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!
• 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/
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:
– On the road:
– At the store:
– In Sports/ Games
Linear Functions: Tables
#
3
2
1
0
-1
-2
-3
5×#-7
•
•
•
•
•
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
x
3
2
1
0
-1
-2
-3
f(x)=5x-7
8
3
-2
-7
-12
-17
-22
•
•
•
•
•
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: 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
x
5x-7
3
8
2
3
1
-2
0
-7
-1
-12
-2
-17
-3
-22
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
-21
-22
Linear on Winplot
• Winplot examples:
–Linear Mapping examples
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 : Constant
m<0 Decreasing
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
2.0
– f(x) = 2 x + 1 = (2x) + 1 :
• g(x) = 2x; h(u)=u+1
• f (0) = 1 slope = 2
1.0
0.0
-1.0
-2.0
-3.0
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
2.0
1.0
0.0
-1.0
-2.0
-3.0
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
2.0
1.0
– f(x) = 2 x + 1 = (2x) + 1 :
• g(x) = 2x; h(u)=u+1
• inverse of f: 1/2(x-1)
0.0
-1.0
-2.0
-3.0
Mapping Figures and Inverses
• Inverse linear functions:
– socks and shoes with mapping figures
– f(x) = 2(x-1) + 1:
2.0
• g(x)=x-1 h(u)=2u; k(t)=t+1
• Inverse of f: 1/2(x-1) +1
1.0
0.0
-1.0
-2.0
-3.0
Thanks
The End!

Questions?
flashman@humboldt.edu
http://www.humboldt.edu/~mef2