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Parabola Equation

Parabola: set of points equidistant from focus F and directrix. Vertex form: y=a(x−h)²+k. Focus (h, k+p), directrix y=k−p, p=1/(4a). Latus rectum LR=4p.

Concept Fundamentals
y=a(x−h)²+k
Vertex
(h, k+p), p=1/(4a)
Focus
y=k−p
Directrix
4p
LR

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a>0: opens up; a<0: opens down. Focus and directrix are p units from vertex. All parabolas are similar (same shape, different scale).

Key quantities
y=a(x−h)²+k
Vertex
Key relation
(h, k+p), p=1/(4a)
Focus
Key relation
y=k−p
Directrix
Key relation
4p
LR
Key relation

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Why: Parabolas appear in projectile motion, reflectors (satellite dishes), and conic sections. Focus-directrix definition: PF = distance to directrix. Used in optics and antenna design.

How: Vertex form: y=a(x−h)²+k. a=1/(4p) where p=focal distance. Focus (h,k+p), directrix y=k−p. Axis x=h. Latus rectum = 4p.

a>0: opens up; a<0: opens down.Focus and directrix are p units from vertex.
Sources:Khan Academy — ParabolaWolfram MathWorld — Parabola

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Parabola EquationEnter vertex (h,k) and parameter a or p

Parabola Parameters

Visualization

Enter parabola parameters to see the visualization

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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y=a(x−h)²+k; focus (h,k+p), p=1/(4a).

— Conic Sections

L

Latus rectum = 4p.

— Property

Key Takeaways

  • • A parabola is the set of points equidistant from a focus and a directrix.
  • • Vertex form: y=a(x−h)2+ky=a(x-h)^2+k (vertical) or x=a(y−k)2+hx=a(y-k)^2+h (horizontal).
  • • The parameter p is the distance from the vertex to the focus (and to the directrix).
  • • Latus rectum = 4p — the chord through the focus parallel to the directrix.
  • • Coefficient a=frac14pa=\\frac{1}{4p}; a > 0 opens up/right, a < 0 opens down/left.

Did You Know?

Projectile Motion

A ball thrown in a vacuum follows a parabolic path. Gravity causes the parabolic trajectory.

Satellite Dishes

Parabolic reflectors focus incoming parallel rays to a single point (the focus).

Suspension Bridges

Cables of suspension bridges form catenaries, but parabolas are often used as approximations.

Headlights

Car headlights use parabolic reflectors to create a focused beam of light.

Conic Sections

A parabola is a conic section with eccentricity e = 1 — the boundary between ellipse and hyperbola.

Architecture

Parabolic arches distribute weight efficiently and have been used since ancient times.

Understanding Parabola Equations

The parabola has a vertex (turning point), a focus (fixed point), and a directrix (fixed line). Every point on the parabola is equidistant from the focus and directrix.

y=a(x−h)2+kquadtextorquadx=a(y−k)2+hy = a(x-h)^2 + k \\quad \\text{or} \\quad x = a(y-k)^2 + h

where a = 1/(4p), (h,k) is the vertex, and p is the focal distance.

Expert Tips

Wide vs Narrow

Larger p gives a wider parabola (smaller |a|). Smaller p gives a narrower parabola (larger |a|).

Focus Location

Focus is p units from the vertex in the direction the parabola opens.

Directrix

The directrix is perpendicular to the axis of symmetry and p units from the vertex.

Standard Form

(x-h)² = 4p(y-k) for vertical; (y-k)² = 4p(x-h) for horizontal parabolas.

Frequently Asked Questions

What is the vertex form of a parabola?

y = a(x-h)² + k for vertical parabolas, or x = a(y-k)² + h for horizontal. (h,k) is the vertex, a = 1/(4p).

What is the parameter p?

p is the distance from the vertex to the focus (and to the directrix). It determines how "wide" or "narrow" the parabola is.

What is the latus rectum?

The latus rectum is the chord through the focus parallel to the directrix. Its length equals 4p.

How do I know which way the parabola opens?

For y = a(x-h)² + k: a > 0 opens up, a < 0 opens down. For x = a(y-k)² + h: a > 0 opens right, a < 0 opens left.

What is the axis of symmetry?

The vertical line x = h for vertical parabolas, or the horizontal line y = k for horizontal parabolas.

Can p be negative?

No. p is a distance and must be positive. The sign of a (and thus opening direction) is determined by orientation.

How is the parabola used in real life?

Parabolas appear in satellite dishes, headlights, projectile motion, suspension bridges, and architectural arches.

How to Use This Calculator

  1. Enter vertex (h, k) and parameter p. Choose axis (x or y) and orientation (positive/negative).
  2. Click a sample example to auto-fill and see results.
  3. View the visualization with vertex, focus, directrix, and parabola curve.
  4. Check the step-by-step solution for vertex form and key properties.
  5. Copy results to share or paste into assignments.

Note: Parameter p must be positive. Results are for educational use.

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