Phantom Graphs


Hey! Hi! If you are here from the video above, thanks for watching! Below are two interactive plots where you can play with general phantom graphs of 2nd- and 3rd-degree polynomials. You’ll have the best user experience on PC but if you insist on mobile, it might work if you open the graphs in a new window or a dedicated GeoGebra app. Here’s a link for the first interactive graph. And here’s a second one. If it doesn’t work as expected, I’m sorry. It’s really great on PC, I promise.

If you haven’t seen the above video, I highly recommend watching it (note I am really biased) and then playing with the graphs below. In the video I explain what phantom graphs are and why literally every graph you plotted in the past, has a hidden part (when plotted in the “phantom” way).

Interactive Phantom Plot for 2nd-Degree Polynomials

The GeoGebra applet below lets you play with a general second-degree polynomial $ax^2 + bx + c$. Move the sliders on the bottom-left to see the corresponding phantom graphs in the 3D plot on the right and their roots in the 2D plot above. The real part of the curve is pictured in blue and the phantom part in green.

Interactive Phantom Plot for 3rd-Degree Polynomials

This applet works the same as the applet above but displays the third-degree polynomial $ax^3 + bx^2 + cx +d$. Have fun!