Analyze parabolas in vertex or standard form — find vertex, focus, directrix, axis of symmetry, latus rectum, and x/y-intercepts with interactive graph.
A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). As one of the four conic sections, parabolas appear throughout science, engineering, and everyday life — from satellite dish shapes and projectile trajectories to headlight reflectors and suspension bridge cables.
This calculator accepts input in vertex form y = a(x−h)² + k or standard form y = ax² + bx + c, and computes all key properties: vertex, focus, directrix, axis of symmetry, latus rectum, discriminant, and x/y-intercepts. The interactive graph plots the parabola curve, focus point, and directrix line, while a properties table provides a comprehensive summary.
The parameter a controls the parabola's width and direction: |a| > 1 makes it narrower, |a| < 1 makes it wider, and the sign determines whether it opens upward (minimum) or downward (maximum). Understanding these properties is essential for algebra, calculus (optimization), physics (projectile motion), and engineering (antenna design, optics).
Parabola analysis involves converting between vertex and standard forms, computing the focus and directrix from 1/(4a), finding x-intercepts via the quadratic formula, and plotting the curve — all of which require multiple steps that are easy to get wrong. This calculator accepts either form, automatically converts to the other, and displays every property: vertex, focus, directrix, axis of symmetry, latus rectum, discriminant, and intercepts. The interactive graph visualizes the parabola, focus, and directrix together, making abstract conic section properties tangible.
Vertex form: y = a(x − h)² + k Standard form: y = ax² + bx + c Focus: (h, k + 1/(4a)) Directrix: y = k − 1/(4a)
Result: Vertex (0, 0), Focus (0, 0.25), Directrix y = −0.25
The standard parabola y = x² has its vertex at the origin, focus at (0, 1/4), and directrix y = −1/4.
A parabola has a remarkable optical property: any ray entering parallel to the axis of symmetry reflects off the curve and passes through the focus. This is why satellite dishes, radio telescopes, and car headlights use parabolic shapes. A satellite dish collects faint signals and concentrates them at the focus-mounted receiver. Conversely, a light source placed at the focus of a parabolic reflector produces a parallel beam — the principle behind flashlights, searchlights, and solar concentrators.
The vertex form y = a(x−h)² + k immediately reveals the vertex (h, k) and direction (sign of a), making it ideal for graphing. The standard form y = ax² + bx + c directly shows the y-intercept (c) and is more convenient for the quadratic formula. Converting between them requires completing the square: factor a from the first two terms, add and subtract (b/2a)² inside the parentheses, and simplify. Mastering this conversion is essential for algebra and pre-calculus courses.
Ignoring air resistance, any object thrown at an angle follows a parabolic trajectory. The projectile's height as a function of horizontal distance is y = x·tan(θ) − (gx²)/(2v₀²cos²(θ)), which is a downward-opening parabola (a < 0). The vertex gives the maximum height, and the x-intercepts give the launch point and landing point. The range is maximized at a 45° launch angle. Understanding parabolas makes projectile motion problems in physics intuitive.
The vertex is the turning point — the minimum (a > 0) or maximum (a < 0) point. In vertex form, it's (h, k).
The focus is a point inside the parabola; the directrix is a line outside. Every point on the parabola is equidistant from both.
The chord through the focus perpendicular to the axis of symmetry. Its length is |1/a|, measuring the parabola's width at focus height.
Complete the square: h = −b/(2a), k = c − b²/(4a). Then y = a(x−h)² + k.
A parabolic reflector focuses all incoming parallel rays to the focus point, maximizing signal strength — this is the optical reflection property. Use this as a practical reminder before finalizing the result.
Yes — equations of the form x = ay² + by + c produce horizontal parabolas. This calculator handles vertical (y = f(x)) parabolas.