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Graphing Hyperbolic Cosine Function

Graphing Hyperbolic Cosine Function
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2018-07-31 07:16
FunctionTrigonometryCoordinate GeometryCalculusGraphing
In collections Phantom Graphs
This example demonstrates how to plot the graph of the hyperbolic cosine function cosh(x^2/2) and visualize its behavior as x changes. Key mathematical concepts include hyperbolic functions, coordinate geometry, and function graphing. The construction utilizes GeoGebra commands and objects such as Plane, CurveCartesian, and Sequence. Suitable for high school and above students to understand the plotting and properties of complex function graphs.

GeoGebra Commands & Parameters

  1. Plane((0, c, 0), (1, c, 0), (0, c, 1)) → p
  2. CurveCartesian(0, cosh(t^(2) / 2), t, t, -5, 5) → a
  3. CurveCartesian(t, cos(t^(2)), t, t, -5, 5) → b
  4. CurveCartesian(t, cos(t^(2)), (-t), t, -5, 5) → d
  5. Sequence((sqrt(acos(c) + ((2 * π) * k)), c, sqrt(acos(c) + ((2 * π) * k))), k, 0, 5) → Pts_1
  6. Sequence(((-sqrt(acos(c) + ((2 * π) * k))), c, sqrt(acos(c) + ((2 * π) * k))), k, 0, 5) → Pts_3
  7. Sequence(((-sqrt(acos(c) + ((2 * π) * k))), c, (-sqrt(acos(c) + ((2 * π) * k)))), k, 0, 5) → Pts_4
  8. Sequence((sqrt(acos(c) + ((2 * π) * k)), c, (-sqrt(acos(c) + ((2 * π) * k)))), k, 0, 5) → Pts_2
  9. Sequence((sqrt((-acos(c)) + ((2 * π) * k)), c, sqrt((-acos(c)) + ((2 * π) * k))), k, 0, 5) → Pts_5
  10. Sequence((sqrt((-acos(c)) + ((2 * π) * k)), c, (-sqrt((-acos(c)) + ((2 * π) * k)))), k, 0, 5) → Pts_6
  11. Sequence(((-sqrt((-acos(c)) + ((2 * π) * k))), c, sqrt((-acos(c)) + ((2 * π) * k))), k, 0, 5) → Pts_7
  12. Sequence(((-sqrt((-acos(c)) + ((2 * π) * k))), c, (-sqrt((-acos(c)) + ((2 * π) * k)))), k, 0, 5) → Pts_8
  13. Sequence((sqrt(acosh(((-1)^(k) * c)) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))), c, (k * π / sqrt(acosh(((-1)^(k) * c)) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))))), k, 1, 5) → Pts_9
  14. Sequence((sqrt(acosh(((-1)^(k) * c)) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))), c, ((-k) * π / sqrt(acosh(((-1)^(k) * c)) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))))), k, 1, 5) → Pts_{10}
  15. Sequence(((-sqrt(acosh(((-1)^(k) * c)) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2))))), c, (k * π / sqrt(acosh(((-1)^(k) * c)) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))))), k, 1, 5) → Pts_{11}
  16. Sequence(((-sqrt(acosh(((-1)^(k) * c)) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2))))), c, ((-k) * π / sqrt(acosh(((-1)^(k) * c)) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))))), k, 1, 5) → Pts_{12}
  17. Sequence((sqrt((-acosh(((-1)^(k) * c))) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))), c, (k * π / sqrt((-acosh(((-1)^(k) * c))) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))))), k, 1, 5) → Pts_{13}
  18. Sequence((sqrt((-acosh(((-1)^(k) * c))) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))), c, ((-k) * π / sqrt((-acosh(((-1)^(k) * c))) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))))), k, 1, 5) → Pts_{14}
  19. Sequence(((-sqrt((-acosh(((-1)^(k) * c))) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2))))), c, (k * π / sqrt((-acosh(((-1)^(k) * c))) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))))), k, 1, 5) → Pts_{15}
  20. Sequence(((-sqrt((-acosh(((-1)^(k) * c))) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2))))), c, ((-k) * π / sqrt((-acosh(((-1)^(k) * c))) + sqrt(acosh(((-1)^(k) * c))^(2) + (k^(2) * π^(2)))))), k, 1, 5) → Pts_{16}
  21. Sequence(CurveCartesian[t, ((-1)^(k) * cosh((0.5 * (t^(2) - (k^(2) * π^(2) / t^(2)))))), (k * π / t), t, -5, 5], k, 1, 7) → Curves1
  22. Sequence(CurveCartesian[t, ((-1)^(k) * cosh((0.5 * (t^(2) - (k^(2) * π^(2) / t^(2)))))), ((-k) * π / t), t, -5, 5], k, 1, 7) → Curves2