Parabola Vertex, Focus & Roots Calculator

Parabola Guide

Forms of a Parabola Equation

Standard form: y = ax² + bx + c. Most common in school maths. a determines opening direction (a > 0: up, a < 0: down) and width (larger |a|: narrower). Vertex form: y = a(x − h)² + k. Directly shows vertex at (h, k). Useful for graphing and transformations. Factored form: y = a(x − r₁)(x − r₂). Shows the roots r₁ and r₂ directly. Only works when roots are real. Converting standard to vertex form: complete the square. y = a(x² + (b/a)x) + c = a(x + b/(2a))² + c − b²/(4a). So vertex: h = −b/(2a),

Vertex, Axis, and Roots

Vertex: the maximum or minimum point. x-coordinate: h = −b/(2a). y-coordinate: substitute h back into equation. Axis of symmetry: vertical line through vertex. Equation: x = −b/(2a). Roots (x-intercepts): solve y = 0. Use quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). Sum of roots: −b/a. Product of roots: c/a. Useful for verifying answers. y-intercept: substitute x = 0. y = c. The y-intercept is always the constant term in standard form. Example: y = x² − 4x + 3. Vertex: h = 4/2 = 2, k = 3 −

Focus and Directrix

A parabola is defined geometrically: every point is equidistant from focus and directrix. Focus: a fixed point inside the parabola curve. Directrix: a line outside, perpendicular to the axis of symmetry. For y = ax² + bx + c with vertex (h, k): focal length p = 1/(4a). Focus: (h, k + p). Directrix: y = k − p. Larger |a|: focus closer to vertex (narrower parabola). Smaller |a|: focus further from vertex (wider). Applications of focus property: parabolic reflectors (satellite dishes, headlamps, mi

Real-World Parabolas

Projectile motion: any thrown object follows a parabolic path (ignoring air resistance). h(t) = h₀ + v₀t − ½gt². Range: maximum at 45° launch angle (in vacuum). With air resistance: paths are slightly distorted but still close to parabolic for moderate speeds. Architecture: arches, bridges, and structural supports often use parabolic curves for efficient load distribution. Hanging cables: form a catenary (very close to but not exactly a parabola). Parabolic bridges (concrete viaducts) load-distr