What Is a Parabola?

How the curve is defined

As a conic section, a parabola is formed when a plane cuts a cone parallel to one of the cone's generating lines. In analytic geometry it is usually read from a vertex-based equation and a visible opening direction.

This page keeps the focus, directrix, vertex, and curve on one graph so the parabola can be understood as a distance rule, not just as a memorised U-shaped picture.

For a vertical parabola, the axis of symmetry runs through the vertex and focus. Points on the curve balance their distance to the focus with their perpendicular distance to the directrix. If the parabola opens downward, the same rule remains true; only the direction changes.

The vertex is the turning point of the graph, but it is not the focus. The focus sits inside the opening and the directrix lies on the opposite side. Keeping those three objects separate makes an equation or graph much easier to label correctly.

• A parabola is the set of points equidistant from a focus and a directrix.
• Every point on the parabola is equally distant from the focus and the directrix.
• The vertex is the turning point nearest the directrix and the focus.
• A parabola has eccentricity 1, which distinguishes it from circles, ellipses, and hyperbolas.

Where parabolas help

• Use parabolas in coordinate-conic problems involving vertex form, focus, and directrix.
• Use them in physics and engineering contexts such as projectile paths and reflective surfaces.
• Use them when identifying a conic with one open branch and one axis of symmetry.

Parabola checks to make

• Do not confuse the focus with the vertex; they are distinct points on the axis of symmetry.
• Do not assume every U-shaped graph opens upward; parabolas can open in other directions.
• Do not mix the focus-directrix equality rule with the two-foci rules used for ellipses and hyperbolas.