Let's start with the function
(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: .
- Mathematica File confirming the formula for the y-value of the vertex.
- Animation of Standard Form.