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Example 2a: Functional Variations

Example 2a: Functional Variations
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2023-03-22 01:54
FunctionDerivativeParametric Equation
In collections Realidades de GeoGebra
This example demonstrates the dynamic changes in functions. Key concepts include functions, derivatives, tangents, and equations. Using GeoGebra commands such as Ray, Segment, Point, and Tangent, dynamic observations of function graphs can be made. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Ray(A, B) → a
  2. Segment(S3_1, S3_2) → s_3
  3. Point(s_3) → C1_3
  4. Segment(S2_1, S2_2) → s_2
  5. Point(s_2) → C1_2
  6. Segment(S1_1, S1_2) → s_1
  7. Point(s_1) → C1_1
  8. Tangent(Azul_1, f) → g
  9. Segment(S4_1, S4_2) → s_4
  10. Point(s_4) → C5_4
  11. Point(s_3) → C5_3
  12. Point(s_2) → C5_2
  13. Point(s_1) → C5_1
  14. Point(s_4) → C4_4
  15. Point(s_2) → C4_2
  16. Point(s_3) → C4_3
  17. Point(s_1) → C4_1
  18. Segment(S7_1, S7_2) → s_7
  19. Point(s_7) → C3_7
  20. Segment(S6_1, S6_2) → s_6
  21. Point(s_6) → C3_6
  22. Segment(S5_1, S5_2) → s_5
  23. Point(s_5) → C3_5
  24. Point(s_4) → C3_4
  25. Point(s_3) → C3_3
  26. Point(s_2) → C3_2
  27. Point(s_1) → C3_1
  28. Point(s_3) → C2_3
  29. Point(s_2) → C2_2
  30. Point(s_1) → C2_1
  31. Point(a) → Rojo
  32. OrthogonalLine(Rojo, xAxis) → t
  33. Tangent((xj, f(xj)), f) → e_1
  34. Slope(e_1) → h
  35. If(abs(h) ≥ 0, 1 / sqrt(1 + (h * h)), 1) → m3
  36. Slope(g) → h2
  37. If(abs(h2) ≥ 0, 1 / sqrt(1 + (h2 * h2)), 1) → m2
  38. OrthogonalLine(Azul_1, g) → d
  39. If(xc > xj, (tiempo2, e_2(tiempo2))) → Verde_1
  40. OrthogonalLine(Verde_1, e_1) → l_1
  41. OrthogonalLine(Rojo, xAxis) → i_1
  42. If(elegida ≟ 2, R1, R1 + (-100, 0)) → R2_1
  43. If(elegida ≟ 2, R2, R2 + (-100, 0)) → R2_2
  44. If(elegida ≟ 2, R3, R3 + (-100, 0)) → R2_3
  45. If(elegida ≟ 1, R1, R1 + (-100, 0)) → R1_1
  46. If(elegida ≟ 1, R2, R2 + (-100, 0)) → R1_2
  47. If(elegida ≟ 1, R3, R3 + (-100, 0)) → R1_3
  48. If(elegida ≟ 3, R1, R1 + (-100, 0)) → R3_1
  49. If(elegida ≟ 3, R2, R2 + (-100, 0)) → R3_2
  50. If(elegida ≟ 3, R3, R3 + (-100, 0)) → R3_3
  51. If(elegida ≟ 3, R4, R4 + (-100, 0)) → R3_4
  52. If(elegida ≟ 3, R5, R5 + (-100, 0)) → R3_5
  53. If(elegida ≟ 3, R6, R6 + (-100, 0)) → R3_6
  54. If(elegida ≟ 3, R7, R7 + (-100, 0)) → R3_7
  55. If(elegida ≟ 4, R1, R1 + (-100, 0)) → R4_1
  56. If(elegida ≟ 4, R2, R2 + (-100, 0)) → R4_2
  57. If(elegida ≟ 4, R3, R3 + (-100, 0)) → R4_3
  58. If(elegida ≟ 4, R4, R4 + (-100, 0)) → R4_4
  59. If(elegida ≟ 5, R1, R1 + (-100, 0)) → R5_1
  60. If(elegida ≟ 5, R2, R2 + (-100, 0)) → R5_2
  61. If(elegida ≟ 5, R3, R3 + (-100, 0)) → R5_3
  62. If(elegida ≟ 5, R4, R4 + (-100, 0)) → R5_4
  63. Polygon(PL_1, PL_2, PL_3, PL_4) → PL, pl_1, pl_2, pl_3, pl_4
  64. Polygon(PO_1, PO_2, PO_3, PO_4) → PO, po_1, po_2, po_3, po_4
  65. Segment(S8_1, S8_2) → s_8
  66. Point(s_8) → v
  67. Segment(Azul_4, Azul_3) → v_g

AI Drawing Prompt Reference

In the same coordinate plane, start with a given function curve f. Choose a tangent line g to the graph and mark one or more tangent points on the curve; at each selected point, construct the corresponding tangent line and consider the local slope. For each tangent slope, draw perpendicular/orthogonal lines relative to the coordinate x-axis and relative to the tangent direction, using their intersections to define several auxiliary points. Connect some of these auxiliary points with line segments and keep also a few rays emanating from given reference points. The final figure should clearly show multiple tangent constructions tied to f, together with the resulting slope and orthogonality relationships, highlighting functional variation.
The prompt above was compiled from the original drawing commands and is for reference only.