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Focus of a Parabola — Definition, Formula & Examples

Focus of a Parabola

The focus of a parabola is a fixed point on the interior of a parabola used in the formal definition of the curve.

A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.

Note: For a parabolic mirror, all rays of light emitting from the focus reflect off the parabola and travel parallel to each other (parallel to the axis of symmetry as well).

 

Upward-opening parabola with vertex (h,k) and focus point p above vertex. Formulas: vertical 4p(y-k)=(x-h)²; horizontal...

Parabola with point P, focus inside curve, directrix below. L1 (P to focus) = L2 (P to directrix), shown with arrows.

 

Example:  

Graph of upward-opening parabola with vertex at (3,-2), focus above vertex, vertical axis of symmetry at x=3, and horizontal...

This is a graph of the parabola Equation of a parabola: y = (1/4)x² − (3/2)x + 1/4 with all its major features labeled: axis of symmetry, focus, vertex, and directrix.

 

 

See also

Conic sections, foci of an ellipse, foci of a hyperbola, focal radius

Key Formula

y=14p(x−h)2+k⟹Focus=(h,  k+p)y = \frac{1}{4p}(x - h)^2 + k \quad \Longrightarrow \quad \text{Focus} = (h,\; k + p)
Where:
  • hh = The x-coordinate of the vertex of the parabola
  • kk = The y-coordinate of the vertex of the parabola
  • pp = The directed distance from the vertex to the focus (positive means the parabola opens upward; negative means it opens downward)

Worked Example

Problem: Find the focus of the parabola y = (1/8)x².
Step 1: Identify the vertex. The equation y = (1/8)x² has the form y = (1/(4p))x² with vertex at the origin.
Vertex=(h,k)=(0,0)\text{Vertex} = (h, k) = (0, 0)
Step 2: Match the coefficient to find p. Set 1/(4p) equal to the coefficient 1/8.
14p=18\frac{1}{4p} = \frac{1}{8}
Step 3: Solve for p by cross-multiplying.
4p=8⟹p=24p = 8 \quad \Longrightarrow \quad p = 2
Step 4: Apply the focus formula. Since p = 2, the focus is p units above the vertex.
Focus=(0,  0+2)=(0,2)\text{Focus} = (0,\; 0 + 2) = (0, 2)
Answer: The focus of the parabola y = (1/8)x² is at the point (0, 2).

Another Example

This example differs by featuring a downward-opening parabola with a non-origin vertex, showing how a negative value of p places the focus below the vertex.

Problem: Find the focus of the parabola y = -2(x − 3)² + 5.
Step 1: Read the vertex directly from vertex form. Here h = 3 and k = 5.
Vertex=(3,5)\text{Vertex} = (3, 5)
Step 2: Set the leading coefficient equal to 1/(4p). The coefficient of (x − 3)² is −2.
14p=−2\frac{1}{4p} = -2
Step 3: Solve for p.
1=−8p⟹p=−181 = -8p \Longrightarrow p = -\dfrac{1}{8}
Step 4: Compute the focus. Because p is negative, the focus lies below the vertex (the parabola opens downward).
Focus=(3,  5+(−18))=(3,  398)\text{Focus} = \left(3,\; 5 + \left(-\tfrac{1}{8}\right)\right) = \left(3,\; \frac{39}{8}\right)
Answer: The focus is at (3, 39/8), which is 1/8 of a unit below the vertex.

Frequently Asked Questions

How do you find the focus of a parabola from standard form y = ax² + bx + c?
First convert to vertex form by completing the square to get y = a(x − h)² + k. Then use 1/(4p) = a to solve for p. The focus is at (h, k + p). If a is positive the parabola opens up, and if a is negative it opens down.
What is the difference between the focus and the directrix of a parabola?
The focus is a point on the interior of the parabola, while the directrix is a line on the exterior. Every point on the parabola is equally distant from both. If the vertex is at (h, k), the focus is at (h, k + p) and the directrix is the horizontal line y = k − p.
What does the focus of a parabola do in real life?
Satellite dishes, flashlights, and car headlights use parabolic shapes because signals or light rays arriving parallel to the axis of symmetry all reflect through the focus. Conversely, a light source placed at the focus produces a parallel beam. This reflective property is the main practical application of the focus.

Focus of a Parabola vs. Directrix of a Parabola

Focus of a ParabolaDirectrix of a Parabola
What it isA fixed point inside the parabolaA fixed line outside the parabola
Location (vertical parabola)(h, k + p)y = k − p
DimensionZero-dimensional (a point)One-dimensional (a line)
Role in definitionDistance from any point on the curve to the focus equals……the perpendicular distance from that same point to the directrix
Reflective propertyRays from the focus reflect off the parabola in parallelNot directly involved in reflection; serves as the geometric reference line

Why It Matters

You encounter the focus of a parabola in Algebra 2 and Precalculus when studying conic sections, and again in physics when analyzing optics and projectile motion. Understanding the focus lets you convert between different forms of quadratic equations and solve real-world design problems involving satellite dishes, telescopes, and headlights. It also lays the groundwork for studying the foci of ellipses and hyperbolas.

Common Mistakes

Mistake: Confusing the coefficient a in y = ax² with the value of p, and writing the focus as (0, a) instead of (0, 1/(4a)).
Correction: Remember that a = 1/(4p), so you must solve for p first: p = 1/(4a). For example, if a = 2 then p = 1/8, not 2.
Mistake: Placing the focus on the wrong side of the vertex when the parabola opens downward or sideways.
Correction: Always check the sign of p. A negative p means the focus is below (or to the left of) the vertex, matching the direction the parabola opens.

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