Iterated Functions in Precalculus:
Doing It Again and Again with Sketchpad
2005 NCTM Annual Meeting Session 440
Scott Steketee
Key Curriculum Press, Emeryville, California
Iteration in Mathematics
There are a number of big ideas in mathematics that form unifying themes across
different branches of mathematics, including functions, transformations, proof, and
data. Iteration is also such a topic, with important applications throughout the
curriculum. Understanding iteration, the way in which systematically repeated simple
operations can build complex structures, can shed light on many important concepts in
arithmetic, algebra, geometry, fractals, calculus, and mathematical modeling.
Iteration is also at the heart of the distinction between discrete and continuous
processes. This distinction begins with the concept of number and comes into play
throughout mathematics. Even the process of graphing a continuous function (whether
it’s done by hand, by graphing calculator, or by Sketchpad) is accomplished in practice
by plotting a discrete set of points and interpolating between data points. This interplay
is fundamental to mathematics: The definition of continuity itself depends on the
concept of a limit, involving ever-smaller discrete intervals.
Because iteration allows us to move from the discrete toward the continuous, it is a
powerful tool for elucidating many mathematical topics, ranging from the counting
numbers to calculus.
Iteration and Technology
Iteration often involves large numbers of operations that would be difficult and timeconsuming to carry out by hand, so it’s particularly suited to the use of technology. The
value of graphing calculators lies in their ability to perform the iterations required to
graph and manipulate functions. Similarly, Sketchpad makes it possible, by means of
iteration, to investigate mathematical phenomena that would be computationally
intractable without technology.
Iteration in Precalculus
In addition to the many topics where iteration is an implicit part of the mathematics
studied in precalculus, there are a number in which iteration plays a more explicit role.
We will look at a few of those today, using prepared student activities and sketches.
Availability of Materials
The activities and sketches used in this talk are available on the web at
http://www.keypress.com/sketchpad/general_resources/recent_talks.
Generating Arithmetic / Geometric Sequences Numerically
Students develop an understanding of arithmetic and geometric sequences by building
and modifying them with Sketchpad.
A Sequence Approach to Logs
In this activity students graph geometric sequences against arithmetic sequences to
obtain good approximations of log curves.
Compound Interest
In this activity students use iterated calculations Students use iterated calculations to
compute and plot the value of a compound interest investment. This is connected to the
constant e and the general formula for continuous compounding.
A Geometric Approach to ei
Students use a limit definition of e, along with multiplication on the complex plane, to
find the value of ei.
The Logistic Function
In this activity students build a Sketchpad model of a population for which the growth
is restrained by some factor. In the process, students explore the sensitivity of the longterm behavior to the initial size of the population and to the parameters that determine
the growth.
The Taylor Series
In this activity students explore how adding terms to a Taylor series approximation
increases the accuracy of the approximation.
The Sierpinski Gasket
In this activity students create various fractal designs (including the Sierpinski triangle
and Mira, Julia, and Mandelbrot fractals) as strange attractors.
Barnsley’s Fern
Students plot points using four pairs of iterated functions. By choosing randomly
among the function pairs, they create a fractal Barnsley’s Fern.
Generating Arithmetic and Geometric Sequences Numerically
In this activity, you’ll build and explore arithmetic and geometric sequences by using
Sketchpad’s iteration feature.
ARITHMETIC SEQUENCES
Open Sequences.gsp in the 8 Sequences and Series folder. This sketch includes a
start value of 2 and a difference of 3. With these two values, you can generate an
arithmetic sequence.
Q1
Look at the number line on the sketch. What arithmetic sequence is shown?
How do the numbers in the sequence relate to the start and difference values?
0
5
10
15
Now you’ll create a table of values that corresponds to the arithmetic sequence on
the number line.
1.
Choose MeasureCalculate to display the Calculator. Click on start in the
sketch, the sign on the keypad, and difference in the sketch to compute
start difference.
The beginning value for
an iteration is often
called the seed, or
the pre-image, of
the iteration.
2.
Select start, and choose TransformIterate. Map start to start difference by
clicking on start difference. Then click Iterate to confirm the mapping.
3.
A table appears with the 2nd through 5th terms in your arithmetic sequence.
To increase the number of terms in your sequence, select the table and press the
key on your keyboard several times. You can decrease the number of terms by
pressing the key.
To change the value of a
parameter, double-click
it with the Arrow tool
and enter a new number.
Q2
Your sequence does not include the term 24. Find two ways to change the
sequence so that it includes 24.
Q3
Below are several arithmetic sequences. For each one, find the start and difference
values that generate them.
Answer this question
without creating the
sequence.
Q4
Chapter 8: Sequences and Series
a.
