Visualizing Linearity: Alternatives to Line Graphs Martin Flashman

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Visualizing Linearity Alternatives to Line
GraphsMartin Flashman

• Professor of MathematicsHumboldt State
  University
• mef2_at_humboldt.edu
• http//www.humboldt.edu/mef2
• Thursday September 3, 2009

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Visualizing Linearity Functions with and without
Graphs!

• Linearity can be interpreted with other ways to
  visualize this important quality line graphs, but
  there are.
• I will discuss alternatives using mapping figures
  and demonstrate how these figures can enhance
  understanding of some key concepts.
• Examples of the utility of mapping figures and
  some important function features (like slope
  and intercepts) will be demonstrated.
• I will demonstrate a variety of visualizations
  of these mappings using Winplot, freeware from
  Peanut Software.
• http//math.exeter.edu/rparris/peanut/
• No special expertise will be presumed beyond
  pre-calculus mathematics.

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Outline

• Linear Functions They are everywhere!
• Traditional Approaches
• 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 Examples on Winplot

• Winplot examples
• Linear Mapping examples

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Visualizing f (x) mx b with a mapping figure
-- four examples

• Example 1 m -2 b 1
• f (x) -2x 1
• Each arrow passes through a single point, which
  is labeled F - 2,1.
• The point F completely determines the function
  f.
• given a point / number, x, on the source line,
• there is a unique arrow passing through F
• meeting the target line at a unique point /
  number, -2x 1,
• which corresponds to the linear functions value
  for the point/number, x.

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Visualizing f (x) mx b with a mapping figure
-- four examples

• Example 2 m 2 b 1
• f (x) 2x 1
• Each arrow passes through a single point, which
  is labeled
• F 2,1.
• The point F completely determines the function
  f.
• given a point / number, x, on the source line,
• there is a unique arrow passing through F
• meeting the target line at a unique point /
  number, 2x 1,
• which corresponds to the linear functions value
  for the point/number, x.

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Visualizing f (x) mx b with a mapping figure
-- four examples

• Example 3 m 1/2 b 1
• f (x) 1/2x 1
• Each arrow passes through a single point, which
  is labeled F 1/2,1.
• The point F completely determines the function
  f.
• given a point / number, x, on the source line,
• there is a unique arrow passing through F
• meeting the target line at a unique point /
  number, 1/2x 1,
• which corresponds to the linear functions value
  for the point/number, x.

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Visualizing f (x) mx b with a mapping figure
-- four examples

• m 0 b 1
• f (x) 0x 1
• Each arrow passes through a single point, which
  is labeled F 0,1.
• The point F completely determines the function
  f.
• given a point / number, x, on the source line,
• there is a unique arrow passing through F
• meeting the target line at a unique point /
  number, f(x)1,
• which corresponds to the linear functions value
  for the point/number, x.

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Visualizing f (x) mx b a special example

• m 1 b 1
• f (x) x 1
• Unlike the previous examples, in this case it is
  not a single point that determines the mapping
  figure, but the single arrow from 0 to 1, which
  we designate as F1,1
• It can also be shown that this single arrow
  completely determines the function.Thus, given a
  point / number, x, on the source line, there is
  a unique arrow passing through x parallel to
  F1,1 meeting the target line a unique point /
  number, x 1, which corresponds to the linear
  functions value for the point/number, x.
• The single arrow completely determines the
  function f.
• given a point / number, x, on the source line,
• there is a unique arrow through x parallel to
  F1,1
• meeting the target line at a unique point /
  number, x 1,
• which corresponds to the linear functions value
  for the point/number, x. x

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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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Final Comment on Duality

• The Principle of Plane Projective Duality
  Suppose S is a statement of plane projective
  geometry and S' is the planar dual statement for
  S. If S is a theorem of projective geometry,
  then S' is also a theorem of plane projective
  geometry.

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Application of duality to linear functions.

• S A linear function is determined by two
  points in the plane.
• S A linear function is determined by two
  lines in the plane.

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