How to Convert From a Standard to a Vertex Form

A polynomial is in standard form if each term has a lower degree than the term before it.
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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.


    • Show all of your work when solving equations.


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

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