Parabola Calculator

Enter coefficients a, b, c for y = ax² + bx + c to find vertex, focus, directrix, axis of symmetry, and intercepts.

Enter Coefficients

y = 1x² + (-4)x + 3

Result

Vertex
--
coordinates

Step-by-Step Solution

Understanding Parabolas

A parabola is the graph of a quadratic function y = ax² + bx + c. It is a U-shaped curve that opens upward when a > 0 and downward when a < 0. The parabola is one of the conic sections, formed by slicing a cone with a plane parallel to one of its sides.

Key Properties of a Parabola

Vertex

The turning point of the parabola, the minimum (a>0) or maximum (a<0).

h = -b/(2a), k = f(h)

Focus

A fixed point inside the parabola that defines its reflective property.

F = (h, k + 1/(4a))

Directrix

A fixed line outside the parabola; every point is equidistant from the focus and directrix.

y = k - 1/(4a)

Axis of Symmetry

The vertical line through the vertex that divides the parabola into mirror images.

x = -b/(2a)

Finding X-Intercepts (Roots)

The x-intercepts are found by solving ax² + bx + c = 0 using the quadratic formula:

x = (-b +/- sqrt(b² - 4ac)) / (2a)

The discriminant D = b² - 4ac determines the number of real roots: two real roots (D > 0), one repeated root (D = 0), or no real roots (D < 0).

Vertex Form

The vertex form of a parabola is y = a(x - h)² + k, where (h, k) is the vertex. Converting from standard form involves completing the square. This form makes it easy to identify the vertex and understand transformations.

Applications of Parabolas

  • Physics: Projectile motion follows a parabolic trajectory.
  • Engineering: Satellite dishes and headlight reflectors use parabolic shapes.
  • Architecture: Suspension bridge cables form parabolic curves.
  • Optics: Parabolic mirrors focus parallel light to a single point (the focus).