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Exponential Growth & Decay Visualization

Exponential Growth & Decay Visualization
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2016-10-04 10:40
FunctionTrigonometryCalculus
This example dynamically illustrates exponential growth and decay. Key mathematical concepts include exponential functions and their graph characteristics. Using GeoGebra commands like Vector, Translate, Polygon, and Dilate, the impact of parameter changes on the function graph can be observed intuitively. Suitable for high school mathematics education, particularly in exponential functions and calculus courses.

GeoGebra Commands & Parameters

  1. Vector((0, 0), (1, 0)) → u
  2. Translate(G, u) → G'
  3. Translate(B, u) → B'
  4. Translate(C, u) → C'
  5. Translate(D, u) → D'
  6. Translate(A, u) → A'
  7. Polygon(A, A', B) → poly1, b, a_1, a'
  8. Polygon(B, B', C) → poly2, c, b_1, b'
  9. Polygon(C, C', D) → poly3, d, c_1, c'
  10. Polygon(D, D', E) → poly4, e, d_1, d'
  11. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], A') → A'_1
  12. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], B') → B'_1
  13. Dilate(C, If[0 < o ≤ 1, 1 - o, 0], C') → C'_1
  14. Dilate(D, If[0 < o ≤ 1, 1 - o, 0], D') → D'_1
  15. Segment(A, A'_1) → p
  16. Segment(B, B'_1) → q
  17. Segment(C, C'_1) → r
  18. Segment(D, D'_1) → s
  19. Dilate(A'_1, If[1 < o ≤ 2, 2 - o, 0], B) → H
  20. Segment(A'_1, H) → t
  21. Dilate(A'_1, If[2 < o ≤ 3, 3 - o, 0], B'_1) → I
  22. Vector(A'_1, I) → v
  23. Translate(A'_1, v) → J
  24. Translate(H, v) → H'
  25. Segment(J, H') → t'
  26. Dilate(B'_1, If[3 < o ≤ 4, 4 - o, 0], C'_1) → K
  27. Vector(B'_1, K) → w
  28. Translate(J, w) → J'
  29. Translate(H', w) → H''
  30. Segment(J', H'') → t''
  31. Dilate(C'_1, If[4 < o ≤ 5, 5 - o, 0], D'_1) → M
  32. Vector(C'_1, M) → u_1
  33. Translate(J', u_1) → J''
  34. Translate(H'', u_1) → H'''
  35. Segment(J'', H''') → t'''
  36. Translate(L, u) → L'
  37. Translate(F, u) → F'
  38. Line(A, B) → f_1
  39. IntegralBetween(f, g, 0, 1) → e_1
  40. Translate(O, u) → O'
  41. IntegralBetween(f, g_1, 1, 2) → h_1
  42. IntegralBetween(f, h, 2, 3) → i_1
  43. Dilate(O, If[0 < p_1 ≤ 1, 1 - p_1, 0], O') → O'_1
  44. Dilate(L, If[0 < p_1 ≤ 1, 1 - p_1, 0], L') → L'_1
  45. Dilate(G, If[0 < p_1 ≤ 1, 1 - p_1, 0], G') → G'_1
  46. Segment(O, O'_1) → l_1
  47. Segment(L, L'_1) → m_1
  48. Dilate(G'_1, If[1 < p_1 ≤ 2, 2 - p_1, 0], O) → P
  49. Segment(G'_1, P) → q_1
  50. Dilate(G'_1, If[2 < p_1 ≤ 3, 3 - p_1, 0], O'_1) → S
  51. Vector(G'_1, S) → v_1
  52. Translate(G'_1, v_1) → T
  53. Translate(P, v_1) → P'
  54. Segment(O'_1, L) → r_1
  55. Dilate(O'_1, If[4 < p_1 ≤ 5, 5 - p_1, 0], L'_1) → W
  56. Vector(O'_1, W) → s_1
  57. Translate(O'_1, s_1) → Z
  58. Translate(L, s_1) → L'_2
  59. Translate(G, u) → G'_2
  60. Translate(C_1, u) → F_1
  61. Translate(D_1, u) → G_1
  62. Dilate(G, If[0 < a_2 ≤ 1, 1 - a_2, 0], G'_2) → G'_3
  63. Dilate(C_1, If[0 < a_2 ≤ 1, 1 - a_2, 0], F_1) → H_1
