Putting a quadratic into standard form

From www.norsemathology.org

Let's start with the function

where (otherwise we'd have a linear function).

Now we factor out an a:

We want to create a perfect square, by replacing that term

How do we do that?

Well, thinking backwards is often useful. If we consider a perfect square and expand it,

,

we see that we can rewrite that as

.

We've solved for the quadratic piece and the linear piece. Now we set this equal to the expression we want to replace.

Setting

or

we can write

.

from which we arrive at

Finally, following a little simplification, we can write f(x) in standard form as

or

The former has the advantage of featuring the discriminant, from the quadratic formula. The simplest way of finding the constant is by evaluating

From this form, we can see that the maximum or minimum of the function occurs at

because this is the value at which the squared term is zero. The value of the function at is

These two special values together are called the vertex: .

Mathematica:

1. Mathematica File confirming the formula for the y-value of the vertex.
2. Animation of Standard Form.