Finding the vertex

Notes

The points $P$ and $Q$ below are called the vertex of their parabolas.

parabolas with vertex

If the parabola is concave, the vertex is the maximum point. This is the case for $P$ .

But if the parabola is convex, the vertex is the minimum point. This is the case for $Q$ .

We have seen how to find the vertex by looking at the midpoint of the roots, but if there are no roots this cannot work. Fortunately, completing the square can help.

Example

Use completing the square to find the coordinates of the vertex of each parabola.

$y = x^{2} - 6 x + 4$
$y = - x^{2} - 8 x - 9$
$y = 3 x^{2} + 6 x - 1$
$y = - 2 x^{2} + 3 x + 1$
$y = 1 - x^{2}$

Finding the vertex ​

Notes ​

Example ​

Exercise ​

Finding the vertex

Notes

Example

Exercise