An Introduction to Parabolas

Date: 02/04/99 at 22:31:54
From: Hayley
Subject: Parabolas

What is a parabola? I have heard my brother, who is in high school, 
use it. I need your help!

Sincerely,
Hayley

Date: 02/05/99 at 12:59:24
From: Doctor Rick
Subject: Re: Parabolas

Hi, Hayley. Thanks for your question!

I did a Web search and here is one nice page I found:

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

A parabola is a curve of a particular shape. You have seen parabolas 
when you watch a stream of water from a hose or fountain, starting 
upward, curving as it nears the peak, and straightening out somewhat 
as it heads back down. It's the path followed by any thrown object, 
but it's easiest to see with water. The path is called a "parabolic 
trajectory."

You can understand some things about a parabola before you learn 
algebra. Parabolas were known long before algebra was invented. The 
Greek mathematician Apollonius wrote a book called "Conic Sections" 
that studied parabolas and related curves. A conic section is what you 
get if you take a cone and slice it with a plane. Look at the Web page 
above for some nice illustrations of this. If the plane is parallel to 
the side of the cone, you get a parabola. If the plane is in other 
directions, you get a circle, an ELLIPSE, or a HYPERBOLA.

Parabolas have some interesting properties. If you draw a line and a 
point somewhere off the line, a parabola is the set of all the points 
in the plane that are the same distance from the point and the line. 
(Distances from a point to a line are measured perpendicular to the 
line.)

      .                                       .       
       .                                     .        
        .                                   .         
         .                                 .
           .              *---------------.            
             .            |\            . |
                .         |   \      .    |
                   .      |      \.       |
                          .       |       |
                          |       |       |
                          |       |       |
                          |       |       |
------------------------------------------------------------

That * in the picture, the point you chose, is called the FOCUS of the
parabola. The line is called the DIRECTRIX. 

If you make a mirror shaped like a parabola (a PARABOLIC MIRROR) and 
put a lightbulb at the focus, the light that reflects from the mirror 
will all be shining in the same direction - which is why flashlights 
have parabolic mirrors. You can run the same thing backward: light (or 
radio waves) coming toward a parabolic mirror from far away will all 
reflect in toward the focus - the light or radio waves are FOCUSED. 
Reflecting telescopes take advantage of this property of parabolas, 
and so do parabolic dish antennas for satellite TV and such.

Parabolas are all over the place, and as you can see, they are 
important. But algebra makes it a lot easier to study and understand 
them. The equations that describe parabolas are called "quadratic 
equations." You'll learn a lot more about this when you get to algebra.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/