Last modified on November 18, 2013
Write-Up
For a write up on the following explorations, I am asking you to pull together and synthisize a number of points that are relevant to understanding parabolas and their construction. I do not apologize for asking you to move away from the traditional curriculum approach of begining with the standard form quadratic function of the parabola
and arriving at all other stuff we want to know about parabolas (usually, this approach is falls far short of any comprehensive overview of the parabola.) Rather, I am looking for a presentation where the standard form of the quadratic function parabola is the end product rather than the beginning.
In what follows, I have made nine suggestions. It is entirely possible that you might want to incorporate other items since this is in no way "comprehensive."
1. One definition of a parabola is the locus of points equidistant from a line (called a directrix) and a point (called a focus). Use this definition to generate an equation.
Define or describe the axis of symmetry of a parabola,
If we take the axis of symmetry be the perpendicular line to the directrix through the focal point and let the distance p in the positive direction be from the vertex to the focal point, then the distance to the directrix from the vertex is negative, or -p
By choosing to show the graph of the parabola in the xy plane with the vertex at (0,0) and the focal point at (0, p) then the directrix is the line y = -p.
Write the equation, square both sides and simplify to get a very basic equation for the parabola.
2.
3. Show that all parabolas are similar. Discuss how to understand his phenomenon in the face of animations and technology that, for many people, is very counter-intuitive.
4. Generalize the equation if the vertex is at (h, k) rather than at (0,0). Discuss the value of a.
5. Rewrite your generalized equation from 4 as
. Discuss the value of a. Express b and c in terms of a, h, and k.
6. Explain why this is a Conic Section and build a case for going from that geometric definition of the parabola to one with any of theses equation.
7. Find an equation for a parabola who axis of symmetry is NOT parallel to either the x-axis or the4 y-axis.
8. Derive an equation of the parabola in terms of POLAR COORDINATES.
9. Explain this:
