Visualizing Linearity: by r5ryMgN

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									Visualizing Linearity:
Alternatives to Line Graphs

Martin Flashman
  Professor of Mathematics
  Humboldt State University
  mef2@humboldt.edu
  http://www.humboldt.edu/~mef2
  Thursday September 3, 2009
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.
Outline
 Linear Functions: They are everywhere!
 Traditional Approaches:
     Tables
     Graphs

 Mapping Figures
 Winplot Examples
 Characteristics and Questions
 Understanding Linear Functions Visually.
Linear Functions:
They are everywhere!
   Where do you find Linear Functions?
     At   home:

     On   the road:

     At   the store:

     In   Sports/ Games
Linear Functions: Tables
    #     5×#-7
                     Complete the table.
    3
    2                x = -3,-2,-1,0,1,2,3
    1                f(x) = 5x – 7
    0
    -1
                     f(0) = ___?
    -2               For which x is f(x)>0?
    -3
Linear Functions: Tables
     x   f(x)=5x-7      Complete the table.
     3           8
     2           3      x = -3,-2,-1,0,1,2,3
     1          -2      f(x) = 5x – 7
     0          -7
                        f(0) = ___?
    -1         -12
    -2         -17      For which x is f(x)>0?
    -3         -22
Linear Functions: On Graph
                   




                                       Plot Points (x , 5x - 7):

                                                 x       5x-7
                           




                   
                                                 3        8
                                                 2        3
                  



                                                 1        -2
                  
                                                 0        -7
                                                 -1       -12
                  



                                                -2        -17
                  
                                                -3        -22
Linear Functions: On Graph
                  
                                       Connect Points
                                        (x , 5x - 7):
                          


                                           x      5x-7
                  

                                           3        8
                                        2        3
                                           1       -2
                 

                                           0       -7
                 
                                           -1      -12
                                           -2      -17
                 

                                           -3      -22
Linear Functions: On Graph
                     




                      
                                          Connect the Points
                             




                     
                                                x    5x-7
                                                3     8
                                                2     3
                    




                                             1     -2
                                                0     -7
                    

                                                -1   -12
                                             -2   -17
                                                -3   -22
Linear Functions:
Mapping 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
Linear Functions:
Mapping Figures       8
                      7
                      6
                      5

   x    5x-7          4
                      3
                      2
   3     8            1
                      0
                     -1
   2     3           -2
                     -3

   1     -2          -4
                     -5
                     -6
   0     -7          -7
                     -8
                     -9
   -1   -12         -10
                    -11

   -2   -17         -12
                    -13
                    -14
   -3   -22         -15
                    -16
                    -17
                    -18
                    -19
                    -20
                    -21
                    -22
Linear Examples on Winplot
 Winplot   examples:
  Linear   Mapping examples
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 function’s
           value for the point/number, x.
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 function’s
         value for the point/number, x.
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 function’s value
           for the point/number, x.
    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].
      Thepoint 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 function’s value
         for the point/number, x.
    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 F[1,1]
   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 F[1,1]
    meeting the target line a unique point / number, x + 1, which
    corresponds to the linear function’s 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 F[1,1]
           meeting the target line at a unique point / number, x + 1,
          which corresponds to the linear function’s value for the
            point/number, x. x
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 > 0 : Increasing   m<0 Decreasing
        m= 0 : Constant
Characteristics and Questions
Characteristics on graphs and mappings figures:
   “fixed   points” : f(x) = x
        Using focus to find.




   Solving    a linear equation:
      -2x+1 = -x + 2
      Using foci.
Compositions are keys!
 Linear Functions can be understood and
 visualized as compositions with mapping
                                    2.0


 figures
                                    1.0

  f(x)   = 2 x + 1 = (2x) + 1 :
     g(x) = 2x; h(u)=u+1           0.0




     f (0) = 1 slope = 2          -1.0




                                   -2.0




                                   -3.0
Compositions are keys!
 Linear Functions can be understood and
 visualized as compositions with mapping
 figures.                            2.0




  f(x)   = 2(x-1) + 1:              1.0




      g(x)=x-1 h(u)=2u; k(t)=t+1    0.0




     f(1)= 1 slope = 2             -1.0




                                    -2.0




                                    -3.0
Mapping Figures and Inverses
   Inverse linear functions:
     socks  and shoes with mapping figures
                                       2.0


     f(x) = 2x; g(x) = 1/2 x
                                       1.0

     f(x) = x + 1 ; g(x) = x - 1
                                       0.0




     f(x)   = 2 x + 1 = (2x) + 1 :   -1.0




         g(x) = 2x; h(u)=u+1         -2.0




        inverse of f: 1/2(x-1)       -3.0
Mapping Figures and Inverses
   Inverse linear functions:
     socks  and shoes with mapping figures
                                        2.0



     f(x) = 2(x-1) + 1:
                                        1.0

        g(x)=x-1 h(u)=2u; k(t)=t+1
        Inverse of f: 1/2(x-1) +1
                                        0.0




                                       -1.0




                                       -2.0




                                       -3.0
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.
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.
           Thanks
          The End!

            
          Questions?
    flashman@humboldt.edu
http://www.humboldt.edu/~mef2

								
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