Visualizing Linearity: Visualizing Linearity: Alternatives to Line GraphsAlternatives to Line Graphs

Martin FlashmanMartin Flashman

Visualizing Linearity: Visualizing Linearity: Alternatives to Line GraphsAlternatives to Line Graphs

Martin FlashmanMartin FlashmanProfessor of MathematicsProfessor of MathematicsHumboldt State University Humboldt State University mef2@humboldt.eduhttp://www.humboldt.edu/~mef2 Thursday September 3, 2009Thursday September 3, 2009

Visualizing Linearity Functions Visualizing Linearity Functions with and without Graphs!with and without Graphs!

Linearity can be interpreted with other ways to visualize this Linearity can be interpreted with other ways to visualize this important quality line graphs, but there are. important quality line graphs, but there are.

I will discuss alternatives using I will discuss alternatives using mapping figuresmapping figures and demonstrate and demonstrate how these figures can enhance understanding of some key how these figures can enhance understanding of some key concepts. 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 No special expertise will be presumed beyond pre-calculus

mathematics.mathematics.

OutlineOutline Linear Functions: They are everywhere! Traditional Approaches:

TablesGraphs

Mapping Figures Winplot Examples Characteristics and Questions Understanding Linear Functions Visually.

Linear Functions: Linear Functions: They are everywhere!They are everywhere! Where do you find Linear Functions?

At home:

On the road:

At the store:

In Sports/ Games

Linear Functions: TablesLinear Functions: Tables

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

Linear Functions: TablesLinear 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)>0?

x f(x)=5x-7

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Linear Functions: On GraphLinear Functions: On Graph

x 5x-7

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Plot Points (x , 5x - 7):

Linear Functions: On GraphLinear Functions: On Graph

x 5x-7

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0 -7

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Connect Points (x , 5x - 7):

Linear Functions: On GraphLinear Functions: On Graph

x 5x-7

3 8

2 3

1 -2

0 -7

-1 -12

-2 -17

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Connect the Points

Linear Functions:Linear Functions:Mapping FiguresMapping Figures Connect point x to point

5x – 7 on axes

x f(x)=5x-7

3 8

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0 -7

-1 -12

-2 -17

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Linear Functions: Linear Functions: Mapping FiguresMapping Figures

x 5x-7

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Linear Examples on WinplotLinear Examples on WinplotWinplot examples:

Linear Mapping examples

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

Example 1: m = -2; b = 1f (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 = 1f (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 = 1f (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 function’s value

for the point/number, x.

Visualizing f (x) = mx + b a special example:

m = 1; b = 1f (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 Characteristics and QuestionsQuestions 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 Characteristics and QuestionsQuestionsCharacteristics 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!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)=u+1 f (0) = 1 slope = 2

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Compositions are keys!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)=t+1 f(1)= 1 slope = 2

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Mapping Figures and InversesMapping 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)=u+1 inverse of f: 1/2(x-1)

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Mapping Figures and InversesMapping 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)=t+1 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.

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.

ThanksThanksThe End!The End!

Questions?Questions?

flashman@humboldt.eduhttp://www.humboldt.edu/~mef2 http://www.humboldt.edu/~mef2