functions can not be seen
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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 - PowerPoint PPT PresentationTRANSCRIPT
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Functions can not be seen
Rainer KaendersUniversity of Cologne
GeoGebra Conference Linz 2011
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Functions can not be seen
• … but can be represented • GeoGebra can make the identification of
functions with their graphs stronger• Many different representations: show that
there is not the representation of a function• Each representation has ist own possibilities
and failures.• GeoGebra as catalyst for mathematics!
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Functions can not be seen
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Nomograms
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Examples
Van Dormolen [Dor78], Malle (z.B. in [Mal93], S. 265 oder [Mal00], Spivak ([Spi67], S. 79, www.dynagraph.de Hans-Jürgen Elschenbroich
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Examples
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Examples
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Examples
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Examples
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Examples
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Composition
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Composition
• Identical Function has a natural appearance• The increase / decrease from x to f(x) becomes visible • Inverse function easy to construct• Involutions, f(x) with f f = id, • Projections, p(x) with p p = p• Composition and Decomposition of two functions • Iterations can be visualized • General notions on mappings of sets (injective,
surjective, sections, …)
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Composition
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Linear Functions
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Linear Functions
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Decomposition
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Linear Approximation
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A Circle as Derivative Curve
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Parabola as Derivative Curve
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Deltoide and
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Run along the Edge
How to use level curves to represent functions?
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Run along the Edge
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Toblerone Graph
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Outlook
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r() = C a
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Thank you for your attention!
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Outline• Functions can not be seen• Nomogrammes
- Composition- Linear Functions- Linear Approximation- Translation of Domain and Value Line- Solving Equations
• Run along the Edge- Toblerone Diagram- Product and Sum of Functions
• Outlook- Various Coordinates