Parabola Vertex, Focus & Roots Calculator

Parabola Guide

What should I know about 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), and k is found by substituting this h value back into the original equation — the same result the completing-the-square method arrives at directly.

What should I know about 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 − 4 = −1, giving a vertex at (2, −1) for this particular parabola.

What should I know about 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, such as satellite dishes, headlamps, and microphone dishes, all exploit the property that signals or light entering parallel to the axis reflect through the single focus point.

What should I know about 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, such as concrete viaducts, use the shape specifically for its load-distributing structural properties, spreading weight evenly along the curve.