Detailed definition
Understanding Parabola
A parabola is the set of points equidistant from a fixed point called the focus and a fixed line called the directrix. The graph has one branch, one vertex, and one axis of symmetry.
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.
Key facts
Important ideas to remember
- 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 it is used
Where parabola shows up
- 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.
Common mistakes
What to watch out for
- 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.