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Sketching Graphs from Derivative Information

Sketching Graphs from Derivative Information
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2017-01-30 06:19
FunctionCalculusDerivativeOptimizationGraphing
This GeoGebra example demonstrates how to sketch a function's graph based on derivative information, such as extrema and inflection points. It illustrates the relationship between the sign of the derivative, monotonicity, concavity, and the shape of the curve. The construction uses commands like FitPoly to fit polynomial segments, Extremum and TurningPoint to automatically locate critical and inflection points, and Text, If, Segment, Point for annotations and connecting segments. Objects include multiple points (A, B, C, etc.), line segments (segA to segK), and parabolic arcs (parABC, parBCD, etc.) that approximate the function between key points. This is ideal for high school or introductory college calculus students to visualize how derivative information guides graph sketching.

GeoGebra Commands & Parameters

  1. Text("This function is initially:") → text8
  2. Text("f'(x) = 0 when x is:") → text4
  3. If(f'(1) < 0, "Decreasing", "Increasing") → text9
  4. Segment((1, -5), (1, 5)) → segA
  5. Segment((2, -5), (2, 5)) → segB
  6. Segment((3, -5), (3, 5)) → segC
  7. Segment((4, -5), (4, 5)) → segD
  8. Segment((5, -5), (5, 5)) → segE
  9. Segment((6, -5), (6, 5)) → segF
  10. Segment((7, -5), (7, 5)) → segG
  11. Segment((8, -5), (8, 5)) → segH
  12. Segment((9, -5), (9, 5)) → segI
  13. Segment((10, -5), (10, 5)) → segJ
  14. Segment((11, -5), (11, 5)) → segK
  15. Point(segA) → ZA
  16. Point(segB) → ZB
  17. Point(segC) → ZC
  18. Point(segD) → ZD
  19. Point(segE) → ZE
  20. Point(segF) → ZF
  21. Point(segG) → ZG
  22. Point(segH) → ZH
  23. Point(segI) → ZI
  24. Point(segJ) → ZJ
  25. Point(segK) → ZK
  26. FitPoly({ZA, ZB, ZC}, 2) → parABC
  27. FitPoly({ZB, ZC, ZD}, 2) → parBCD
  28. FitPoly({ZC, ZD, ZE}, 2) → parCDE
  29. FitPoly({ZD, ZE, ZF}, 2) → parDEF
  30. FitPoly({ZE, ZF, ZG}, 2) → parEFG
  31. FitPoly({ZF, ZG, ZH}, 2) → parFGH
  32. FitPoly({ZG, ZH, ZI}, 2) → parGHI
  33. FitPoly({ZH, ZI, ZJ}, 2) → parHIJ
  34. FitPoly({ZI, ZJ, ZK}, 2) → parIJK
  35. Text("Tick the box below to check your solution!") → text11
  36. Text("Passes through points at:") → text1
  37. Text("Drag the red points to match the graph described on the left") → text10
  38. FitPoly({A, B, C, D, E}, 4) → p
  39. Extremum(p, 1, 11) → TP_1, TP_2, TP_3
  40. Text(x(TP_1)) → stattext1
  41. TurningPoint(p) → IP_1, IP_2
  42. Text(IP_1) → inftext1
  43. Text(IP_2) → inftext2
  44. Text(x(TP_2)) → stattext2
  45. Text(x(TP_3)) → stattext3