Mathematics · Geometry · Analytic Geometry
Parabola Calculator
Calculates the vertex, focus, directrix, axis of symmetry, and latus rectum of a parabola from its standard quadratic equation coefficients.
Calculator
Formula
Given the standard quadratic form y = ax² + bx + c, the vertex (h, k) is at h = -b/(2a) and k = c - b²/(4a). The focal length p = 1/(4a), placing the focus at (h, k + p) and the directrix at y = k - p. The axis of symmetry is the vertical line x = h, and the latus rectum has length |1/a|.
Source: Anton, H., Bivens, I., Davis, S. — Calculus: Early Transcendentals, 10th Ed.; standard analytic geometry reference.
How it works
A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. When expressed in the standard quadratic form y = ax² + bx + c, the parabola opens upward if a > 0 and downward if a < 0. The magnitude of a controls how narrow or wide the parabola is — larger |a| produces a narrower curve, while smaller |a| produces a wider one.
The vertex is the turning point of the parabola, located at (h, k) where h = −b/(2a) and k = c − b²/(4a). This can also be interpreted as completing the square on the quadratic expression. The focal length p = 1/(4a) is the signed distance from the vertex to the focus. The focus sits at (h, k + p), and the directrix is the horizontal line y = k − p. The axis of symmetry is the vertical line x = h, which bisects the parabola perfectly. The latus rectum is the chord through the focus parallel to the directrix; its length equals |1/a|, providing a measure of the parabola's width at the focus.
Practical applications are widespread. Parabolic reflectors in satellite dishes, car headlights, and radio telescopes use the focusing property — any ray parallel to the axis reflects through the focus. Projectile motion under gravity traces a parabolic arc, and suspension cables of parabolic bridges carry uniformly distributed loads optimally. Understanding these properties is also foundational for multivariable calculus, where paraboloids appear in optimization problems and surface integrals.
Worked example
Suppose the parabola is defined by a = 1, b = −4, c = 3, giving y = x² − 4x + 3.
Step 1 — Vertex x-coordinate: h = −b/(2a) = −(−4)/(2 × 1) = 2
Step 2 — Vertex y-coordinate: k = c − b²/(4a) = 3 − 16/4 = 3 − 4 = −1. Vertex = (2, −1).
Step 3 — Focal length: p = 1/(4a) = 1/4 = 0.25.
Step 4 — Focus: Located at (h, k + p) = (2, −1 + 0.25) = (2, −0.75).
Step 5 — Directrix: y = k − p = −1 − 0.25 = y = −1.25.
Step 6 — Axis of symmetry: x = h = x = 2.
Step 7 — Latus rectum: |1/a| = |1/1| = 1. The chord through the focus parallel to the directrix has length 1 unit.
This parabola opens upward with a relatively narrow shape (a = 1) and its minimum point is at (2, −1).
Limitations & notes
This calculator assumes a vertical parabola of the form y = ax² + bx + c. It does not handle horizontal parabolas (x = ay² + by + c) or rotated conics. The coefficient a must be non-zero — if a = 0, the equation becomes linear and no parabola exists. Extremely small values of |a| produce very large focal lengths and wide parabolas, which may indicate a near-degenerate or impractical physical scenario. All calculations assume exact real-valued arithmetic; floating-point rounding may introduce minor errors for very large or very small coefficient values. This tool covers only real parabolas in the Cartesian plane and does not address parabolas in polar or parametric form.
Frequently asked questions
What does the coefficient 'a' tell you about a parabola?
The coefficient a determines both the direction and width of the parabola. If a > 0, the parabola opens upward; if a < 0, it opens downward. A larger absolute value of a makes the parabola narrower, while a smaller absolute value makes it wider and more spread out.
What is the focus of a parabola and why does it matter?
The focus is a fixed interior point such that every point on the parabola is equidistant from the focus and the directrix. It matters because parabolic reflectors (satellite dishes, headlights, telescopes) direct all parallel incoming signals or light rays to converge at the focus, making it the functional hot spot of the shape.
How is the directrix related to the focus?
The directrix is a horizontal line located on the opposite side of the vertex from the focus, at the same distance p from the vertex. Together, the focus and directrix define the parabola: any point on the curve is exactly equidistant from both. The vertex sits exactly halfway between the focus and directrix.
What is the latus rectum of a parabola?
The latus rectum is the chord of the parabola that passes through the focus and is perpendicular to the axis of symmetry. Its length is |1/a| for a parabola in the form y = ax² + bx + c. It provides a convenient measure of the parabola's 'width' at the level of the focus.
Can this calculator handle parabolas that open left or right?
No — this calculator handles only vertical parabolas of the form y = ax² + bx + c, which open upward or downward. Horizontal parabolas have the form x = ay² + by + c. To analyze a horizontal parabola, you would swap the roles of x and y in all formulas, using the analogous expressions with the axis of symmetry being horizontal.
Last updated: 2025-01-15 · Formula verified against primary sources.