How to Convert an Equation Into Vertex Form

By Pamela Dorr
Parabolas are all around us.

Parabola equations are written in the standard form of y=ax^2+bx+c. This form can tell you if the parabola opens up or down and, with a simple calculation, can tell you what the axis of symmetry is. While this is a common form to see an equation for a parabola in, there is another form that can give you a little more information about the parabola. The vertex form tells you the vertex of the parabola, which way it opens, and whether it is a wide or narrow parabola.

Using the standard equation of y=ax^2+bx+c, find the x value of the vertex point by plugging the a and b coefficients into the formula x= -b/2a.

For example:

y=3x^2+6x+8 x= -6/(2*3) = -6/6 = -1

Substitute the found value of x into the original equation to find the value of y.

y= 3(-1)^2+6(-1)+8 y= 3-6+8 y= 5

The values of x and y are the coordinates of the vertex. In this case, the vertex is at (-1,5).

Insert the vertex coordinates into the equation y= a(x-h)^2 + k, where h is the x-value and k is the y-value. The value of a comes from the original equation.

y = 3(x+1)^2+5 This is the vertex form of the parabola's equation.

(The h is a +1 in the equation because a negative in front of the -1 makes it positive.)

To convert the vertex form back to standard form, simply square the binomial, distribute a, and add the constants.

y=3(x+1)^2+5 y=3(x^2+2x+1)+5 y=3x^2+6x+3+5 y=3x^2+6x+8

This is the original standard form of the equation.

Tip

If a is positive, the parabola opens up. If a is negative, the parabola opens down. If |a|>1, the parabola is wide. If |a|<1, the parabola is narrow.

Warning

Watch the negative signs. Forgetting a negative is one of the most common mistakes. Copy the original problem carefully. Another common mistake is miscopying the original problem.

About the Author

Pamela Dorr began writing professionally in 2010. She's published several educational articles and tutorials on the University of Houston - Victoria Academic Center's website. Dorr is currently pursuing a Bachelor of Science in mathematics from the University of Houston - Victoria.