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Formula of a Parabola

Formula of a Parabola
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2015-12-14 23:09
Coordinate GeometryCalculusParametric EquationGeometric Construction
In collections Conic Section: Parabola
This GeoGebra example demonstrates the geometric definition and formula derivation of a parabola, visually illustrating how it is constructed as the set of points equidistant from a focus and a directrix. The core mathematical concepts involved include parabola properties, definite integrals, real numbers, volume calculations, parametric curves, and cuboid references, highlighting connections across analytic geometry and calculus. The construction employs commands such as Point to place essential elements like Focus and Directrix, Intersect to find key intersections, Segment and Ray to draw geometric components; objects including p_1, c, A, B, and multiple text labels (text1 to text17) provide annotations and step-by-step explanations. This example is ideal for high school or college students studying coordinate geometry, function graphing, or calculus, offering an engaging way to grasp parabola fundamentals through interactive visualization.

GeoGebra Commands & Parameters

  1. Point(yAxis) → Focus
  2. Point(c) → A
  3. Intersect(c, xAxis, 1) → B
  4. Segment(A, Focus) → d
  5. Intersect(Directrix, yAxis) → C
  6. Segment(Focus, B) → p
  7. Segment(B, C) → d_1
  8. Point(xAxis) → D
  9. Ray(A, D) → e
  10. Intersect(e, xAxis) → E
  11. Segment(A, E) → a

AI Drawing Prompt Reference

Draw a parabola with a focus on the y-axis, using point A and the intersection of the directrix with the x-axis, B, to construct the basic shape. First, place the focus on the y-axis, then place point A on the parabola and draw segment d through A and the intersection B with the x-axis. Next, draw segment p from the focus to B and segment d1 from B to the intersection C of the directrix with the y-axis. Place point D on the x-axis and draw ray e through A and D, then find the intersection E of ray e with the x-axis, and finally draw segment AE.
The prompt above was compiled from the original drawing commands and is for reference only.