Water, Water, Everywhere, and Parabolas to Boot

I’ve done this activity before, but never with GeoGebra, the dynamic geometry and function graphing program. So for those of you who would like to try it with that open-source (meaning, it’s completely free!) program, here are the steps I’ve found to work best.

First, you need a picture of water that is following a parabolic path. Fortunately, that happens at most drinking fountains:

Digital StillCameraNow we need to paste this into GeoGebra. You can drag it directly into the graphics pane of GeoGebra, or go to Edit/Insert Image From…File:


Now you can resize and drag the picture around any way you like. (Helpful tip: It really simplifies things if you make the water stream pass through the origin.) Hit the little Graphics toggle at the top left of the Graphics pane, and click on the grid button:


You’ll notice that you can’t see the axes or the grid behind the picture, so click on the picture to select it, then right-click on it to choose “Object Properties”. Under the Color tab, you can then change the opacity of the picture. Something around 50% usually works fine. (If you don’t see an Opacity slider, click on the pic1 label in the left pane of the Preferences window).


You can also change the color of the grid by going to Options/Advanced, select the “Preferences – Graphics” button (it has a little circle and triangle) and clicking on the Grid tab:


Now you’re ready to actually model the water’s path! Place the picture so that it passes as close as possible to a couple of grid intersections. Put three points on the water’s path using the point tool. In the Algebra pane on the left, you’ll notice that the points’ coordinates are automatically calculated.


Using your three sets of x and y-values, you can set up a system of three equations in the form of y=a{{x}^{2}}+bx+c. Students can use any method they wish to solve that system. Mine were comfortable enough with matrices and graphing calculators to use them to solve the system:


Enter the quadratic function with the calculated a, b, and c values into the Input bar at the bottom of GeoGebra. Then, see your function model the water’s path perfectly!