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Odd Functions: Another Look (Take 2)!

Odd Functions: Another Look (Take 2)!
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2016-11-25 22:16
FunctionTransformationGeometric ConstructionSlider AnimationGraphing
① This interactive GeoGebra example dynamically illustrates the geometric property of odd functions: for any point on the graph, its reflection through the origin also lies on the graph. ② Core mathematical concepts include the definition of odd functions, point symmetry, and function transformations. ③ Construction highlights: uses commands such as PointIn to select points on the function, Mirror to create the origin-symmetric point, FitPoly to generate a modifiable polynomial odd function, and includes sliders and buttons for dynamic interaction; objects include points, functions, segments, vectors, rotations, and translations. ④ Suitable for high school and above, ideal for exploring function properties, symmetry, and for classroom demonstrations.

GeoGebra Commands & Parameters

  1. PointIn(a) → B
  2. PointIn(b) → C
  3. PointIn(c) → D
  4. PointIn(d) → E
  5. FitPoly(list, 4) → f
  6. Mirror(g, x = 0) → g_1
  7. Point(g) → F
  8. Intersect(h, y = 0) → G
  9. Mirror(G, x = 0) → G'
  10. Mirror(F, A) → F'
  11. Dilate(H, If[0 < e ≤ 1, 1 - e, 0], G) → H'
  12. Segment(H, H') → i
  13. Dilate(G, If[1 < e ≤ 2, 2 - e, 0], F) → G'_1
  14. Segment(G, G'_1) → j
  15. Rotate(H, If[2 < e ≤ 3, ((-(e - 2)) * π), (-π)], H) → H'_1
  16. Rotate(H', If[2 < e ≤ 3, ((-(e - 2)) * π), (-π)], H) → H''
  17. Dilate(F, If[4 < e ≤ 5, 5 - e, 0], F') → F'_1
  18. Vector(F, F'_1) → u
  19. Translate(G, u) → G'_2
  20. Translate(G'_1, u) → I
  21. Mirror(g, A) → g_2
  22. Rotate(G, If[4 < e ≤ 5, ((-(e - 4)) * π), (-π)], G) → G'_3
  23. Rotate(G'_1, If[4 < e ≤ 5, ((-(e - 4)) * π), (-π)], G) → J
  24. Dilate(G, If[5 < e ≤ 6, 6 - e, 0], G') → G'_4
  25. Vector(G, G'_4) → v
  26. Translate(G'_3, v) → K
  27. Translate(J, v) → J'
  28. Dilate(A, If[2 < e ≤ 3, 3 - e, 0], F) → A'
  29. Segment(A, A') → l
  30. Rotate(A, If[5 < e ≤ 6, ((e - 5) * π), π], A) → A'_1
  31. Rotate(A', If[5 < e ≤ 6, ((e - 5) * π), π], A) → A''
  32. Rotate(F, If[3 < e ≤ 4, ((e - 3) * π), π], A) → F'_2
  33. Rotate(G, If[5 < e ≤ 6, ((e - 5) * π), π], A) → G'_5
  34. Rotate(G'_1, If[5 < e ≤ 6, ((e - 5) * π), π], A) → L
  35. Segment(G, F) → m
  36. Rotate(G, If[3 < e ≤ 4, ((e - 3) * π), π], A) → G'_6
  37. Rotate(G'_1, If[3 < e ≤ 4, ((e - 3) * π), π], A) → M
  38. Segment(G'_6, M) → j'
  39. Rotate(H, If[3 < e ≤ 4, ((e - 3) * π), π], A) → H'_2
  40. Rotate(H', If[3 < e ≤ 4, ((e - 3) * π), π], A) → H''_1
  41. Segment(H'_2, H''_1) → i'
  42. Angle(F, A, F'_2) → α
  43. Rotate(g, If[4.5 < e ≤ 5.5, ((e - 4.5) * π), π], A) → g_3
  44. Rotate(F, If[4.5 < e ≤ 5.5, ((e - 4.5) * π), π], A) → F'_3
  45. Angle(F, A, F'_3) → β
  46. Rotate(A, If[3 < e ≤ 4, ((e - 3) * π), π], A) → A'_2
  47. Rotate(A', If[3 < e ≤ 4, ((e - 3) * π), π], A) → A''_1
  48. Segment(A'_2, A''_1) → l'
  49. Rotate(A, If[4.5 < e ≤ 5.5, ((e - 4.5) * π), π], A) → A'_3
  50. Rotate(A', If[4.5 < e ≤ 5.5, ((e - 4.5) * π), π], A) → A''_2
  51. Segment(A'_3, A''_2) → l'_1

AI Drawing Prompt Reference

Draw a dynamic illustration of the geometric property of odd functions: generate a modifiable odd function, select any point on the graph, reflect it through the origin, and show that the reflected point remains on the function. Include points, function curves, segments, vectors, rotations, and dilations, with sliders and buttons for dynamic interaction to explore point symmetry and function transformations.
The prompt above was compiled from the original drawing commands and is for reference only.