# Consolidating three descriptions of a parabola in precalculus

I want to present these three descriptions of a parabolic curve to my precalculus class:

1. The graph of a quadratic function $f(x) = ax^2+bx+c$.

2. Given a line called the directrix and a point called the focus, the set of all points in the plane that are equidistant to the directrix and focus.

3. The intersection of a cone with a plane that is parallel to a tangent-plane to the cone.

Showing students that the first and second descriptions actually give us the same curve isn't too bad: just draw a focus and directrix in the plane, take an arbitrary point that is equidistant to them, and show that the $y$-coordinate is quadratic in the $x$-coordinate.

But showing that these first two descriptions are the same as the third seems quite a bit harder. Does anyone know an intuitive way to show that the conic section description of a parabola gives us the same curve as the other two descriptions? I really want something brief but satisfying (so not necessarily thorough) so that my students don't just tune out. But every method I've seen so far has been a bit too involved to present to a precalculus class (like here), and I don't see an easy way to prune out some details of these methods.

• have you tried to find any animations or videos that can help to visualize what is going on with the conic? I have found that for issues like this a quick 20 second animation can take the place of a 30 minute lesson. – celeriko Dec 20 '16 at 18:52
• @celeriko No I haven't. That might be a good idea. I'm worried that a video would be too long of an explanation in my particular case though (I'm a TA doing an additional lecture section, and I usually have to pack the class with material). And I shouldn't ask them to watch a video for homework; none of them will. – Mike Pierce Dec 20 '16 at 19:11
• understood, however does not need to be a long video, you can definitely find really nice videos/animations that are less than 5 minutes in length. Might be worth a shot, I totally get having to pack in too much material for the time period though, its tough. – celeriko Dec 20 '16 at 19:41

This 5-minute video from Creative Learning is effective: Conic Section 3D Animation.

But it might be too long (one of your worries). In which case, I would use the Wolfram Demo Conic Sections: The Double Cone, in which you interactively manipulate the plane and rotate the image in 3D, and can spend as little or as much time as you'd like:

• The video explains the conic-section definition of a parabola, but doesn't explain how it's gives the same curve as the other two descriptions. It's clearly gives us a "U-shaped" curve, but how do we know it's the same U-shaped curve as the graph of a quadratic polynomial? I keep looking at this explanation, but I'm not sure how to present that to my class in a less complicated way. – Mike Pierce Dec 20 '16 at 20:21
• @MikePierce: Good point. If you don't want to derive it in detail, perhaps just this high-level view suffices: The implicit equation of a cone in 3D, in terms of $x,y,z$ is quadratic in those three variables. Intersecting with a plane leaves an implicit quadratic equation in $x,y,z$. Quadratic equations lead to parabolas, ellipses, hyperbolas. – Joseph O'Rourke Dec 20 '16 at 20:31
• I like that idea. Maybe write down the equation of the double-cone in $3$-space, convince them that the equation does correspond to that cone, and then write down the equation of a "tilted cone" (which I think they'll just accept) so the plane we intersect with we can just think of the $x,y$-plane. Then just notice that if we let $z = 0$ the result is a quadratic. – Mike Pierce Dec 20 '16 at 20:50
• @MikePierce: Yes. The 3D cone is $x^2 + y^2 = (r/h)^2 z^2$ under certain assumptions ($r/h$ radius/height ratio, apex @origin). Nice idea to tilt. – Joseph O'Rourke Dec 20 '16 at 21:05