# Parabolas

## Common Parabola Features

There are two standard forms for the equation of a parabola, shown below.

Parabolas all share the following properties.

• One focus and one directrix

• Eccentricity: e = 1

• This means that every point on the parabola is an equal distance from the focus and from the directrix.

• One vertex: Located at the midpoint between the focus and the directrix

• One axis of symmetry: Passing through the vertex (and focus) and perpendicular to the directrix

We can divide parabolas into two types: those that open up or down, and those that open left or right.

The information we can find from the equation of each type of parabola is summarized in the table below.

## Graphs of Parabolas

### Parabolas Opening Up or Down

A parabola with an equation of this form opens up or down. Notice that y is written as a function of x.

Use the sliders below to change the values of a, h, and k, and see different parabolas that open up or down. The parabola is graphed along with its focus, directrix, and axis of symmetry.

### Parabolas Opening Right or Left

A parabola with an equation of this form opens right or left. Notice that x is written as a function of y.

Use the sliders below to change the values of a, h, and k, and see different parabolas that open right or left. The parabola is graphed along with its focus, directrix, and axis of symmetry.