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General Quadratic Equation (Parabola): Imaginary Roots

General Quadratic Equation (Parabola): Imaginary Roots
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2020-01-26 16:49
FunctionCoordinate GeometryComplex Number
This example demonstrates the general quadratic equation with imaginary roots, illustrating the properties of a parabola. Key mathematical concepts include quadratic equations, complex roots, and parabolas. Using GeoGebra commands such as Point, Line, and Intersect, the parabola is constructed and its properties are observed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Point({F_{x}, F_{y}}) → F
  2. Line(F, yAxis) → f
  3. Intersect(d, f) → B
  4. OrthogonalLine(B, d) → i
  5. Intersect(i, xAxis) → D
  6. Line(D, yAxis) → l
  7. Parabola(F, d) → c
  8. Intersect(c, i) → T_{x},
  9. OrthogonalLine(T_{x}, yAxis) → n
  10. Intersect(n, yAxis) → V''
  11. OrthogonalLine(T_{x}, xAxis) → j
  12. Intersect(d, j) → H
  13. OrthogonalLine(H, yAxis) → m
  14. Intersect(m, yAxis) → F''
  15. Distance(V'', F'') → fl''
  16. OrthogonalLine(Point[{0, y(V'') - (sgn(y(F'') - y(V'')) * fl'')}], yAxis) → d''
  17. Parabola(F'', d'') → c''
  18. Intersect(c'', xAxis) → J, K
  19. Line(Point[{0, x(J)}], xAxis) → r_{1}
  20. Line(F, xAxis) → e
  21. If(a ≠ 0, Intersect[d, e], Point[{Infinity, F_{x}}]) → A
  22. OrthogonalLine(A, d) → h
  23. If(a ≠ 0, Intersect[c, h], Point[{Infinity, Infinity}]) → T_{y}
  24. Point({x(T_{y}), 0}) → F'
  25. Point({x(T_{x}), 0}) → V'
  26. Distance(V', F') → fl'
  27. OrthogonalLine(Point[{x(V') - (sgn(x(F') - x(V')) * fl'), 0}], xAxis) → d'
  28. If(a ≠ 0, Parabola[F', d'], f) → c'
  29. OrthogonalLine(T_{y}, xAxis) → k
  30. Intersect(k, xAxis) → E
  31. Intersect(j, xAxis) → G
  32. Segment(F, A) → s_{a}
  33. Segment(A, T_{y}) → s_{c}
  34. Segment(T_{y}, E) → s_{f}
  35. Segment(F, B) → s_{b}
  36. Segment(B, T_{x}) → s_{d}
  37. Segment(T_{x}, G) → s_{e}
  38. Point(c') → z_{1}
  39. If(a ≟ 0, par0(z_{1}), If[imaginary(par1(z_{1})) ≟ 0, par1(z_{1}), par2(z_{1})]) → z'_{1}
  40. Segment(B, D) → s_{g}
  41. Intersect(r_{1}, l) → R_1
  42. Segment(D, R_1) → s_{h}
  43. Angle(d, h) → α
  44. Angle(d, i) → β
  45. Segment(T_{x}, Point[{0, y(T_{x})}]) → s_{j}
  46. Point({imaginary(z_{1}), fr(imaginary(z_{1}))}) → z''_{1}
  47. Segment(G, H) → s_{k}
  48. Segment(H, F'') → s_{l}
  49. Intersect(d, k) → I
  50. Segment(I, V'') → s_{m}
  51. Segment(I, F') → s_{n}
  52. Segment(T_{x}, T_{y}) → s_{o}
  53. CircleArc(Point[{0, 0}], Point[{0, (-x(J))}], Point[{0, x(J)}]) → p
  54. Segment(Point[{0, x(J)}], R_1) → s_{p1}
  55. Line(Point[{0, (-x(J))}], xAxis) → r_{2}
  56. Intersect(r_{2}, l) → R_{2}
  57. Segment(Point[{0, (-x(J))}], R_{2}) → s_{p2}
  58. Segment(D, R_{2}) → s_{i}
  59. Vertex(c) → V, , ,
  60. FirstAxis(c) → g