3, 6, 9, 12, 15, . . .
b.
10, 14, 18, 22, 26, . . .
c.
1, 1, 1, 1, 1, . . .
d.
0.5, 0.75. 1.0, 1.25, 1.5, . . .
Suppose the start value of your sequence is 4 and the difference is 6. Will there be
a term in your sequence between 2000 and 2010?
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
129
Generating Arithmetic and Geometric Sequences Numerically
(continued)
GEOMETRIC SEQUENCES
Page 2 of Sequences.gsp includes a start value of 1 and a ratio of 3. With these two
values, you can generate a geometric sequence.
4.
Using the directions for creating an arithmetic sequence as a guide, create a table
that corresponds to the geometric sequence shown on the number line.
Q5
Your sequence does not include the term 24. Describe two ways to change the
sequence so that it includes 24.
Q6
Below are several geometric sequences. For each one, find the start and ratio
values that generate them.
Q7
a.
2, 8, 32, 128, 512, . . .
b.
32, 16, 8, 4, 2, . . .
c.
1, 1, 1, 1, 1, . . .
d.
1, 1, 1, 1, 1, . . .
Change your sequence so that start 1 and ratio 3. How many copies of
the 2nd arc (between 3 and 9) can fit into the 3rd arc? How many copies of the
3rd arc can fit into the 4th arc? Does this pattern continue?
EXPLORE MORE
Hint: Your iteration
requires using two
pre-image parameters.
130
Q8
The Fibonacci sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, . . . , where each term is
the sum of the preceding two terms. Use the seed values on page 3 of the sketch
to generate an iterated table of Fibonacci values.
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
Chapter 8: Sequences and Series
A Sequence Approach to Logs
John Napier (1550–1617) of Scotland is credited with the invention of logarithms.
Napier studied logs by making tables that placed geometric sequences and arithmetic
sequences side by side. In this activity, you’ll explore Napier’s idea and use it to build a
good approximation of a log graph.
EXAMINE A TABLE
Remember,
logc r n n logc r.
The table at right shows a geometric sequence in the left
column with common ratio r and an arithmetic sequence
in the right column with common difference d. When you
plot the (x, y) points, they lie on the graph of a logarithm.
x
y
r3
3d
r2
2d
Suppose the point (r, d) lies on the curve y logc x for a
particular value of a constant c. This means that d logc r.
r1
d
1
0
r
d
r2
2d
r3
3d
Q1
Explain why all the other points in the table must also
lie on this log curve.
CONSTRUCT A LOG CURVE
Open the sketch Logs.gsp in the 4 Other Functions folder. You’ll see a point F with
coordinates (1, 0). The sketch contains two adjustable sliders with values labeled d
and r.
These were the points
from the preceding table.
This is the point (r, d ).
By plotting and connecting the points in the table, you’ll obtain a good approximation
of a logarithmic curve.
1.
Use Sketchpad’s calculator to compute the two values xF r and yF d.
2.
Plot point G at xF r, yF d . Draw a segment connecting points F and G.
3.
With point F selected, choose TransformIterate. In the Iteration dialog box,
designate point G as the image point by clicking on point G in the sketch.
4.
Click Iterate in the dialog box to iterate the segment three times. The
x-coordinates of all five points form a geometric sequence with common
ratio r, and the y-coordinates form an arithmetic sequence with common
difference d.
5.
With the iterated image selected, press the key on your keyboard at least
15 times to increase the number of iterations.
Now that you’ve created the points (r, d), r 2, 2d , r 3, 3d , r 4, 4d , . . . , you’ll
use the same technique to create the points to the left of (1, 0). These points are
(1/r, d), 1/r 2, 2d , 1/r 3, 3d , . . . .
64
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
Chapter 4: Other Functions
A Sequence Approach to Logs
(continued)
6.
Use Sketchpad’s Calculator to compute the two values xF /r and yF d.
7.
Plot point H at xF /r, yF d . Draw a segment connecting points F and H.
8.
Repeat steps 3–5 with points F and H to obtain the remainder of the curve.
MANIPULATE THE CURVE
Now that your construction is complete, you’ll change the values of d and r to obtain
curves passing through specific points.
9.
You might need to
increase the number
of iterations if your
curve shrinks.
To extend the function,
drag the arrows on
its endpoints.
Choose GraphPlot Points, and plot the point (4, 2).
10.
Use the sliders to adjust the values of d and r so that the curve passes through
(4, 2). There are many values that will work; try to find values so that the
curve looks smooth.
Q2
The points on your curve (those that you created through iteration) satisfy the
equation y logc x. What is the value of c?
11.
Technically, your curve is really not a curve—it’s composed of many individual
line segments. To see how closely your curve approximates the equation from
Q2, graph the function. To do so, you’ll need to remember this log identity:
logab log(b)/log(a).
12.
After you’ve graphed the function, scroll to the right to see how well your stitched
curve approximates the actual function. Notice how flat the log function is!
EXPLORE MORE
Plot the point (9, 2). Adjust the sliders so that the stitched curve passes through this
point. Change the equation of the function accordingly.
Chapter 4: Other Functions
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
65
Compound Interest
It’s impossible to pinpoint the moment of discovery of the constant e. Several
branches of mathematics converged on the number from different directions.
One of these was the applied mathematics of finance.
A SIMPLISTIC INVESTMENT
When an investment is compounded, the interest is paid periodically, and each time
the interest itself is invested so that it can start accumulating interest as well. The more
frequently the interest is paid and invested, the greater the advantage for the investor.
Depending on where
you live, this interest
rate may be illegal (and
is certainly unlikely).
Consider a very simple investment, one dollar at 100% interest for one year. Watch
what happens when it is compounded.
1.
Create a new sketch and make three parameters:
t 0 (the time in years)
P 1 (the principal)
k 2 (the number of compounding periods per year)
To set the precision,
select the calculation
and choose Edit
PropertiesValue.
2.
The length of each time period will be 1/k. Create a calculation to find the
beginning of the next time period, by adding the length of one period to the
original time t.
3.
The interest for the first time period is P/k, so the value of the investment at the
end of this period will be P P/k. Express this value in factored form, and create
a calculation for it. Set the precision of this calculation to hundred thousandths.
4.
Plot these two points and connect them with a
line segment:
(t, P)
Q1
5.
2
t + 1 = 0.50
k
t = 0.00
P = 1.00
[(t 1/k), P (1 1/k)]
( )
P ⋅ 1 + 1 = 1.50000
k
What do these points represent in terms of
the investment?
Hide the two points, leaving only the segment to show the growth.
The calculations you just completed follow the investment for only one of the k
compounding periods. You must repeat the calculations one more time to get to the
end of the year.
6.
Chapter 4: Other Functions
Calculate (k 1). Label the calculation depth.
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
59
Compound Interest
(continued)
7.
You may need to move
the table so that you can
see the bottom row.
Q2
Q3
Select in order t, P, and depth. Press the Shift key
and choose TransformIterate To Depth. Map
the two parameters to their respective calculations
as shown in the table at right.
Pre-image First Image
t
t 1/k
P
P (1 1/k)
Edit parameter k to change the number of compounding
periods. The bottom row of the table shows the value of
the investment at the end of the year. What value must
you use for k to compound the interest quarterly? What
is the value of the investment after one year if it is
compounded quarterly? Monthly? Weekly? Daily?
2
k=4
Increasing the compounding frequency always results in more money for the
investor, but, as you can see, there seems to be a limit. What appears to be the
limit of the value of the investment at the end of the year?
The value of this limit is known as the number e. Mathematically, you could express
the limit this way:
e
1
lim 1 k
k→
k
A MORE REALISTIC INVESTMENT
As k grows, the investment modeled in the preceding example approaches what is
called continuous compounding. In practice, daily compounding comes so close that
the difference is negligible. Now, make some changes to your sketch and model a
more realistic investment. This time, it will be $100 at 8.5% over a term of 5 years.
8.
Create these new parameters:
r 0.085 (8.5% interest, displayed to the thousandth)
term 5 (investment term in years)
Is the graph off the
screen? Use GraphGrid
Form to change the grid
form to Rectangular and
then rescale the axes.
60
9.
Edit parameter P to make the starting principal $100 instead of $1. Set k to 12
for monthly compounding.
10.
The interest for the first time period is (P r)/k, so the value after the first time
period is P (P r)/k. Express this in factored form, and edit the existing
calculation to match.
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
Chapter 4: Other Functions
Compound Interest
(continued)
11.
The total number of periods should now be the number of periods in a year
multiplied by the number of years. Edit the depth calculation to achieve this.
Remember to subtract one, because you’ve already calculated the result for
one period.
Q4
What is the investment worth at the end of the term if the interest is
compounded annually? Daily?
The function being modeled by this iterated calculation is the compound interest
formula: A(t) P(1 r/k)kt. The iteration involves repeatedly multiplying the
previous result by the same factor, (1 r/k). This is an exponential function, and
you write it using any base you choose. If you use e as the base, you could write it
as a e bt.
P(1 r/k)kt ae bt
Clearly, the coefficient a must equal P, but determining b is another matter.
After setting the
keyboard adjustment,
you can select the
parameter and press
the + or – key on the
keyboard to change the
value of the parameter
by the specified amount.
