Parabola Magic

I worked on more calculus exercises today, then did some experimenting with graphs using Desmos. What I discovered is that adding a line to a parabola will always produce a congruent parabola. So far, I haven’t been able to find any other type of curve that has this property. For all other curves, adding a line also adds “tiltiness” to the graph, so that the result is not congruent to the original.

Here is an example using a parabola. The original parabola appears in green, the line in red, and their sum in blue. Click here for an interactive version of this figure.

We know that these parabolas are congruent because the coefficient of the $x^2$ term does not change. (It is called $a$ in the interactive graph.) This coefficient is equal to $\frac{1}{4p}$, where $p$ is the distance of the vertex of the parabola from both the focus and the directrix. It is this distance that determines the curvature of the parabola.

(To see that $a$ does equal $\frac{1}{4p}$, try converting the general equation for a parabola $(x-h)^2=4p(y-k)$ into the general equation $y=ax^2+bx+c$. That’s another thing I did today.)

Below is a gallery of some of the other curves I tried. Again, the original graphs are in green, the line in red, and their sum in blue. You can see how each result has greater tiltiness than the original in one way or another.

I haven’t yet been able to explain why parabolas have this congruence property while other types of curve do not nor tried to understand what other implications it may have, but I find it magical.¹

1. To do. ↩︎