Parabola Given Three Points

Date: 03/27/2002 at 23:15:10
From: Samuel Burke
Subject: How do you find the equation for a parabola given three 
points on it?

Dear Dr. Math,

How can I find the equation for a parabola given three points on it? I 
have no idea where to start. Help me!

Date: 03/28/2002 at 08:54:23
From: Doctor Rick
Subject: Re: How do you find the equation for a parabola given three 
points on it?

Hi, Samuel.

This problem can be a bit tricky to understand because it involves a 
sort of mental switch in your thinking about constants and variables.

You know that the parabola must have the form

  y = ax^2 + bx + c

What you need to find out is the values of the "constants" a, b, 
and c. Because you don't know them yet, you need to treat a, b, and c 
as variables.

What DO you know? You know three points (x,y) on the parabola. Let's 
say one of these points is (2,3). Then you can plug these values into 
the general equation for the parabola:

  3 = a*2^2 + b*2 + c

  3 = 4a + 2b + c

We have made the "variables" x and y into constants. Now, if you do 
the same thing with the other two points on the parabola, you'll have 
three linear equations in three unknowns. Do you know what to do with 
them?

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