Parabola Calculator
Calculate vertex, focus, directrix, and axis of symmetry for any parabola
Parabola Properties
Standard Form: y = ax² + bx + c
Frequently Asked Questions
What is a parabola?
A parabola is a U-shaped curve that represents a quadratic function. It's the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). Parabolas appear in projectile motion, satellite dishes, and headlight reflectors.
What is the focus?
The focus is a point inside the parabola. All points on the parabola are equidistant from the focus and the directrix. Light rays parallel to the axis reflect through the focus, which is why parabolic mirrors concentrate light.
What is the directrix?
The directrix is a line outside the parabola. It's perpendicular to the axis of symmetry and helps define the parabola's shape. The distance from any point on the parabola to the focus equals its distance to the directrix.
How do you find the focus and directrix?
For y = ax² + bx + c, first find the vertex (h, k). Then p = 1/(4a). The focus is at (h, k + p) and the directrix is the line y = k - p. The value p represents the distance from vertex to focus.
What determines if it opens up or down?
The sign of coefficient 'a' determines direction. If a > 0, the parabola opens upward (vertex is minimum). If a < 0, it opens downward (vertex is maximum). The magnitude of |a| affects how wide or narrow the parabola is.
What is the latus rectum?
The latus rectum is a line segment through the focus, perpendicular to the axis of symmetry, with endpoints on the parabola. Its length is |1/a|. It helps in sketching the parabola accurately.