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Exponential Growth vs. Linear Growth

Exponential Growth vs. Linear Growth
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2020-10-09 17:23
FunctionGeometric Construction
This example dynamically illustrates the difference between exponential and linear growth. Key mathematical concepts include exponential functions, linear equations, and geometric transformations. Construction involves GeoGebra commands like Vector, Translate, Polygon, Dilate, etc. Suitable for high school students learning function graphs and geometric transformations.

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(G, G'_1) → k_1
  47. Segment(O, O'_1) → l_1
  48. Segment(L, L'_1) → m_1
  49. Dilate(G'_1, If[1 < p_1 ≤ 2, 2 - p_1, 0], O) → P
  50. Segment(G'_1, P) → q_1
  51. Midpoint(O'_1, L) → Q
  52. Midpoint(L'_1, F) → R
  53. Dilate(G'_1, If[2 < p_1 ≤ 3, 3 - p_1, 0], O'_1) → S
  54. Vector(G'_1, S) → v_1
  55. Translate(G'_1, v_1) → T
  56. Translate(P, v_1) → P'
  57. Segment(T, P') → j_1
  58. Dilate(G'_1, If[3 < p_1 ≤ 4, 4 - p_1, 0], Q) → U
  59. Vector(G'_1, U) → w_1
  60. Translate(G'_1, w_1) → V
  61. Translate(P, w_1) → P'_1
  62. Segment(V, P'_1) → n_1
  63. Segment(O'_1, L) → r_1
  64. Dilate(O'_1, If[4 < p_1 ≤ 5, 5 - p_1, 0], L'_1) → W
  65. Vector(O'_1, W) → s_1
  66. Translate(O'_1, s_1) → Z
  67. Translate(L, s_1) → L'_2
  68. Segment(Z, L'_2) → t_1
  69. Dilate(O'_1, If[5 < p_1 ≤ 6, 6 - p_1, 0], R) → A_1
  70. Vector(O'_1, A_1) → u_2
  71. Translate(O'_1, u_2) → B_1
  72. Translate(L, u_2) → L'_3
  73. Segment(B_1, L'_3) → f_2
  74. Translate(G, u) → G'_2
  75. Translate(C_1, u) → F_1
  76. Translate(D_1, u) → G_1
  77. Dilate(G, If[0 < a_2 ≤ 1, 1 - a_2, 0], G'_2) → G'_3
  78. Dilate(C_1, If[0 < a_2 ≤ 1, 1 - a_2, 0], F_1) → H_1
  79. Dilate(D_1, If[0 < a_2 ≤ 1, 1 - a_2, 0], G_1) → I_1
  80. Segment(G, G'_3) → g_2
  81. Segment(C_1, H_1) → h_2
  82. Segment(D_1, I_1) → i_2
  83. Dilate(G'_2, If[1 < a_2 ≤ 2, 2 - a_2, 0], C_1) → J_1
  84. Segment(G'_2, J_1) → j_2
  85. IntegralBetween(f, g, 0, 1) → b_2
  86. IntegralBetween(f, g_1, 1, 2) → c_2
  87. IntegralBetween(f, h, 2, 3) → d_2
  88. Segment(D_1, F_1) → k_2
  89. Circle(F_1, e_2) → p_2
  90. Intersect(p_2, k_2) → K_1,
  91. Circle(K_1, e_2) → q_2
  92. Intersect(q_2, k_2) → L_1, M_1
  93. Dilate(G'_2, If[2 < a_2 ≤ 3, 3 - a_2, 0], M_1) → N_1
  94. Vector(G'_2, N_1) → v_2
  95. Translate(G'_2, v_2) → O_1
  96. Translate(J_1, v_2) → P_1
  97. Segment(O_1, P_1) → l_2
  98. Rotate(O_1, If[3 < a_2 ≤ 4, ((a_2 - 3) * π), π], K_1) → R_1
  99. Rotate(P_1, If[3 < a_2 ≤ 4, ((a_2 - 3) * π), π], K_1) → S_1
  100. Segment(R_1, S_1) → m_2
  101. Rotate(R_1, If[4 < a_2 ≤ 5, ((a_2 - 4) * π), π], L_1) → T_1
  102. Rotate(S_1, If[4 < a_2 ≤ 5, ((a_2 - 4) * π), π], L_1) → U_1
  103. Segment(T_1, U_1) → n_2
  104. Segment(M_1, D_1) → r_2
  105. Dilate(M_1, If[5 < a_2 ≤ 6, 6 - a_2, 0], G_1) → Q_1
  106. Vector(M_1, Q_1) → w_2
  107. Translate(M_1, w_2) → V_1
  108. Translate(D_1, w_2) → W_1
  109. Segment(V_1, W_1) → s_2
  110. Midpoint(W_1, E_1) → Z_1
  111. Rotate(V_1, If[6 < a_2 ≤ 7, ((a_2 - 6) * π), π], W_1) → A_2
  112. Rotate(W_1, If[6 < a_2 ≤ 7, ((a_2 - 6) * π), π], W_1) → B_2
  113. Segment(A_2, B_2) → t_2
  114. Rotate(A_2, If[7 < a_2 ≤ 8, ((a_2 - 7) * π), π], Z_1) → C_2
  115. Rotate(B_2, If[7 < a_2 ≤ 8, ((a_2 - 7) * π), π], Z_1) → D_2
  116. Segment(C_2, D_2) → f_3

AI Drawing Prompt Reference

Create a dynamic illustration comparing exponential growth versus linear growth. Show a series of stepped polygons rising from left to right, with each step increasing in height to demonstrate accelerating exponential growth, alongside another set of parallel stepped polygons with uniform step heights showing steady linear growth. Include shaded integral areas bounded by function curves beneath the steps. As the animation progresses from start to finish, each step reveals progressively while the integral areas change, clearly showing the difference between the two growth patterns. The overall appearance is that of a mathematical chart with axes, curves, and filled polygon regions.
The prompt above was compiled from the original drawing commands and is for reference only.