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Complex Roots and Rotated Parabola Geometry

Complex Roots and Rotated Parabola Geometry
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2017-01-31 17:34
Coordinate GeometryDerivativeComplex NumberTransformation
This GeoGebra example demonstrates the geometric relationship between complex roots and rotated parabolas. Key concepts include complex numbers, quadratic functions, rotation transformations, and calculus derivatives. Construction utilizes commands like ComplexRoot, Rotate, Parabola, Conic, and Intersect to visualize how complex roots influence the orientation and shape of the parabola. It is designed for high school or university students studying analytic geometry and complex analysis, providing an interactive tool to explore the connection between algebraic roots and geometric curves.

GeoGebra Commands & Parameters

  1. ComplexRoot(x^(2) + x + 6) → z_1OriginalRoot, z_2OriginalRoot2
  2. Midpoint(z_2OriginalRoot2, z_1OriginalRoot) → I
  3. Min(ParabolaOriginal_1, -10, 10) → J
  4. Circle(I, J) → p
  5. Intersect(ParabolaOriginal_1, xAxis, 1) → C
  6. Intersect(ParabolaOriginal_1, xAxis, 2) → D
  7. Rotate(ParabolaOriginal_1, α, I) → CurveRotated
  8. Point(CurveRotated) → Z
  9. Point(CurveRotated) → A_1
  10. Point(CurveRotated) → B_1
  11. Point(CurveRotated) → C_1
  12. Rotate(J, α, I) → J'
  13. Conic(Z, A_1, J', B_1, C_1) → e
  14. Segment(z_1OriginalRoot, z_2OriginalRoot2) → g
  15. Rotate(z_1OriginalRoot, α, I) → z_4RotatedRoot
  16. Rotate(z_2OriginalRoot2, α, I) → z_3RotatedRoot
  17. Segment(z_4RotatedRoot, z_3RotatedRoot) → g'
  18. CircleArc(I, z_1OriginalRoot, z_4RotatedRoot) → h
  19. Angle(z_1OriginalRoot, I, z_4RotatedRoot) → β
  20. Focus(e) → B,
  21. Directrix(e) → f,
  22. Parabola(B, f) → ParabolaRotated_1
  23. Segment(J', z_4RotatedRoot) → i
  24. Intersect(f, i) → E
  25. Intersect(ParabolaRotated_1, p) → F, G, ,
  26. Parabola(B, f) → q
  27. LaTeX(ParabolaRotated_1) → ParabolaRotated
  28. LaTeX(ParabolaRotated, true, true) → text3
  29. LaTeX(ParabolaOriginal_1) → ParabolaOriginal
  30. LaTeX(ParabolaOriginal, true, true) → text2
  31. Point(f) → H
  32. Point(f) → K
  33. Segment(H, K) → n
  34. Midpoint(K, H) → L