How to Convert From a Standard to a Vertex Form

By Lindsay Howell; Updated April 24, 2017
A polynomial is in standard form if each term has a lower degree than the term before it.

Standard and vertex forms are mathematical equations used to describe the curve of a parabola. The vertex form can be thought of as a compressed parabolic equation, whereas the standard form is the longer, expanded version of the same equation. With a basic understanding of high school level algebra, you can convert the standard form to the vertex form.

Start with the standard form of the parabolic equation; for example, y = (x + 3)² + 4. When plotted on a graph, the parabola will have a vertex of 3, 4.

Expand the polynomial within the parentheses: (x+3)(x+3). Add the 4 back into the equation; you will now have (x+3)(x+3) + 4.

Factor the polynomial. Start with the first X in the first parenthesis and multiply it by both the numbers in the second parenthesis: x² + 3x. Now take the 3 in the first parenthesis and multiply it by the numbers in the second: 3x + 9. Add the 4 into the equation so you have x² + 3x + 3x + 9 + 4.

Combine like factors: x² has no like factor, so it stays as it is. There are two numbers with x, so add them as the equation states: 6x. Now add the 9 and the 4 so you have 13. Your final equation will be y = x² + 6x + 13.

Tip

Show all of your work when solving equations.

Warning

Factoring the polynomials out of order will lead to the wrong results.

About the Author

Lindsay Howell has been writing since 2003. Her works have been featured in "Bittersweet," her campus literary magazine. Howell has a Bachelor of Arts in English literature from Frostburg State University.