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Visualizing Linear Functions with and without Graphs! Martin Flashman

Visualizing Linear Functions with and without Graphs! Martin Flashman. Professor of Mathematics Humboldt State University mef2@humboldt.edu http://www.humboldt.edu/~mef2 Saturday October 25, 2008 11:30- 12:20. Visualizing Linear Functions with and without Graphs!.

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Visualizing Linear Functions with and without Graphs! Martin Flashman

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  1. Visualizing Linear Functions with and without Graphs!Martin Flashman Professor of MathematicsHumboldt State University mef2@humboldt.edu http://www.humboldt.edu/~mef2 Saturday October 25, 2008 11:30- 12:20

  2. Visualizing Linear Functions with and without Graphs! • Linear functions are both necessary, and understandable- even without considering their graphs. • A sensible way to visualize them will be given without using graphs. • Examples of their utility and some important function features (like slope and intercepts) will be demonstrated with and without graphs. • Activities for students that involve them in understanding the function and linearity concepts will be illustrated. • The author will demonstrate a variety of visualizations of these mappings using Winplot, freeware from Peanut Software. • http://math.exeter.edu/rparris/peanut/

  3. Outline • Linear Functions: They are everywhere! • Tables • Graphs • Mapping Figures • Winplot Examples • Characteristics and Questions • Understanding Linear Functions Visually.

  4. Linear Functions: They are everywhere! • Where do you find Linear Functions? • At home: • On the road: • At the store: • In Sports/ Games

  5. 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?

  6. 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-7 3 8 2 3 1 -2 0 -7 -1 -12 -2 -17 -3 -22

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

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

  9. Linear Functions: On Graph Connect the Points

  10. 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

  11. Linear Functions: Mapping Figures

  12. Linear on Winplot • Winplot examples: • Linear Mapping examples

  13. 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

  14. 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.

  15. 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

  16. 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

  17. 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)

  18. 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

  19. ThanksThe End! Questions?flashman@humboldt.eduhttp://www.humboldt.edu/~mef2

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