Parabolas
A satellite dish and a flashlight reflector share the same shape for one reason: every point on a parabola sits the same distance from a single point (the focus) as it does from a fixed line (the directrix). That equal-distance rule is the whole definition, and it tells you where to aim.
By the end of this page you can:
- State the focus-directrix definition of a parabola.
- Read the vertex, focus, and directrix off an equation in standard form.
- Write the equation of a parabola from its focus and directrix.
The floor. For a parabola with vertex at the origin opening up, the equation is x squared equals 4py, the focus is (0, p), and the directrix is the line y equals negative p.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson 7e |
| Chapter | 11 |
| Section | 11.1 Parabolas |
| Open alternate | OpenStax Precalculus 2e, 12.3 The Parabola |
| Difficulty | core |
| Time | 30 min |
Before You Start
The Idea
Fix a point F (the focus) and a line not through it (the directrix). The parabola is every point whose distance to F equals its distance to the directrix. Setting those two distances equal and simplifying gives a tidy equation. When the vertex sits at the origin and the parabola opens up or down, only one number controls everything: p, the directed distance from the vertex to the focus. The focus lands p units from the vertex on one side, and the directrix is the line the same distance on the other side.
The sign of p decides direction. A positive p opens the parabola toward the focus (up, for the vertical case); a negative p opens it the other way. Watch that p is a single distance, not its square. The 4p coefficient is where students most often slip.
Key Formulas
A parabola with vertex at the origin and vertical axis has the standard form:
\[ x^2 = 4py \]
Its focus is the point (0, p) and its directrix is the line y = -p.
\[ \text{focus} = (0, p), \qquad \text{directrix: } y = -p \]
For the horizontal case (axis along the x-axis), the form is y squared equals 4px, with focus (p, 0) and directrix x = -p.
Source: Stewart 7e, Section 11.1 Parabolas.
Worked Example
Find the focus and directrix of the parabola x squared equals 12y, then sketch which way it opens.
Match x squared equals 12y to the standard form x squared equals 4py. So 4p equals 12, which gives p equals 3. Because p is positive and the squared variable is x, the parabola opens upward.
Show the answer
From 4p = 12 we get p = 3.
Focus: (0, p) = (0, 3).
Directrix: y = -p, so y = -3.
The parabola opens upward (vertex at the origin, focus above it). Check one point: at y = 3, x squared = 36, so x = 6 or x = -6. The point (6, 3) is distance 6 from the focus (0, 3) and distance 3 - (-3) = 6 from the directrix y = -3. Equal distances confirm the result.
Connections
This builds on coordinate-plane distance, since the focus-directrix definition is just two distances set equal. It leads into shifted conics, where the same parabola is moved so its vertex sits at a general point (h, k) instead of the origin. The parabola is one of the three conic sections; the ellipse and the hyperbola use two foci and a different distance rule, and each lives in its own node.
Resources
- Primary: Stewart, Redlin, Watson 7e, Section 11.1 Parabolas.
- Alternate: OpenStax Precalculus 2e, Section 12.3 The Parabola.
Last updated: 2026-06-24