# How to Write a Parabolic Equation in Vertex Form

*E. S. Kim*

In algebra, there are three ways to write the equation of a parabola (2^{nd} degree polynomial) in Cartesian coordinates. The standard form of a parabola is*y* = *f*(*x*) = *ax*^{2} + *bx* + *c*.

The factored form is

*y* = *f*(*x*) = *a*(*x - r*)(*x - s*),

where *r* and *s* are the roots of the function, possibly complex numbers. The vertex form of a parabola is

*y* = *f*(*x*) = *a*(*x - h*)^{2} + *k*,

where *h* is the *x*-coordinate of vertex and *k* is the *y*-coordinate. In all three forms, the coefficient *a* governs the shape of the parabola. When *a* is positive, the curve is concave up, and when *a* is negative the curve is concave down. If *a* = 0 the function *f*(*x*) is a line. See figure below:

### How to Convert from Standard Form to Vertex Form

The standard form is the most compact way to write the three terms of a second degree polynomial. However, if you need to draw the graph of a parabola, the vertex form is more convenient. The vertex is located at the point (*h*,

*k*) and the parameter

*a*indicates the general shape of the parabola. Once you have the equation in the form

*f*(

*x*) =

*a*(

*x - h*)

^{2}+

*k*, you have all the information you need to draw the curve.

You can convert the standard form of a parabola to vertex form by completing the square. Doing this yields the relation:

*ax*

^{2}+

*bx*+

*c*=

*a*(

*x*+

^{b}/

_{2a})

^{2}+

*c*-

^{b²}/

_{4a}.

So

*h*= -

^{b}/

_{2a}and

*k*=

*c*-

^{b²}/

_{4a}. You can also use the converter calculator above to quickly transform a parabolic equation into vertex form.

**Example:**Convert the parabola

*f*(

*x*) = 5

*x*

^{2}- 20

*x*+ 1 to vertex form. Solution: Since we have

*a*= 5,

*b*= 20, and

*c*= 1, we plug these values into the equivalence relation above. This gives us

5

*x*

^{2}- 20

*x*+ 1 = 5(

*x*- 20/(2*5))

^{2}+ 1 - (20

^{2})/(4*5)

= 5(

*x*- 2)

^{2}- 19

This means the vertex of the parabola is at the point (2, -19) and the parabola is concave up.

© *Had2Know 2010
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