Visualizing Linear Functions with and without Graphs Martin Flashman

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Visualizing Linear Functions with and without
Graphs!Martin Flashman

• Professor of MathematicsHumboldt State
  University
• mef2_at_humboldt.edu
• http//www.humboldt.edu/mef2
• Saturday October 25, 2008
• 1130- 1220

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Visualizing Linear Functions with and without
Graphs!

• Linear functions are both necessary, and
  understandable- even without considering their
  graphs.
• A sensible way to visualize them will be given
  without using graphs.
• Examples of their utility and some important
  function features (like slope and intercepts)
  will be demonstrated with and without graphs.
• Activities for students that involve them in
  understanding the function and linearity concepts
  will be illustrated.
• The author will demonstrate a variety of
  visualizations of these mappings using Winplot,
  freeware from Peanut Software.
• http//math.exeter.edu/rparris/peanut/

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Outline

• Linear Functions They are everywhere!
• Tables
• Graphs
• Mapping Figures
• Winplot Examples
• Characteristics and Questions
• Understanding Linear Functions Visually.

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Linear Functions They are everywhere!

• Where do you find Linear Functions?
• At home
• On the road
• At the store
• In Sports/ Games

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Linear Functions Tables

• Complete the table.
• x -3,-2,-1,0,1,2,3
• f(x) 5x 7
• f(0) ___?
• For which x is f(x)gt0?

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Linear Functions Tables

• Complete the table.
• x -3,-2,-1,0,1,2,3
• f(x) 5x 7
• f(0) ___?
• For which x is f(x)gt0?

• x f(x)5x-7
• 3 8
• 2 3
• 1 -2
• 0 -7
• -1 -12
• -2 -17
• -3 -22

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Linear Functions On Graph

• Plot Points (x , 5x - 7)

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Linear Functions On Graph

• Connect Points (x , 5x - 7)

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Linear Functions On Graph

• Connect the Points

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Linear FunctionsMapping Figures

• Connect point x to point 5x 7 on axes
• x f(x)5x-7
• 3 8
• 2 3
• 1 -2
• 0 -7
• -1 -12
• -2 -17
• -3 -22

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Linear Functions Mapping Figures
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Linear on Winplot

• Winplot examples
• Linear Mapping examples

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Characteristics and Questions

• Simple Examples are important!
• f(x) x C added value
• f(x) mx slope or rate or magnification
• Linear Focus point
• Slope m
• m gt 0 Increasing mlt0 Decreasing
• m 0 Constant

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Characteristics and Questions

• Characteristics on graphs and mappings figures
• fixed points f(x) x
• Using focus to find.
• Solving a linear equation
• -2x1 -x 2
• Using foci.

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Compositions are keys!

• Linear Functions can be understood and
  visualized as compositions with mapping figures
• f(x) 2 x 1 (2x) 1
• g(x) 2x h(u)u1
• f (0) 1 slope 2

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Compositions are keys!

• Linear Functions can be understood and
  visualized as compositions with mapping figures.
• f(x) 2(x-1) 1
• g(x)x-1 h(u)2u k(t)t1
• f(1) 1 slope 2

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Mapping Figures and Inverses

• Inverse linear functions
• socks and shoes with mapping figures
• f(x) 2x g(x) 1/2 x
• f(x) x 1 g(x) x - 1
• f(x) 2 x 1 (2x) 1
• g(x) 2x h(u)u1
• inverse of f 1/2(x-1)

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Mapping Figures and Inverses

• Inverse linear functions
• socks and shoes with mapping figures
• f(x) 2(x-1) 1
• g(x)x-1 h(u)2u k(t)t1
• Inverse of f 1/2(x-1) 1

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ThanksThe End!? Questions? flashman_at_humboldt.e
duhttp//www.humboldt.edu/mef2