Parabolas are a central topic in high school algebra classes, but, perhaps because of the rigid separation between algebra and geometry classes in the US secondary curriculum, we do not usually treat them as geometric objects. While most teachers are aware of some of the parabola’s geometric properties, few of us are familiar with the proofs of those properties.
I have long had a page on my Web site about the basic geometry of the parabola:
- Geometric Definition
- Reflection Property
- All Parabolas are Similar
This last point clashes with the idea that some parabolas are “pointier”. My friend and colleague Rachel Chou suggests that it is much easier to see this by shading a finite “cup” on each parabola, and she sent me a Geometer’s Sketchpad file to make her point. This led me to revise the final GeoGebra figure on my Geometry of the Parabola page:
I agree: this does help. Go there, read all about it, and download the .gsp file, or the .ggb file.
PS: See also an illustrated proof that a parabola is indeed a conic section.