Q5
Why must a equal P?
12.
Create new parameter b. Use EditPropertiesValue to set its precision to
thousandths, and use EditPropertiesParameter to set the keyboard
adjustments to 0.001.
13.
Define and plot the function A(x) Pe bx.
Q6
Set k to a very high number so that the iteration
approximates continuous compounding. Adjust the
value of parameter b so that the function graph aligns
with the iterated point plot. What function models
the current value of $100 compounded continuously
at 8.5%?
Q7
Chapter 4: Other Functions
200
b = 0.108
A(x) = P ⋅ eb⋅x
100
P = 100.00
r = 0.085
k = 365
5
What is the general function for the value of an investment of principal P, at
interest rate r, compounded continuously for x years? Test your answer by
changing the equation for A(x) in the sketch. Make sure the graph of A(x) always
matches the iteration for different values of P and r.
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
61
A Geometric Approach to e i
One of the most beautiful—and mysterious—mathematical discoveries has to do
with the value of e i. What does it mean to raise the number e to an imaginary power?
In this activity, you’ll explore e i through a geometric approach.
GETTING STARTED
A good place to begin your investigation is with the mathematical constant e.
Consider these calculations:
1
1 1
0
10
1
, 1
100
100
1,000
1
, 1
1,000
1
, 1
10,000
10,000
,...
1.
Open a new sketch. Choose EditPreferences from the Edit menu. Set the
Scalars precision to hundred-thousandths.
2.
Choose MeasureCalculate. Calculate, one at a time, the values of the four
expressions above.
Q1
3.
Q2
What do you notice about your four calculations?
Calculate four more expressions that continue the sequence above.
n
Based on your calculations, approximate the value of 1 n1 to several decimal
places as n grows ever larger.
n
The mathematical constant e is defined as the limiting value of 1 n1 as n
approaches infinity. Raising e to a power, like e 2 or e 3, involves a similar definition:
x
e x the limiting value of 1 n
Q3
n
as n approaches infinity
Use the definition of e x and a large n to approximate the value of e 3.
SKETCH AND INVESTIGATE
Now that you know how to approximate e x when x is a real number, you’re ready to
consider e i. Raising e to an imaginary power certainly seems strange, but let’s use
our definition of e x with x i and see what happens:
i
e i the limiting value of 1 n
n
as n approaches infinity
As before, we can get a sense of how this expression behaves by starting with a small
10
value of n. When n 10, we must evaluate 1 1i0 .
Chapter 7: Polar Coordinates and Complex Numbers
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
123
A Geometric Approach to e i
(continued)
This is a good time to
review the activity
Multiplication of
Complex Numbers.
4.
Open the sketch eipi.gsp in the 7 Polar Coords and Complex Nos folder. The
axes represent the complex plane with real numbers on the horizontal axis and
imaginary numbers on the vertical axis. Point A is at (1, 0) and represents the
value 1. Point B is at 1, 10 and represents the value 1 1i0 .
Q4
The sketch provides two pieces of information about right triangle OAB: the
measure of ∠AOB and the length of OB. Describe geometrically what it means
to multiply the complex number 1 1i0 by itself. Describe geometrically what
it means to raise 1 1i0 to the tenth power.
Sketchpad makes the process you described in Q4 simple to carry out.
5.
Select point A and n 1, and hold down the Shift key. Choose
TransformIterate to Depth.
6.
Click on point B to map point A to point B. Click Iterate to confirm your
mappings. The iterated triangles appear, all of which are similar to OAB.
Q5
7.
Select the iterated point image and
choose TransformTerminal Point.
Label the terminal point P.
8.
With point P selected, choose
MeasureCoordinates.
Q6
The third page of the
sketch provides an
intuitive explanation of
what you’re observing.
2
9.
Q7
3
9
10
Identify the locations of 1 1i0 , 1 1i0 , . . . , 1 1i0 , and 1 1i0 .
B
P
O
10
What is the value of 1 1i0 ?
A
Drag slider point n to the right to increase the value of n.
What do you notice about the value of 1 in as n grows larger?
n
EXPLORE MORE
You can generalize the method of finding e i to compute e raised to the i power,
where is any number.
10.
Open page 2 of eipi.gsp.
This sketch computes e raised to the power ik for any value of k.
124
11.
Double-click the parameter k and change its value to 2.
Q8
Use the iteration process from page 1 of the sketch to approximate the value
of e i/2.
Q9
Approximate the imaginary powers of e for values of k such as 3 and 4.
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
Chapter 7: Polar Coordinates and Complex Numbers
A Geometric Approach to e i
(continued)
Q10
For each of your approximations, what do you notice about its distance from
the origin?
