Explore the geometry of quadratic functions. Enter coefficients a, b, c for y = ax² + bx + c. Instantly compute vertex, focus, directrix, axis of symmetry, intercepts, and visualize the parabola with its key elements.
A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). For a vertical parabola given by y = ax² + bx + c (a ≠ 0), the vertex represents the extreme point, while the focus and directrix encode the reflective property central to satellite dishes, headlights, and architectural arches.
Given y = ax² + bx + c, complete the square: y = a(x - h)² + k where
h = -b/(2a), k = c - b²/(4a).
Focus: (h, k + 1/(4a)) Directrix: y = k - 1/(4a)
For a quadratic in standard form y = ax² + bx + c, the vertex form y = a(x - h)² + k is obtained via completing the square. The distance from vertex to focus is |p| = 1/|4a|, with p = 1/(4a). The focus lies on the axis of symmetry (vertical line x = h) at (h, k + p). The directrix is the horizontal line y = k - p. If a > 0, parabola opens upward (vertex minimum); if a < 0, opens downward (vertex maximum). The axis of symmetry is x = h. The discriminant Δ = b² - 4ac determines x-intercepts: two real roots if Δ > 0, one double root if Δ = 0, none if Δ < 0.
The reflective property of a parabola states that any ray parallel to the axis of symmetry reflects off the curve toward the focus. This principle underlies satellite dishes, solar concentrators, and parabolic microphones.
Engineers designing a solar thermal collector use a parabolic cross-section y = 0.25x² (a = 0.25). The vertex at (0,0) and focus at (0,1) concentrates sunlight onto a receiver tube. Using our calculator, with a=0.25, b=0, c=0, we find vertex (0,0), focus (0,1), directrix y = -1. The focal length p = 1 ensures optimal heat collection. This precise relationship enables efficient energy conversion and showcases the power of analytic geometry.
Under uniform gravity, a projectile’s trajectory follows a parabolic path y = tan(θ₀)·x - (g/(2v₀²cos²θ₀))·x². By identifying coefficients, one can compute maximum height (vertex y‑coordinate), range (x‑intercepts), and optimal launch angle. Our calculator provides immediate numerical insight into such physics problems.
| Quadratic (a,b,c) | Vertex (h,k) | Focus | Directrix | Intercepts (real) |
|---|---|---|---|---|
| y = x² - 4x + 3 | (2, -1) | (2, -0.75) | y = -1.25 | x=1, x=3 |
| y = -2x² + 8x - 5 | (2, 3) | (2, 2.875) | y = 3.125 | none (Δ<0) |
| y = 0.5x² + x + 1 | (-1, 0.5) | (-1, 1.0) | y = 0.0 | none |