How to Write Quadratic Equations in Vertex Form

A calculator can aid in the process of writing equations in vertex form.
••• Hemera Technologies/PhotoObjects.net/Getty Images

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.

    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.

    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.

    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).

    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).

    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).

    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.

    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).

Related Articles

How to Identify a Numerical Coefficient of a Term
How to Convert From a Standard to a Vertex Form
How to Complete the Square
How to Graph Parabolas on a TI-84 Calculator
How to Solve for a Variable in a Trig Function
How to Factorise a Quadratic Expression
How to Solve a Parabola
How to Solve Equations for the Indicated Variable
How to Factor Prime Trinomials
How to Find the Volume of a Sphere in Terms of Pi
How to Convert Quadratic Equations From Standard to...
How to Convert an Equation Into Vertex Form
How to Find X in an Algebra Question
How to Find Terms in an Algebra Expression
How to Find Slope From an Equation
How to Find B in Y=Mx + B
How to Do Multiplying & Factoring Polynomials
How to Factor Equations
How to Simplify Algebraic Expressions
How to Use Elimination to Solve the Linear Equation