Q11
When k 2, what angle does the point representing e i/2 make with the x-axis?
Answer this question for k 3 and for k 4.
Q12
The eighteenth-century mathematician Leonard Euler developed the identity
e i cos i sin as a way to compute the value of e raised to any imaginary
power. Explain how this identity makes sense based on your answers to Q10
and Q11.
Q13
Substitute into the identity in Q12. What do you get?
Chapter 7: Polar Coordinates and Complex Numbers
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
125
The Logistic Function
Scientists often use mathematical functions to understand natural processes. For
instance, biologists and environmentalists often use mathematics to study how
animal or plant populations grow larger or become smaller.
In this example, you will use Sketchpad to model a population of animals that
reproduce at a certain rate, but whose numbers are limited by the available food.
DEVELOP THE EQUATION
Your mathematical model of population growth will use a function to calculate the
number of animals in each generation. The input value to this function is the number
of animals in the current generation, and the output value is the number in the next
generation. The function includes two factors: a growth factor and a limiting factor.
For instance, if the
population doubles in
each generation, the
value of k is 2.
Q1
For the growth factor, assume that on average each individual in one generation
produces k individuals in the next generation. The value of k takes into account
normal births and deaths, but not the effect of limited resources. Considering
only normal growth, if the population is p in one generation, write a formula, in
terms of p and k, for the population in the next generation.
If the population depends only on the growth factor, it will increase exponentially.
But as the population becomes larger, food becomes scarce and population growth
is limited by the lack of food. Use n to stand for the population at which the current
generation doesn’t have enough food to survive, and dies without producing offspring.
Taking this value n into account, a reasonable function f(p) for determining the
population in the next generation is f(p) kp(n p)/n. The term kp is the growth
factor, and the term (n p)/n is the limiting factor.
Q2
When p n, what is the value of the limiting factor? How big will the next
generation be?
Q3
When p is very small compared to n, what is the approximate value of the
limiting factor? How does the limiting factor affect f(p)?
Q4
If n 1,000,000 and k 1.5, find the size of the next generation for each of the
following current populations: p 5,000, p 50,000, and p 500,000.
You don’t have to measure the population by counting individuals; you can measure
it in any units you want. The mathematics will be simplest if you measure the
population as a fraction of n. So, if the population is 500,000 and n 1,000,000,
you can define x p/n and record the population as x 0.5.
Chapter 4: Other Functions
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
73
The Logistic Function
(continued)
Q5
If the population is measured in this way, how should you rewrite the limiting
factor? Write a new formula g(x) using this way of measuring population.
To construct a Sketchpad model, you will create an initial size for the population,
evaluate your function, and use the result as the size of the next generation. You will
apply the function repeatedly to observe the behavior of the population over time.
FIRST GENERATION
To enter the parameter k,
choose New Parameter
from the Calculator’s
Value pop-up.
To scale the axes, drag
the unit point, or drag
one of the numeric labels
on the axes.
1.
Start with a new sketch, and choose GraphPlot New Function. Plot the
function g(x) k x · (1 x).
2.
Choose GraphPlot Points, and plot the point (1, 1). Construct a diagonal
segment through the origin and this plotted point. This diagonal segment
represents y x.
3.
Move the origin near the bottom left of your window, and scale the axes so that
the point (1, 0) is near the right edge of your sketch window.
4.
Construct point P on the x-axis between 0 and 1, and measure its abscissa. This
point represents the size of the initial population.
5.
Calculate g xP—the size of the next generation—and plot the point xP , g xP.
6.
Construct a segment connecting point P and the plotted point.
The calculated value of g xP is the output (y-value) of the function in the first
generation, but you must use it as the input (x-value) for the function in the next
generation. The next step provides a geometric way of turning the existing y-value
into an x-value.
The x-value of the
intersection point
is the input for the
next generation.
7.
Q6
74
Construct a line through xP , g xP
parallel to the x-axis, and construct the
intersection Q of this parallel with the
diagonal line. Hide the parallel line, and
construct a segment connecting the
plotted point and Q.
Why is x Q equal to the value of the
function at xP ? In other words, why is
x Q g xP true?
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
1
k = 1.00
f (x) = kx(1− x)
xP = 0.68
0.5
f ( xP) = 0.22
Q
P
1
Chapter 4: Other Functions
The Logistic Function
(continued)
This completes the construction for a single generation, because the x-value of point
Q is g xP, the population at the start of the second generation. In the next step, you
will iterate this process, by using the same construction to find the population at the
start of the third generation, and then the fourth generation, and so on.
NEXT GENERATIONS
You can also use
EditProperties
Iteration to change the
number of iterations.
You may need to
increase the number
of iterations to answer
this question.
