This parabola calculator unpacks y = ax² + bx + c into its geometry: vertex, focus, directrix, axis of symmetry, focal length and both intercepts.
Vertex from h = −b/2a, k = c − b²/4a (that is, completing the square — the standalone tool sits in the Math hub). Focal length p = 1/(4a): focus sits p above the vertex (for a > 0), directrix p below. Sign of a decides up/down; |a| decides narrow/wide — a = 1/(4p) ties shape directly to the reflective geometry that makes satellite dishes work.
From Coefficients to Conic Geometry
A parabola is the set of points equidistant from focus and directrix — the algebraic (a, b, c) and geometric (vertex, p) descriptions translate through vertex form, which our Completing the Square Calculator builds step by step. Part of our Math Calculators collection.
Worked Example (Verified)
- y = x² − 4x + 3 (defaults): vertex (2, −1); axis x = 2; focus (2, −0.75); directrix y = −1.25; y-intercept 3; x-intercepts 1 and 3 ✔ ((x−1)(x−3) agrees)
- Downward case: a = −1 flips everything — vertex becomes the maximum, focus below, directrix above; projectile arcs live here (gravity supplies a < 0)
- Narrowness: a = 4 → p = 1/16 = 0.0625 — focus hugs the vertex and the arms rocket upward; a = 0.25 → p = 1, a lazy wide bowl (satellite-dish territory)
- Tangency: discriminant 0 (e.g. x² − 4x + 4) → one double intercept at the vertex (2, 0) — the parabola kisses the axis
Why the Focus Matters Outside Class
- Reflective property: rays parallel to the axis reflect through the focus — dishes, headlights, solar cookers, whispering galleries.
- Projectile motion: constant gravity traces parabolas; vertex = apex, intercepts = launch/landing — sports analytics in conic form.
- Suspension cables approximate parabolas under uniform horizontal load (catenaries under self-weight — a famous near-miss distinction).
- Optimization: vertex k is the max/min value quadratics are forever being asked for — profit, area, height problems all end here.
Related Tools
- Completing the Square Calculator — vertex form, manually
- Quadratic Formula Calculator — the intercepts' engine
- Math Calculators — the full collection
Frequently Asked Questions
How do you find the vertex of a parabola?
h = −b/(2a), then k = ah² + bh + c (equivalently c − b²/4a). For x² − 4x + 3: h = 2, k = −1 → vertex (2, −1), the minimum when a > 0 and the maximum when a < 0. This calculator also returns the focus, directrix, axis and intercepts.
What are the focus and directrix?
The defining pair: every point on a parabola is equidistant from the focus (a point) and the directrix (a line). For vertex (h, k) and p = 1/(4a): focus = (h, k + p), directrix y = k − p. Defaults: p = 0.25, focus (2, −0.75), directrix y = −1.25.
How do I know if a parabola opens up or down?
The sign of a: positive opens upward (vertex is the minimum), negative downward (maximum). Magnitude controls width — large |a| narrows, small |a| widens — because a = 1/(4p) ties steepness to the focal distance.
What if the x-intercepts don't exist?
Then the discriminant b² − 4ac is negative: the parabola floats entirely above (a > 0) or below (a < 0) the axis, with a complex-conjugate root pair. The vertex and geometry still compute normally — real intercepts are a crossing question, not an existence question for the curve.
Parabola vs quadratic — same thing?
The parabola is the graph; the quadratic is the function drawing it. Vertex/focus/directrix belong to the geometric view (conic sections), coefficients/roots to the algebraic view; this page ferries between them, which is most of what conic-sections coursework asks.
Is this parabola calculator free?
Yes — free, instant, no sign-up.