MATHEMATICS LAB · GEOMETRY

Study of the Parabola

Change the equation to explore the vertex, focus, directrix, axis, and intersections. Drag points on the Cartesian plane to investigate each construction.

The parameter p must be nonzero; its sign determines the opening direction. The lab supports axis-aligned parabolas opening horizontally or vertically (no rotation).

1. Define the parabola

Orientation and equationGo to graph ↓

Parabola axisEquation form

Examples:

Display options

Controls and information

Grid Geometric constructions

Drag V to move the vertex, F to change parameter p, and T to move the tangency point along the parabola. Use +, −, the mouse wheel, or Fit to adjust the view.

Parabola and vertex V Focus F Directrix Line ℓ Tangent and point T

3. Additional options

Add intersections and tangents to the graph

Intersection with a line ℓLine equation

Tangent and normal at point T on the parabolax-coordinate of T Also show the normal at T

Options update the graph; disabled constructions are neither shown nor calculated.

2. Interactive parabola graph

Axes at 1:1 scale

Parabola, focus, and directrix

4. Results

Updated values

5. Geometric construction

Meaning of focus and directrix

6. Explained calculations

Formulas and numerical substitutions

What you can explore in this version

Equations and properties

Vertex and polynomial forms, opening direction, focus, directrix, axis, and latus rectum.

Intersections

Intersections of the parabola with the axes and with any straight line, including tangent cases.

Tangent and normal

Equations of the tangent and normal at a point T chosen directly on the graph.

Real nondegenerate parabolas with axes parallel to the coordinate axes are supported. Rotated conics and degenerate parabolas are not yet available.