How To Write Quadratic Equations In Vertex Form

Converting an equation to vertex form can be tedious and require an extensive degree of algebraic background knowledge, including weighty topics such as factoring. The vertex form of a quadratic equation is y = a(x – h)^2 + k, where "x" and "y" are variables and "a," "h" and k are numbers. In this form, the vertex is denoted by (h, k). The vertex of a quadratic equation is the highest or lowest point on its graph, which is known as a parabola.

Step 1

Ensure that your equation is written in standard form. The standard form of a quadratic equation is y = ax^2 + bx + c, where "x" and "y" are variables and "a," "b" and "c" are integers. For instance, y = 2x^2 + 8x – 10 is in standard form, whereas y – 8x = 2x^2 – 10 is not. In the latter equation, add 8x to both sides to put it in standard form, rendering y = 2x^2 + 8x – 10.

Step 2

Move the constant to the left side of the equals sign by adding or subtracting it. A constant is a number lacking an attached variable. In y = 2x^2 + 8x – 10, the constant is -10. Since it is negative, add it, rendering y + 10 = 2x^2 + 8x.

Step 3

Factor out "a," which is the coefficient of the squared term. A coefficient is a number written on the variable's left-hand side. In y + 10 = 2x^2 + 8x, the coefficient of the squared term is 2. Factoring it out yields y + 10 = 2(x^2 + 4x).

Step 4

Rewrite the equation, leaving an empty space on the right side of the equation after the "x" term but before the end parenthesis. Divide the coefficient of the "x" term by 2. In y + 10 = 2(x^2 + 4x), divide 4 by 2 to get 2. Square this result. In the example, square 2, producing 4. Place this number, preceded by its sign, in the empty space. The example becomes y + 10 = 2(x^2 + 4x + 4).

Step 5

Multiply "a," the number you factored out in Step 3, by the result of Step 4. In the example, multiply 2*4 to get 8. Add this to the constant on the left side of the equation. In y + 10 = 2(x^2 + 4x + 4), add 8 + 10, rendering y + 18 = 2(x^2 + 4x + 4).

Step 6

Factor the quadratic inside the parentheses, which is a perfect square. In y + 18 = 2(x^2 + 4x + 4), factoring x^2 + 4x + 4 yields (x + 2)^2, so the example becomes y + 18 = 2(x + 2)^2.

Step 7

Move the constant on the left-hand side of the equation back over to the right by adding or subtracting it. In the example, subtract 18 from both sides, producing y = 2(x + 2)^2 – 18. The equation is now in vertex form. In y = 2(x + 2)^2 – 18, h = -2 and k = -18, so the vertex is (-2, -18).