The vertex of a quadratic curve is the maximum or minimum. A quadratic curve has either a maximum or minimum, and only has one. Completing the square makes it possible to find the maximum or minimum value of a quadratic function without sketching it. We start with an expression
which we want to express in the form
The vertex is then
If
this is a minimum and if
it is a maximum.
Example: find the vertex of the graph for the expression
Equating coefficients of
Equating coefficients of
Equating constant terms:
Hence
in completed square form is
The vertex is
This is shown below.
