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Non-Algebraic Explanation of a Parabola


Date: 05/18/2000 at 09:09:25
From: Donelle
Subject: Parabola

I need to know specific details about a Parabola in more simple words. 
They can be math words but more simple, please.

Thank you.


Date: 05/18/2000 at 12:05:24
From: Doctor Peterson
Subject: Re: Parabola

Hi, Donelle.

Okay, I'll give it a try. I love putting math into simple terms, but 
it's very hard to do that when I don't know what you need. The more 
you can tell me about your needs, the better I can help.

First, let me point you to a page in our archives that I think you 
might find useful as a starter, since it's written for a younger 
student and avoids algebra:

   An Introduction to Parabolas
   http://mathforum.org/dr.math/problems/hayley2.4.99.html   

A parabola can be defined in several ways. First, it is a conic 
section; if you take a cone and cut a slice off parallel to its slope, 
like this, the curve you get will be a parabola:

                          /\
                       /\/  \cutting plane
                      / /\   \
                     / /  \   \
                    / /    \   \
                   / /      \   \
                  / /      * *   \
                 / /     *  /*\   \
                / /    *   / * \   \
     parabola-------->*   /  *  \  /
              / /       s/  *    \/
             / /    *  i/        /\
            / /       x/   *    /  \
           /  \   *  a/        /    \cone
          /    \     /   *    /      \
         /    ++*+++/++++++++/++++    \
        /+++++   \ /   *    /     +++++\
       +          +   *    /            +
        ++++++     \ *    /       ++++++
              ++++++*++++/++++++++
                     \  /
                      \/

Second, it can be defined in terms of the "focus" and "directrix." We 
choose a point in a plane that will be the focus, and a line (usually 
parallel to the x- or y-axis, but it could be any line) for the 
directrix. The parabola is the set of all points that are the same 
distance from the focus as from the directrix. Since the distance from 
a point to a line is the distance along the perpendicular, that looks 
like this:


    *                                           *


                        Focus
           *              +              *
                             \ a
                                \
                  *               * P
                          *       |
                                  |b
                                  |
   -------------------------------+---------------
          Directrix

The distance a from the focus to point P is the same as the distance b 
from point P to the directrix, so P is on the parabola.

If you want a better picture, look at the page Dr. Rick referred to in 
his archived answer.

   Dave's Math Tables: Conic Sections - David Manura
   http://math2.org/math/algebra/conics.htm      

Now, you can use this definition to derive the equation of a parabola. 
I won't get into that, because once we do that, I can't avoid digging 
into algebra. If you want to know how to write the equation, or how to 
find the focus and directrix from the equation, and so on, please 
write back and tell me what you need to know.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Conic Sections/Circles
High School Equations, Graphs, Translations
High School Geometry

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