sure that it is in some way an enhancement over what one can accomplish via the textbook and/or with pencil/paper. In this case the applet has a lot to offer. As students think through the ideas they can quickly make corrections to their graph without the hassle of erasing. There is also a built in starting point in the case that a student doesn't remember what a cubic function is.
What are some other ways we could use this applet?
Links you may find useful:
Geogebratube Book for Playing with Quadratics, Cubics, and Quartics
Some Things I Wish I Knew When I Started Using Geogebra.
Transformations and Geogebra
Practice with Transformations