Convert Parabolic Equations to the Standard Form

By Mary Jane Sterling

When the equation of a parabola appears in standard form, you have all the information you need to graph it or to determine some of its characteristics, such as direction or size.

Not all equations come packaged that way, though. You may have to do some work on the equation first to be able to identify anything about the parabola.

The standard form of a parabola is (x – h)2 = a(y – k) or (y – k)2 = a(x h), where (h, k) is the vertex.

The methods used here to rewrite the equation of a parabola into its standard form also apply when rewriting equations of circles, ellipses, and hyperbolas. The standard forms for conic sections are factored forms that allow you to immediately identify needed information. Different algebra situations call for different standard forms — the form just depends on what you need from the equation.

For instance, if you want to convert the equation x2 + 10x – 2y + 23 = 0 into the standard form, you perform the following steps, which contain a method called completing the square (a method you use to solve quadratic equations):

  1. Rewrite the equation with the x2 and x terms (or the y2 and y terms) on one side of the equation and the rest of the terms on the other side.

    x2 + 10x = 2y – 23

  2. Add a number to each side to make the side with the squared term into a perfect square trinomial (thus completing the square).

    In this case, you add 25 to each side. x2 + 10x + 25 = 2y – 23 + 25 simplifies to x2 + 10x + 25 = 2y + 2.

  3. Rewrite the perfect square trinomial in factored form, and factor the terms on the other side by the coefficient of the variable.

    (x + 5)2 = 2(y + 1)

You now have the equation in standard form. The vertex is at (–5, –1); if you were to graph it, you would see that it opens upward and is fairly wide.