Parabola - Definition, Formula, Graph, Examples

Parabola - Definition, Formula, Graph, Examples

TL;DR

A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). It's one of four classical conic sections — created by slicing a cone with a plane parallel to its slant side. The standard equation is y² = 4ax (horizontal opening) or (x − h)² = 4p(y − k) (vertex form).

What Is a Parabola?

A parabola is the locus (set of points) in a plane equidistant from a fixed point — the focus — and a fixed line — the directrix. This definition is purely geometric — no algebra required.

Equivalently, a parabola is a conic section: the curve formed when a flat plane slices through a cone parallel to the cone's slant side. The other three conic sections are the circle, the ellipse, and the hyperbola.

Algebraically, the standard parabola opens horizontally to the right:

y² = 4ax

with focus at (a,0) and directrix x = -a.

What Is the Parabola Equation?

Parabolas come in three standard algebraic forms.

Standard Form (Opens Up)

y = ax² + bx + c, a ≠ 0

The parabola opens upward when a > 0, downward when a < 0.

Vertex Form

y = a(x − h)² + k

The vertex sits at (h,k). The axis of symmetry is x = h.

Conic Form (Vertex at Origin)

y² = 4ax (opens right), x² = 4ay (opens up)

The number a is the distance from the vertex to the focus. The directrix is x = -a or y = -a respectively.

What Are the Parts of a Parabola?

Part Definition
Vertex The "tip" of the parabola — point where it changes direction
Focus A fixed point inside the curve
Directrix A fixed line outside the curve
Axis of symmetry Line through vertex and focus, perpendicular to directrix
Latus rectum Chord through the focus perpendicular to the axis — length 4a

What Are the Properties of a Parabola?

  1. Symmetry. A parabola is symmetric about its axis of symmetry.
  2. One axis of symmetry only. A parabola has exactly one axis.
  3. Single vertex. The unique point where the curve meets its axis of symmetry.
  4. Eccentricity = 1. Ratio of distance-to-focus over distance-to-directrix is equal by definition.
  5. Latus rectum length 4a. Useful for sketching and checking the standard form.
  6. Reflective property. Rays entering parallel to the axis reflect through the focus.

How Do You Derive the Parabola Equation?

The algebraic equation y² = 4ax comes directly from the focus-directrix definition. Here's the derivation:

Setup. Place the vertex at the origin, focus at (a,0), and directrix as the vertical line x = -a.

Distance condition. Distance from P to focus equals distance from P to directrix:

√((x − a)² + y²) = |x + a|

Square both sides:

(x − a)² + y² = (x + a)²

Expand:

x² − 2ax + a² + y² = x² + 2ax + a²

Cancel the x² and a²:

y² = 4ax

How Do You Graph a Parabola?

For a parabola y = ax² + bx + c:

  1. Find the vertex using x = −b/2a, then plug into the equation to find y.
  2. Determine direction. If a > 0, opens up; if a < 0, opens down.
  3. Find the y-intercept by setting x = 0.
  4. Find the x-intercepts using the quadratic formula.
  5. Plot the vertex, intercepts, and a few other points then draw the curve.

Why Are Parabolas Important?

Parabolas are important in nature and engineering because they describe the trajectory of objects under uniform gravity and have a fundamental reflective property. Applications include:

The term parabola comes from the Greek parabolē, named by Apollonius of Perga, who studied conics without algebra.

Common Mistakes With Parabolas

Mistake 1: Confusing the y-intercept with the vertex's y-coordinate.

Mistake 2: Forgetting to take the sign of a into account when sketching.

Mistake 3: Confusing focus and vertex in conic form.

The Mathematicians Who Shaped Parabola Theory