Just as a quadratic equation can map a parabola, the parabola's points can help write a corresponding quadratic equation. Parabolas have two equation forms – standard and vertex. In the vertex form, y = a(x - h)2 + k, the variables h and k are the coordinates of the parabola's vertex. In the standard form, y = ax2 + bx + c, a parabolic equation resembles a classic quadratic equation. With just two of the parabola's points, its vertex and one other, you can find a parabolic equation's vertex and standard forms and write the parabola algebraically.
Substitute in Coordinates for the Vertex
Substitute in Coordinates for the Point
Solve for a
Convert to Standard Form
Set either form to zero and solve the equation to find the points where the parabola crosses the x-axis.
Substitute the vertex's coordinates for h and k in the vertex form. For an example, let the vertex be (2, 3). Substituting 2 for h and 3 for k into y = a(x - h)2 + k results in y = a(x - 2)2 + 3.
Substitute the point's coordinates for x and y in the equation. In this example, let the point be (3, 8). Substituting 3 for x and 8 for y in y = a(x - 2)2 + 3 results in 8 = a(3 - 2)2 + 3 or 8 = a(1)2 + 3, which is 8 = a + 3.
Solve the equation for a. In this example, solving for a results in 8 - 3 = a - 3, which becomes a = 5.
Substitute the value of a into the equation from Step 1. In this example, substituting a into y = a(x - 2)2 + 3 results in y = 5(x - 2)2 + 3.
Square the expression inside the parentheses, multiply the terms by a's value and combine like terms to convert the equation to standard form. Concluding this example, squaring (x - 2) results in x2 - 4_x_ + 4, which multiplied by 5 results in 5_x_2 - 20_x_ + 20. The equation now reads as y = 5_x_2 - 20_x_ + 20 + 3, which becomes y = 5_x_2 - 20_x_ + 23 after combining like terms.