I am a big fan of empowering students to look for and make use of structure in Algebra 2. This is most true for me as we work with functions, parabolas, and quadratics. I’m writing this post about what I’m finding to be an indispensable tool for helping students quickly learn about the structure of the equations of quadratic functions. This tool is easy to use. Simply project the Desmos calculator (use the links below) and activate the sliders.

One of the many great things about Desmos is some of their built in functions on the calculator. Like this one, using vertex form of a parabola:

Link 1: Vertex Form of a Quadratic Function

In this window, you can activate the sliders* individually to demonstrate to students (and share with your math team) how **a**, **h**, and **k** affect the parabola. You may want to stop the slider and manually slide **a** to values you want to emphasize with the class (a = 2, 1, 1/2, 1/3, 0, -1/3, -1/2, -1, -2 for instance).

**To activate the sliders, click on the arrow buttons in rows 2, 3 and 4. To stop them, click again, or manually move the slider to any spot.*

Next, move to standard form, which is really interesting.

Link 2: Standard Form of a Quadratic Function

I suggest you first let **c** slide and have students watch as the parabola moves up and down. Ask them whether the shape is changing. Some will think it is, but it’s just an optical illusion. Tell them to look again.

Then, stop **c** and let **a** slide. Kids can see how the parabola stretches, shrinks and reflects just as it did with vertex form.

Last, the fun one. Ask them to **predict** and then tell their partner/group what they think will happen when **b** slides. Will the shape change? Will it move up, down, left, right? Then, activate the slider.

This is where the math just gets cool. Ask them, as they watch the motion, “What is the path of the vertex?” (it travels along a parabolic path); “What is happening to shape of the graph?” (nothing, it stays the same); and, “What is happening to the y-intercept?’ (the parabola travels through the point (0, c) and the intercept doesn’t change).

I found this to be so helpful to me as a teacher and to students to see quickly what the **structure of these equations** do. To get to them and many others, just click on the bars at the top left corner of the window for the desmos calculator. There are all kinds of great functions to work with. Here’s a picture I made in paint – screen shot, save in paint, edit with brush – to help you find the drop down menu.

P.S. I need to create a note sheet for this where they summarize these structures and the impact of the key components. Next week. Yep, next week. 🙂