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Functions can not be seen

Functions can not be seen. Rainer Kaenders University of Cologne GeoGebra Conference Linz 2011. Functions can not be seen. … but can be represented GeoGebra can make the identification of functions with their graphs stronger

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Functions can not be seen

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  1. Functionscan not beseen Rainer Kaenders University of Cologne GeoGebra Conference Linz 2011

  2. Functionscan not beseen • … but canberepresented • GeoGebra canmaketheidentificationoffunctionswiththeirgraphsstronger • Many different representations: showthatthereis not therepresentationof a function • Eachrepresentationhas ist ownpossibilitiesandfailures. • GeoGebra ascatalystformathematics!

  3. Functionscan not beseen

  4. Nomograms

  5. Examples Van Dormolen[Dor78], Malle (z.B. in [Mal93], S. 265 oder [Mal00], Spivak ([Spi67], S. 79, www.dynagraph.de Hans-Jürgen Elschenbroich

  6. Examples

  7. Examples

  8. Examples

  9. Examples

  10. Examples

  11. Composition

  12. Composition • IdenticalFunctionhas a naturalappearance • The increase / decreasefrom x to f(x) becomesvisible • Inverse function easy toconstruct • Involutions, f(x) with f  f = id, • Projections, p(x) with p  p = p • CompositionandDecompositionoftwofunctions • Iterationscanbevisualized • General notions on mappingsofsets (injective, surjective, sections, …)

  13. Composition

  14. Linear Functions

  15. Linear Functions

  16. Decomposition

  17. Linear Approximation

  18. A Circle as Derivative Curve

  19. Parabolaas Derivative Curve

  20. Deltoide and

  21. Run alongthe Edge Howtouselevelcurvestorepresentfunctions?

  22. Run alongthe Edge

  23. Toblerone Graph

  24. Outlook

  25. r() = C a

  26. Thankyouforyourattention!

  27. Outline • Functionscan not beseen • Nomogrammes - Composition - Linear Functions - Linear Approximation - Translation of Domain and Value Line - SolvingEquations • Run alongthe Edge - TobleroneDiagram - ProductandSumofFunctions • Outlook - VariousCoordinates

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