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Name: Gordon
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Question:
dealing with a parabola of the form y=ax^2+bx+c that passes through 3 given points, I need expressions for a b and c such that a b and c can be found for any 3 points. This is similar to a previous question "Finding the vertex of a parabola with given points" unfortunatley I have been unable to derive the desired expressions from the solution provided to that question.



Replies:
If you simply want the vertex, that is the place were dy/dx=0=2ax+b or x=-b/2a. Knowing a, b and c is more information than you need.

Dr. Myron


What you have, in disguise, is a problem of finding the values of three unknowns from three linear equations. Your three equations stem from the three points (x1,y1), (x2,y2), and (x3,y3).

y1 = ax1^2 + bx1 + c
y2 = ax2^2 + bx2 + c
y3 = ax3^2 + bx3 + c

In matrix notation, this is Y = Xb, where Y = (y1, y2, y3)',b = (a, b, c)', and

x1^2 x1 1
X = x2^2 x2 1
x3^2 x3 1

(X is a matrix, although I can't draw the proper brackets around it.) You need to solve this equation for b. There are several ways to do this, which I believe you are capable of doing. One is to invert the matrix X: b = X^(-1)Y, where X^(-1) is the inverse of X. (This is not the most efficient way to solve the equations, but it is the easiest for me to denote.)

Richard E. Barrans Jr., Ph.D.
Assistant Director
PG Research Foundation, Darien, Illinois


If you substitute the coordinates of the three points you have into the given equation, you will obtain three equations for the three unknowns a, b, and c. Next step is quite simple. Please consult an introductory algebra book as to how to determine these values from the three equations. Look at the heading "simultaneous equations" in the text book.

AK

Dr. Ali Khounsary
Advanced Photon Source
Argonne National Laboratory
Argonne, IL 60439



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