Parabola Calculator
Solve vertical and horizontal parabolas for vertex $(h, k)$, focus coordinates, directrix line, focal parameter ($p$), latus rectum ($|4p|$), and axis of symmetry with dynamic SVG coordinate visualizers.
Parabola Definition & Standard Formulas
A parabola is the conic section with eccentricity e = 1, formed by the locus of points equidistant from a focus point and a directrix line. Its standard vertex equation relates squared coordinates to 4p times linear offset.
Parabola Conic Definition (Focus-Directrix Equidistance Locus)
Geometrically, a parabola is defined as the set of all points $P(x, y)$ in a plane whose distance to a fixed point $F$ (the focus) equals its perpendicular distance to a fixed straight line $D$ (the directrix):
The vertex $(h, k)$ lies exactly midway between the focus and the directrix. The distance from the vertex to the focus is the focal length $|p|$.
Standard Equations: Vertical vs Horizontal Forms
- Opens upward if $p > 0$; downward if $p < 0$.
- Focus: $F(h, k + p)$
- Directrix: $y = k - p$
- Axis of Symmetry: $x = h$
- Opens rightward if $p > 0$; leftward if $p < 0$.
- Focus: $F(h + p, k)$
- Directrix: $x = h - p$
- Axis of Symmetry: $y = k$
Focal Parameter (p), Latus Rectum (|4p|) & Directrix
Signed distance from vertex to focus. Relates to quadratic stretch $a$ via $p = \frac{1}{4a}$.
Total chord length passing through the focus perpendicular to the axis of symmetry.
External reference baseline perpendicular to the axis of symmetry at distance $|p|$ behind vertex.
Converting General Quadratic Form to Vertex Form
For a quadratic function $y = ax^2 + bx + c$, complete the square to find vertex coordinates $(h, k)$ and focal length $p$:
The Parabolic Reflective Property & Engineering Optics
Satellite Antennas & Radio Telescopes
Parallel radio signals from deep space reflect off parabolic dish surfaces to concentrate on the receiver horn positioned at the focus $F$.
Automotive Headlights & Spotlights
An LED or halogen emitter placed at the focus point reflects off the parabolic mirror into a focused, non-diverging parallel forward beam.
Ballistic Trajectories (Gravity)
Under constant downward gravitational acceleration $g$, projectile flight paths in vacuum form exact inverted parabolas $y(x) = x\tan\theta - \frac{g x^2}{2v_0^2\cos^2\theta}$.
Graded Step-by-Step Numerical Solutions
Find the vertex, focus, directrix, and latus rectum for $(y - 3)^2 = -12(x + 2)$.
1. Identify vertex $(h, k)$: $h = -2, k = 3 \implies V(-2, 3)$.
2. Solve for $p$: $4p = -12 \implies p = -3$ (opens leftward).
3. Focus: $F(h + p, k) = (-2 + (-3), 3) = F(-5, 3)$.
4. Directrix: $x = h - p = -2 - (-3) \implies x = 1$.
Latus Rectum $= |4p| = 12$, Axis of Symmetry: $y = 3$.
Common Pitfalls & Sign of p Mistakes
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Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.