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### 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

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

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