Visualizing Linearity: Alternatives to Line Graphs Martin Flashman Professor of Mathematics Humboldt State University 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. 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. 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. No special expertise will be presumed beyond pre-calculus mathematics. 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×# 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: 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

Linear Functions: On Graph x5x Plot Points (x, 5x - 7):

Linear Functions: On Graph x5x Connect Points (x, 5x - 7):

Linear Functions: On Graph x5x Connect the Points

Linear Functions: Mapping Figures Connect point x to point 5x – 7 on axes x f(x)=5x

Linear Functions: Mapping Figures x 5x

Linear Examples on Winplot Winplot examples:  Linear Mapping 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].  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 = 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 figures  f(x) = 2 x + 1 = (2x) + 1 : g(x) = 2x; h(u)=u+1 f (0) = 1 slope = 2

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

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)=u+1 inverse of f: 1/2(x-1)

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)=t+1 Inverse of f: 1/2(x-1) +1

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?