When you plot a quadratic y = ax^2 + bx + c as a graph, the points trace out a smooth, symmetric U-shaped curve called a parabola. Every quadratic makes one — only its width, position, and direction change.

The roots — where it crosses the x-axis

The points where the parabola meets the x-axis are the roots. There y = 0, so the roots are exactly the solutions of ax^2 + bx + c = 0. A factorised quadratic shows them at a glance:

A parabola can have two roots, one (it just touches the axis), or none (it floats clear of it).

The turning point and the line of symmetry

A parabola has a single turning point — its vertex — the lowest point when it opens up, or the highest point when it opens down. A vertical line of symmetry runs straight through the vertex, so the two halves of the curve are mirror images. When there are two roots, the line of symmetry sits exactly halfway between them.

Which way does it open?

The sign of a decides the direction:

• a > 0: the parabola opens up (a valley with a minimum).
• a < 0: the parabola opens down (a hill with a maximum).

The larger |a| is, the narrower the curve.

Try it live

Here is the graph of y = ax^2 + bx + c. Slide the controls and watch the parabola redraw: flip a negative to turn the valley into a hill, and see how b and c slide the roots and vertex around.

Khan Academy graphs a parabola from its roots and vertex here: