📐 Geometry & Conic Sections
Quadratic Parabola Vertex & Focus Calculator
Compute parabola vertex $(h, k)$, focal point, directrix line ($y = k - \frac{1}{4a}$), axis of symmetry ($x = h$), and discriminant roots for quadratic functions ($y = ax^2 + bx + c$).
Coefficients ($y = ax^2 + bx + c$)
Parabola Vertex (h, k)
(3.000, -4.000)
Focus Point $(F)$
(3.000, -3.750)
Directrix Line
y = -4.250
Axis of Symmetry
x = 3.000
💡 Real X-Intercept Roots:
Discriminant $\Delta = 16.00 \implies x_1 = 5.000, \quad x_2 = 1.000$
Geometric Properties of a Parabola
A parabola is the locus of points equidistant from a point (focus) and a line (directrix):
- Vertex: $$h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a}$$
- Focal Length ($p$): $$p = \frac{1}{4a}$$
- Focus: $$(h, k + p) = \left(h, k + \frac{1}{4a}\right)$$
- Directrix: $$y = k - p = k - \frac{1}{4a}$$