  64. Dilate(D_1, If[0 < a_2 ≤ 1, 1 - a_2, 0], G_1) → I_1
  65. Segment(G, G'_3) → g_2
  66. Segment(C_1, H_1) → h_2
  67. Segment(D_1, I_1) → i_2
  68. Dilate(G'_2, If[1 < a_2 ≤ 2, 2 - a_2, 0], C_1) → J_1
  69. Segment(G'_2, J_1) → j_2
  70. IntegralBetween(f, g, 0, 1) → b_2
  71. IntegralBetween(f, g_1, 1, 2) → c_2
  72. IntegralBetween(f, h, 2, 3) → d_2
  73. Segment(D_1, F_1) → k_2
  74. Circle(F_1, e_2) → p_2
  75. Intersect(p_2, k_2) → K_1,
  76. Circle(K_1, e_2) → q_2
  77. Intersect(q_2, k_2) → L_1, M_1
  78. Dilate(G'_2, If[2 < a_2 ≤ 3, 3 - a_2, 0], M_1) → N_1
  79. Vector(G'_2, N_1) → v_2
  80. Translate(G'_2, v_2) → O_1
  81. Translate(J_1, v_2) → P_1
  82. Segment(O_1, P_1) → l_2
  83. Rotate(O_1, If[3 < a_2 ≤ 4, ((a_2 - 3) * π), π], K_1) → R_1
  84. Rotate(P_1, If[3 < a_2 ≤ 4, ((a_2 - 3) * π), π], K_1) → S_1
  85. Segment(R_1, S_1) → m_2
  86. Rotate(R_1, If[4 < a_2 ≤ 5, ((a_2 - 4) * π), π], L_1) → T_1
  87. Rotate(S_1, If[4 < a_2 ≤ 5, ((a_2 - 4) * π), π], L_1) → U_1
  88. Segment(T_1, U_1) → n_2
  89. Segment(M_1, D_1) → r_2
  90. Dilate(M_1, If[5 < a_2 ≤ 6, 6 - a_2, 0], G_1) → Q_1
  91. Vector(M_1, Q_1) → w_2
  92. Translate(M_1, w_2) → V_1
  93. Translate(D_1, w_2) → W_1
  94. Segment(V_1, W_1) → s_2
  95. Midpoint(W_1, E_1) → Z_1
  96. Rotate(V_1, If[6 < a_2 ≤ 7, ((a_2 - 6) * π), π], W_1) → A_2
  97. Rotate(W_1, If[6 < a_2 ≤ 7, ((a_2 - 6) * π), π], W_1) → B_2
  98. Segment(A_2, B_2) → t_2
  99. Rotate(A_2, If[7 < a_2 ≤ 8, ((a_2 - 7) * π), π], Z_1) → C_2
  100. Rotate(B_2, If[7 < a_2 ≤ 8, ((a_2 - 7) * π), π], Z_1) → D_2
  101. Segment(C_2, D_2) → f_3
  102. Translate(F, u) → F'_1
  103. Intersect(f, n, (4, 12.713506250000002)) → E_2
  104. Dilate(F, If[0 < p_1 ≤ 1, 1 - p_1, 0], F'_1) → F'_2
  105. Segment(F, F'_2) → g_3
  106. Dilate(O'_1, If[2 < p_1 ≤ 3, 3 - p_1, 0], L) → Q
  107. Segment(O'_1, Q) → j_1
  108. Distance(O'_1, L) → distanceO'1L
  109. Dilate(L'_1, If[3 < p_1 ≤ 4, 4 - p_1, 0], F) → R
  110. Segment(L'_1, R) → t_1
  111. Distance(L'_1, F) → distanceL'1F
  112. Dilate(F'_1, If[4 < p_1 ≤ 5, 5 - p_1, 0], N) → U
  113. Segment(F'_1, U) → f_2
  114. IntegralBetween(f, k_1, x(F), x(F'_1)) → w_1
  115. Distance(F'_1, N) → distanceF'1N
  116. Textfield(base) → InputBox1
  117. Segment(G, G'_1) → i_3

AI Drawing Prompt Reference

Create a dynamic visualization of exponential growth and decay, featuring multiple polygons and lines. Start by drawing a horizontal vector, then translate four points based on this vector to get four new points. Form four adjacent polygons using these eight points, observing how parameter changes affect their shapes. Next, scale each polygon and connect the scaled vertices to the original vertices with lines. Finally, draw several lines connecting specific points and calculate the integral of a function over given intervals, representing the results with different colors or styles.
The prompt above was compiled from the original drawing commands and is for reference only.