8.
Select point P and choose TransformIterate. In the Iterate dialog box, match
point P to point Q by clicking on point Q in the sketch. In the Structure pop-up,
make sure that Tabulate Iterated Values is checked. Click the iterate button in the
dialog box to show the first three steps of the iteration.
9.
Increase the number of iterations to 20 by pressing the key on the keyboard.
In the questions that follow, you are asked to observe the long-term behavior of the
population under various circumstances. You can observe the long-term behavior of
the population by looking at the iterated segments or by observing the values near
the bottom of the table. Be sure to look at the long-term behavior, ignoring the first
few generations.
Q7
Move point P to about 0.5. Does the population ever stabilize? If so, at what
value does it stabilize?
Q8
Drag point P back and forth between 0 and 1, and observe the behavior. Does
the long-term behavior of the population depend on the initial position of P? If
so, in what way does it depend on P? If not, explain why the long-term behavior
is the same no matter where P is.
Q9
Change the value of k to 2.5. What’s the long-term behavior of the population
now? Does it depend on the initial position of point P?
Q10
Change the value of k to 3.1. What’s the long-term behavior of the population
now? Explain this behavior. Does it depend on the initial position of point P?
10.
Change the properties of parameter k so that the keyboard adjustment is 0.01.
Q11
Chapter 4: Other Functions
Select parameter k and use the key on your keyboard to gradually increase the
value of k from 2.5 to 4.0. Record your observations. For particular values of k,
drag P to look at the sensitivity of the long-term population to the initial
population. Also observe the stability of the long-term population as the value
of k is changed slightly.
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
75
The Logistic Function
(continued)
EXPLORE MORE
To actually derive one
of these forms of the
logistic equation from the
other requires calculus.
The function used in this activity produces results at discrete intervals of time,
generation by generation. This discrete form is sometimes called a logistic map. You
can also express the logistic function continuously, in a form that gives the size of
the population as a function of time. Here’s a form that uses k in a similar way:
k1
p(t) k c e(k1)t
Plot this function, using parameters for c and k. Experiment with different values of
the parameters, and observe how the behavior of this form of the logistic function
relates to the behavior of the Sketchpad model you have built. What similarities do
you observe? How can you account for the differences?
76
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
Chapter 4: Other Functions
Taylor Series
You can compute the value of a polynomial function directly and easily for any
particular value of x using multiplication and addition. But values of other
functions, such as the sine function, are much more difficult to compute.
You can learn how to
derive the Taylor series
when you study calculus.
In this activity, you’ll approximate the sine
function using a series called a Taylor series and
observe the behavior of the partial sums when
the series is evaluated to various depths. The
Taylor series approximation for sin(x) is
x 1 x 3 x 5 x 7 x 9 x 11
f(x) 1! 3! 5! 7! 9! 11! . . .
SKETCH AND INVESTIGATE
Parameters must be
independent values in
order to be iterated, so
you can’t set the initial
values of the parameter
that depends on x until
after you construct the
iteration.
1.
In a new sketch, create a square grid, construct point A on the x-axis, and
measure the point’s x-value. Label the x-value x.
2.
Create four parameters to use in iterating the series. Label them i, num, den,
and sum.
Q1
Parameter i represents the index for the terms, following the sequence 1, 3, 5, . . . .
What rule can you apply to one element of this sequence to calculate the next?
Q2
Parameter num represents the numerator, taking on values x, x 3, x 5, x 7,
and so forth. What is the rule to calculate a value of this sequence from the
previous value?
Q3
Parameter den represents the denominator, taking on values 1!, 3!, 5!, and so
forth. What’s the rule to calculate the next value of this sequence? (Express
your answer in terms of the previous values of den and i.)
Q4
Parameter sum represents the sum of all the terms from the first term through
the ith term. What value should you use as the initial value of the sum, before
adding the very first term? What’s the rule to calculate one sum from the
previous sum?
3.
All but one of these parameters have constant initial values that you can assign
now. (The initial value for the other isn’t constant, but depends on the value of
x.) Assign appropriate initial values to the parameters that don’t depend on x.
Assign an initial value to the other parameter as though the value of x were 2.
Q5
Chapter 8: Sequences and Series
What initial values did you assign to the parameters?
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
139
Taylor Series
(continued)
4.
For each of the four parameters, use the rule you described above to calculate the
next value of the quantity it represents. (Your calculations should involve only
the values of the four parameters and the value of x.)
5.
Plot the point (x, sum). The iterated image of this plotted point will allow you to
see the graph of each successive expansion of the Taylor series.
6.
Iterate each of the parameters to its calculated next value.
7.
