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Parabola Equation Anatomy via Locus Construction

Parabola Equation Anatomy via Locus Construction
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2023-11-15 10:19
FunctionCoordinate GeometryGeometric ConstructionLocusGraphing
This example demonstrates how to visually “anatomize” a parabola equation by building geometric constraints and turning parameter changes into a locus (track) of constructed points. It centers on core ideas from analytic geometry and functions, including the geometric characterization of a parabola (focus/directrix style reasoning) and the relationship to a quadratic equation. The construction uses GeoGebra tools to encode distance or proportional relations through vectors and segments. Key build steps include creating foundational objects with Point, Line, Vector, and Segment; enforcing constraints with OrthogonalLine; extracting relationships using Intersect and Angle; applying geometric transformations via Dilate and Rotate; and organizing annotations with RotateText. A computed equality (e.g., eq1/eq2) and parameter-driven points are then used to produce the locus behavior. Suitable for secondary school through pre-university instruction, as well as classroom exploration of how algebraic equations emerge from geometric constructions.

GeoGebra Commands & Parameters

  1. Point(Segment[(h, y(C) + 10), (h, y(C) - 10)]) → D
  2. Line(A, B) → f
  3. Vector(A, C) → u
  4. Polygon(A, C, B) → t1, b, a_1, c
  5. RotateText(text2, pi / 2) → text3
  6. Segment(A, C) → h_{1}
  7. Vector(C, D) → v
  8. Vector((h, 0), A) → w
  9. Vector((0, 0), (h, 0)) → j
  10. Point(Segment[(x(C) - 10, k), (x(C) + 10, k)]) → E
  11. Vector(A, E) → p
  12. Point(eq1) → F
  13. OrthogonalLine(F, q) → r
  14. Intersect(q, r) → G
  15. Point(q) → H
  16. Angle(H, G, F) → α
  17. Dilate(D, If[0 < s ≤ 1, 1 - s, 0], F) → D'
  18. Segment(D, D') → t
  19. Angle(G, F, D) → β
  20. Rotate(D, If[1.5 < s ≤ 2.5, ((-(s - 1.5)) * β), (-β)], F) → D'_{1}
  21. Rotate(D', If[1.5 < s ≤ 2.5, ((-(s - 1.5)) * β), (-β)], F) → D''
  22. Segment(D'_{1}, D'') → t'
  23. Point(eq2) → I
  24. Dilate(E, If[0 < s ≤ 1, 1 - s, 0], I) → E'
  25. Segment(E, E') → f_1
  26. OrthogonalLine(I, eq3) → g_1
  27. Intersect(g_1, eq3) → J
  28. Point(eq3) → K
  29. Angle(K, J, I) → γ
  30. Angle(J, I, E) → δ
  31. Rotate(E, If[1.5 < s ≤ 2.5, ((-(s - 1.5)) * δ), (-δ)], I) → E'_{1}
  32. Rotate(E', If[1.5 < s ≤ 2.5, ((-(s - 1.5)) * δ), (-δ)], I) → E''
  33. Segment(E'_{1}, E'') → i_1

AI Drawing Prompt Reference

This prompt aims to demonstrate the trajectory of a parabola equation through geometric construction. The core involves using basic objects like points, lines, vectors, and segments to establish geometric constraints, and introducing concepts such as focus and directrix to explain the geometric characteristics of a parabola. By extracting parameter changes through orthogonal lines, intersections, and angles, and applying transformations like scaling and rotation, trajectory points are generated. Finally, using computed equalities and parameter-driven points, the dynamic behavior of the parabola equation is presented, helping to understand how algebraic equations emerge from geometric constructions.
The prompt above was compiled from the original drawing commands and is for reference only.