this might help

The Quadratic Formula: The Discriminant and Graphs

http://www.purplemath.com/modules/parabola.htm

Graphing Quadratic Functions: The Leading Coefficient / The Vertex

Each of those has a few pages worth of information.

If the leading coefficient is positive, it opens up.

If the leading coefficient is negative, it opens down.

If the discriminant is 0 you have one real root,

If the discriminant is positive you have two real roots,

If the discriminant is negative you have no real roots.

Complete the square to find the vertex.