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.
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.
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)',
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.
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.
Dr. Ali Khounsary
Advanced Photon Source
Argonne National Laboratory
Argonne, IL 60439
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Update: June 2012