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Function Machine Visualization: Substitute Values to Reveal Outputs

Function Machine Visualization: Substitute Values to Reveal Outputs
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2021-03-24 01:37
VectorFunctionCalculusGeometric ConstructionGraphing
In collections Introduction to Linear Functions and Graphs
This example demonstrates a “function machine” that transforms input values into output values using a specified rule, while making the substitution process visually explicit. It focuses on core ideas such as function evaluation by substitution, linear functions and ratio/proportional relationships, and supporting mathematical interpretation via constructed geometric elements (with connections to integral-calculus style visualization depending on the displayed objects). Construction highlights include drawing Cartesian curves with CurveCartesian, using Circle and Semicircle for circular/arc components, and building structural shapes with Polygon and Segment. Key input/output points are marked with Point, while Vector is used to emphasize directional relationships in the transformation. The example also incorporates Mod for modular or cyclical mapping behavior and Element to retrieve derived geometric objects. For dynamic labeling and formula display, it uses Textfield, Text, LaTeX, and TextToUnicode, while Remove and RemoveUndefined keep the scene consistent by deleting invalid objects. Overall, it is well suited for middle-school and early high-school students learning what it means to apply a function rule, and it works as an interactive classroom demonstration for understanding function mappings and related algebraic reasoning.

GeoGebra Commands & Parameters

  1. CurveCartesian(x(A) + ((r * (a + (1 / b * tanh((b * sin((n * t) + v)))))) * cos(t + v)), y(A) + ((r * (a + (1 / b * tanh((b * sin((n * t) + v)))))) * sin(t + v)), t, 0, 6.28319) → c
  2. CurveCartesian(x(B) + ((R * (a + (1 / b * tanh((b * sin((n * t) - v)))))) * cos(t - v)), y(B) + ((R * (a + (1 / b * tanh((b * sin((n * t) - v)))))) * sin(t - v)), t, 0, 6.28319) → d
  3. Circle(B, R + 0.9) → e
  4. Circle(A, r + 0.9) → g
  5. Circle(A, r + 0.9) → h
  6. CurveCartesian(x(A) + ((r * (a + (1 / b * tanh((b * sin((n * t) + v)))))) * cos(t + v)), y(A) + ((r * (a + (1 / b * tanh((b * sin((n * t) + v)))))) * sin(t + v)), t, 0, 6.28319) → i
  7. CurveCartesian(x(C) + ((R * (a + (1 / b * tanh((b * sin((n * t) - v)))))) * cos(t - v)), y(C) + ((R * (a + (1 / b * tanh((b * sin((n * t) - v)))))) * sin(t - v)), t, 0, 6.28319) → j
  8. CurveCartesian(x(C) + ((r / 2 * (a + (1 / b * tanh((b * sin((n * t) + v)))))) * cos(t + v)), y(C) + ((r / 2 * (a + (1 / b * tanh((b * sin((n * t) + v)))))) * sin(t + v)), t, 0, 6.28319) → o
  9. Polygon(D, E, F, G) → poly1, d_1, e_1, f_1, g_1
  10. Point(g_1) → H
  11. Point(g_1) → I
  12. Polygon(H, I, J, K) → poly2, h_1, i_1, j_1, k
  13. Polygon(L, K, J, M) → poly3, l, k_1, j_2, m
  14. Point(f_1) → N
  15. Polygon(N, O, P, F) → poly4, n_1, o_1, p, f_2
  16. Textfield(s) → ChampTexte1
  17. Vector(Q, S) → u
  18. Textfield(f) → ChampTexte2
  19. Vector(U, V) → w
  20. Vector(W, Z) → z
  21. Polygon(C_1, M, L, D_1) → poly5, c_1, m_1, l_1, d_2
  22. Polygon(E_1, F_1, G_1, H_1) → poly6, e_2, f_3, g_2, h_2
  23. Semicircle(F_1, G_1) → q
  24. Mod(v / °, 2) → t_1
  25. Semicircle(F_1, G_1) → p_1
  26. Segment(A_1, B_1) → a_1
  27. LaTeX(f) → textFromFunction
  28. TextToUnicode(textFromFunction) → L_3
  29. Text("(" + s + ")") → textFormulaValue
  30. TextToUnicode(textFormulaValue) → L_n
  31. Remove(Insert[L_n, L_3, IndexOf[120, L_3]], {120}) → L_4
  32. Remove(Insert[L_n, L_4, IndexOf[120, L_4]], {120}) → L_5
  33. Remove(Insert[L_n, L_5, IndexOf[120, L_5]], {120}) → L_6
  34. Remove(Insert[L_n, L_6, IndexOf[120, L_6]], {120}) → L_7
  35. Remove(Insert[L_n, L_7, IndexOf[120, L_7]], {120}) → L_8
  36. Remove(Insert[L_n, L_8, IndexOf[120, L_8]], {120}) → L_9
  37. RemoveUndefined({L_3, L_4, L_5, L_6, L_7, L_8, L_9}) → L_{10}
  38. Element(L_{10}, Length[L_{10}]) → L_{11}
  39. LaTeX("= " + (UnicodeToText[L_{11}]), true) → textEnumFunction

AI Drawing Prompt Reference

This is a schematic of a function machine showing how input values transform into output values. The diagram includes marked input/output points, curves, circles, arcs, and polygons. The construction order is: first draw curves and circles, then add arcs and polygons, and finally use line segments and vectors to emphasize directional relationships. The overall presentation illustrates function mapping in an intuitive way, suitable for understanding the application of function rules.
The prompt above was compiled from the original drawing commands and is for reference only.