Constructing a quadratic equation from three given points

Given points (x1, y1), (x2, y2) and (x3, y3), three quadratic functions of the form ax2 + bx + c = y can be formed:
  • y1 = a(x1)2 + b(x1) + c
  • y2 = a(x2)2 + b(x2) + c
  • y3 = a(x33)2 + b(x3) + c
These equations can be solved simultaneously for a, b and c. The resulting values of a, b and c obtained from solving the above three equations are put in the quadratic function y = ax2 + bx + c and thus the quadratic function, whose graph passes through (x1, y1), (x2, y2) and (x3, y3), is obtained. For example, given three points (1, 1), (-2, 1) and (-1, -1), the following three equations can be formed:

  • 1 = a(1)2 + b(1) + c
  • 1 = a(-2)2 + b(-2) + c
  • -1 = a(-1)2 + b(-1) + c
on solving the above equations simultaneously, the values of a, b and c are obtained as follows:
  • a = 1
  • b = 1
  • c = -1
Substituting the values of a, b and c in the quadratic function y = ax^2 + bx +c, the function y = x^2 + x - 1 is obtained. The following graph shows the parabola of y = x^2 + x - 1 and the points (1, 1), (-2, 1) and (-1, -1) on it.


In short, the process of obtaining a quadratic equation from its three given points involves three steps:
  • Form three equations (one equation from each point) form ax2 + bx + c = 0
  • Solve the three equations simultaneously to get the values of a, b and c
  • Put the values of a, b and c in the equation ax2 + bx + c = 0 to obtain the quadratic equation

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