Putting a quadratic into standard form

From Norsemathology
Revision as of 16:17, 2 September 2016 by Fubini (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

Let's start with the function

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

Now we factor out an :

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.



we can write


from which we arrive at

Finally, following a little simplification, we can write in standard form as


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: .

Example from Stewart


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