# 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!
 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:

 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
 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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