A
“Lulu” of a Function
Go
to url: http://tapor1.mcmaster.ca/~sgs/cgi-bin/Maths/maths.cgi?theme=none&lang=en&do=activity&activity=lulu&level=0
“Basic”
problem: Think of “your
actions” as inputs to a function, and Lulu’s actions as output. Your actions might be to move left, up, down, or right (use
L, U, D, or R). Notice how Lulu
responds to each of these actions, and write an algebraic formula
to describe the function that is Lulu’s response, using f as the
name of your function. For
example, if “you” move one to the right, notice that Lulu moves two to
the left. Thus, you might write
f(R) = 2L. Is f(R
+ U) = f(U + R)? Can you
show that moves are (or are not) commutative, in the sense that it does (or
doesn’t) matter in which order you move?
Describe the domain and the range of your function.
Variation
1: Click on the “What if you
and Lulu start at different places” prompt. Now,
think of your location and Lulu’s location as two inputs, and the location
of the meeting spot as the output. Is
this a function? That is,
for any starting positions, is there only one possible meeting place? You might wish to add “no meeting possible” to the
range… would you then have a function?
Your challenge: write
down a formula (actually, a pair of formulas… one for each
coordinate) for predicting where the rendezvous takes place.
Use the “randomize” feature to check your formula.
If your work does result in a function, describe its domain and
range. If it doesn’t, explain
why it doesn’t.
Variation
2: Once you have investigated
Variation 1, click on the “What if Lulu uses a different move rule?”
prompt. Answer the questions in
both the problems above (the movement formula and the meeting place
formula). For both, describe
domain and range or explain why no function exists.
Variation
3: Investigate the “What if
you can choose Lulu’s rule?” prompt.
Lulu has three different pre-programmed moves; find the movement
formula and meeting place formula for the “new” rule.
For both, describe domain and range or explain why no function
exists.
Variation 4:
Investigate the “What if you can create your own movement rules for
Lulu?” prompt. You can try
using the “equation builder,” or you can try typing in formulas.
See if you can make rules to cause the meeting place to be in a
particular spot. For both,
describe domain and range or explain why no function exists.
For a .pdf file, go to http://www.uni.uiuc.edu/~hcrussel/LuluFunctions.pdf