1. Press Y= and enter the following sequence:
nMin = 1
u(n) = 2.700u(n-1)*(1-u(n-1))
u(nMin) = {0.4}
Use the arrow keys to go to left of the u(n) and press Enter until you see \ displayed. This will graph your sequence as a line graph instead of a sequence of dots.
2. Press WINDOW and enter:
nMin = 1
Xmin = 0
Ymin = 0
nMax = 25
Xmax = 25
Ymax = 1
PlotStart = 1
Xscl = 5
Yscl = .25
PlotStep = 1
Press GRAPH. Sketch the resulting graph on a piece of graph paper. Describe the graph (does it increase or decrease, does it repeat, what other patterns do you observe).
3. Press Y= and change 2.700 to 3.000, then press GRAPH. Sketch the resulting graph. Describe the graph. How is it different than before?
4. Replace the 3.000 with each of the following numbers. In each case, sketch the resulting graph, and describe it, including how it is similar or different from the others. When the numbers are very close, are the resulting graphs very similar?
3.500
3.949
3.990
3.996
3.750
3.950
3.991
3.997
3.951
5. Press FORMAT (2nd and ZOOM). Choose Web from the first line:
Time
Web
uv
vw
uw
6. Press Y= and put the original 2.700 in place of whatever number you used last.
Press WINDOW and enter the following:
nMin = 1
Xmin = 0
Ymin = -0.25
nMax = 25
Xmax = 1
Ymax = 1.25
PlotStart = 1
Xscl = .25
Yscl = .25
PlotStep = 1
7. Press GRAPH. You will see two graphs. One is a graph representing your sequence, the other is a graph of the equation y = x. Press TRACE. You should see values for n, x, and y. A web graph gives you a way of seeing the big picture for recursive functions. It can help tell you whether your sequence will converge on a certain value, alternate between two or three or four values, or behave chaotically.
Press the right arrow key several times. What happens?
8. Make a web graph for each of the values you used earlier. For each value, carefully sketch the graph and describe it and how it relates to your line graph for that same value.