Ask A Scientist Mathematics Archive

Question:  This is a question pertaining to parabolas.  Suppose you are 
given three points on a parabola, for example (2,1), (4,8), and 
(-2,-1) (just random points).  How could you determine the vertex if
given this information?
 philip rust

Answer:
The general form of a parabola is given by the equation:
  A * x^2 + B * x + C = y where A, B, and C are arbitrary Real constants.
You have three pairs of points that are (x,y) ordered pairs. Substitute
the x and y values of each point into the equation for a parabola. You
will get three LINEAR equations in three unknowns, the three constants.
You can then easily solve this system of three equations for the values
of A, B, and C, and you'll have the equation of the parabola that
intersects your 3 points. The vertex is where the first derivative is 0,
a little algebra gives: ( -B/2A , C - B^2/4A ) for the vertex.
 hawley