How to Find Equation of a Parabola

By Tricia Lobo; Updated April 24, 2017
This red parabola corresponds to a second-degree polynomial equation.

Frequently, in Algebra II and upper-level math classes, you will be given the graph of a parabola and asked to find its equation. Parabolas are graphs described by the equation y = ax^2 + bx + c, in which a, b, and c are real-number coefficients. Alternatively, you can describe a parabola with the equation y = a(x - h)^2 + k, in which the vertex is the point (h, k) and "a" is a real-number coefficient. You can use these two equations, together with the graph of the parabola, to come up with the equation of the parabola.

Determine, from the graph, what the coordinates of the parabola's vertex are. The vertex is the lowest point on a parabola that opens upward.

Plug the vertex coordinates into the parabola vertex formula, y = a(x - h)^2 + k. If the vertex is at (1, 1), this equation becomes y = a(x - 1)^2 + 1.

Find any other point on the parabola, and plug it into your equation in Step 2. If (3, 9) is a point, plugging it in yields 9 = a(3 - 1)^2 + 1.

Solve the equation in Step 3 for a. The equation, simplified, becomes 9 = a*4 + 1, or 8 = 4a, so a = 2.

Plug your value for "a" into the equation in Step 2, to obtain y = 2(x - 1)^2 + 1. You can simplify this equation, if you wish, to give the more standard parabola form. Simplified, the equation becomes y = 2(x^2 - 2x + 1) + 1, or y = 2x^2 - 4x + 3.

About the Author

Tricia Lobo has been writing since 2006. Her biomedical engineering research, "Biocompatible and pH sensitive PLGA encapsulated MnO nanocrystals for molecular and cellular MRI," was accepted in 2010 for publication in the journal "Nanoletters." Lobo earned her Bachelor of Science in biomedical engineering, with distinction, from Yale in 2010.