How to Find Equation of a Parabola

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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.

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