The parameter num doesn’t yet have a correct initial value, because the initial
value depends on x. Select num, choose EditEdit Parameter, and calculate the
initial value so that it depends correctly on x.
8.
Drag point A left and right on the x-axis, observing the values in the table and
the positions of the plotted points.
The last row of the table
should contain n 1.
9.
Select the iterated image of the plotted point and press the key on the
keyboard twice to set the depth of iteration to 1.
The terminal point is the
very last image of the
iterated point, based
on the current depth
of iteration.
10.
With the iterated image of the plotted point still selected, choose Transform
Terminal Point. Then construct the locus of the terminal point as A moves
along the axis.
Q6
What is the shape of the locus? Which terms contribute to this shape?
11.
Set the depth of iteration to 2.
Q7
How does this change the shape of the locus? Which terms contribute now?
12.
Increase the depth to 3. Turn on tracing
for both the iterated image of the
plotted point and the locus. Animate
point A, and observe the behavior of
the point images.
Make sure the calculated
values are what you
expect.
Select all four parameters,
and choose Transform
Iterate. Then match each
parameter to its next value.
To increase the depth,
select either iterated
image (the table or the
image of the plotted
point), and press the
key on the keyboard.
Q8
You may have to move
the origin and change the
domain of the locus to
see two full periods.
140
What shapes do the iterated point
images trace? Sketch their shapes and
explain the role of each trace based on
the terms of the series.
2
A
-num⋅x2 - 2den⋅(i+1)⋅(i+2)
num
sum+ den
n
i+2
0
3.00
-8.00
6.00
2.00
1
5.00
32.00
120.00
0.67
2
7.00
3
9.00
-128.00 - 4
512.00
5040.00
0.93
362880.00
0.91
13.
While point A is moving, increase
the depth until the locus accurately
approximates the sine curve for at least two periods.
Q9
How many terms are required to give a reasonable approximation for the
first period of the sine function? For the first two periods?
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
Chapter 8: Sequences and Series
Taylor Series
(continued)
EXPLORE MORE
The Taylor series for the cosine function is as follows:
x 0 x 2 x 4 x 6 x 8 x 10
f(x) 1 2! 4! 6! 8! 10! . . .
You can change the initial values of your existing iteration to calculate this series.
Decide which parameters to change, and calculate and plot the modified series.
Chapter 8: Sequences and Series
Exploring Precalculus with The Geometer’s Sketchpad
© 2005 Key Curriculum Press
141
Chaos
Dynamical systems, fractals, and chaos are relatively new fields of mathematics.
Many of the most important concepts were not discovered until high-speed
computers made it easier to visualize iterative mappings.
A dynamical system is one that changes over time, with its state at any moment
computed from the state of the system at the previous moment. As the dynamical
system evolves, sometimes its limiting state may approaches a specific fixed state (a
point attractor), or may alternate among a set of fixed states (a periodic attractor). A
limiting state that is neither a point attractor nor a periodic attractor is called a
strange attractor.
SIERPIŃSKI’S TRIANGLE AS AN ATTRACTOR
1. Open the sketch Chaos1.gsp in the folder Supplemental Activities | Chaos.
Select the parameter depth and increase it using the + key. Be careful not to go
too far. A depth of 7 should take the image to the limit of the screen resolution.
B
A
C
This figure may look familiar to you. It is Sierpiński’s triangle, a fractal. It is selfsimilar and infinitely complex. As you will see, it can also act as a strange attractor.
Q1 Let p be the perimeter of the of the midpoint triangle added in the first iteration.
What is the sum of the perimeters of the three triangles added in the second
iteration? What is the sum of the perimeters of the triangles added in the third
iteration? As the number of iterations grows, is there a limit to the sum of the
perimeters added at each iteration?
Repelling
2. Open the Repel page of Chaos1.gsp. Point P' is the dilation of point P by a
factor of 2 about the nearest vertex.
The orbit of a object is
the set of all iterated
images of that object.
Q2 If this dilation is defined as an iteration rule, and point P is on the interior of the
triangle, do you think the orbit will eventually leave the triangle? If so, how many
iterations can remain in the triangle before the next iteration leaves?
3. Select point P and the parameter depth. Hold the Shift key while choosing
Transform | Iterate By Depth. Map point P to P’. Click the Iterate button in
the dialog box to create the iteration.
Exploring Precalculus with The Geometer’s Sketchpad
©2005 Key Curriculum Press
Supplemental Activities: Chaos • 1
Chaos (continued)
4. Construct the terminal point of the iterated point image by selecting the image
and choosing Transform | Terminal Point. Label this point T.
B
T
P
A
P'
C
Q3 Drag point P around to test your conjecture from Q2. Adjust the depth and see
how many iterations you can keep inside the triangle.
Q4 Once an iterated point has left the triangle, is it possible for some subsequent
iteration to return to the triangle?
In fact, Sierpiński’s triangle is a repeller for this iteration rule. What that means is
that any point that is actually on the fractal will have an orbit that never leaves the
fractal. Any other point, however near, will eventually be cast out.
Attracting
Since the iteration rule avoids the fractal, perhaps running this iteration in reverse
would attract the orbit to the fractal. A given orbit point is created by doubling the
distance from the nearest vertex. We can get back to the previous point by halving the
distance. However, there is no way of knowing which vertex was the center of dilation,
so you’ll pick a vertex at random and halve the distance to that vertex.
5. Open the Attract1 page of Chaos1.gsp. The image of point P has three possible
locations, depending on the position of point R. Drag point R to see each of the
possible locations of P'.
You will iterate P to P'. For each iteration, you will randomly choose the vertex toward
which to dilate.
6. Select points P and R and the parameter depth. While holding the Shift key,
choose Transform | Iterate To Depth. Map P to P' and R to itself. In order to
make point R choose a new random vertex for each iteration, choose To New
Random Locations from the Structure menu of the dialog box.
To make the iterated
image points small,
select them and choose
Display | Line Width |
Dashed.
7. Make the iterated image points small.
Q5 The initial value of depth is 100. Describe the pattern formed by the iterated
points.
Q6 Select the depth parameter and press the + key to increase the depth to about
3000. Describe the pattern now.
2 • Supplemental Activities: Chaos
Exploring Precalculus with The Geometer’s Sketchpad
©2005 Key Curriculum Press
Chaos (continued)
Q7 Drag point P to change the initial conditions of the iteration. What changes can
you observe in the pattern?
Q8 At any particular step of the iteration, can you predict where the next point will
be? Does this iteration have a point attractor, a periodic attractor, or a strange
attractor? Explain your answer.
To change the
properties, select the
iterated image and
choose Edit |
Properties. On the
Iteration panel, click the
button To Same
Location Relative to
Original.
Q9 Change the properties of the iterated image so that the iterated images of point R
move to the same location relative to the original, rather than to random
locations. How does this change the image? Does it now have a point attractor, a
periodic attractor, or a strange attractor? What happens if you drag point R?
With a small change, you can produce the same pattern with far fewer iterations.
8. Open the Attract2 page of Chaos1.gsp. On this page all three possible image
points have been constructed.
9. Select point P and the parameter depth. While holding the Shift key, choose
Transform | Iterate To Depth. Map P to one of the image points. Choose
Add New Map from the Structure menu, and map P to one of the other image
points. Choose Add New Map again, and map P to the third point. Finally, click
Iterate.
10. Make the iterated points small. Hide the initial images of P, the three black
points. Increase the depth—but do not enter too high a value, because the
number of iterated objects increases exponentially.
B
P
A
C
Q10 Are all of the iterated points on Sierpiński’s triangle? Are any of them? Try
starting with point P near the middle of the triangle, clearly not on the fractal.
Summarize
You used two mappings here. The first mapping repels points from Sierpiński’s
triangle, and the second and third attract points. In either case, given a starting point
on or within a triangle, these two behaviors hold true:
A point that begins on the fractal will stay on the fractal.
A point that is not on the fractal will never reach the fractal (though the attraction
mapping will cause it to approach the fractal).
Q11 Prove or demonstrate both statements. It is sufficient to show that for both
mappings, if a point is on the fractal, so is the next point(s).
Exploring Precalculus with The Geometer’s Sketchpad
©2005 by Key Curriculum Press
Supplemental Activities: Chaos • 3
Chaos (continued)
MIRA, JULIA, AND MANDELBROT SETS
Mira Sets
1. Open the sketch Chaos2.gsp in the folder Supplemental Activities | Chaos.
The Mira page contains an iteration based on a function f(x), using these equations:
f ( x ) = ax +
2 x 2 − 2ax 2
1 + x2
,
x' = y + f(x), y' = f(x') - bx
Point P' is the first iterated image of P, plotted at (x', y'). Point T is the terminal point
of the iteration.
If you select either a or
b, pressing the + or –
key will change the
parameter’s value
by 0.01.
2. Experiment with small changes in parameters a and b. It is interesting to see
that very small changes in the parameters can produce radical changes in the
orbit. Drag point P to see what effect the initial seed has.
Q1 The initial settings for this sketch are a = 0.20, b = 1.00, and P (8.61, 2.97). With
these settings, the orbit appears to be stable. As you experiment, record the
settings for interesting figures you find. Which settings attract the orbit to a
single point? Which make it periodically go between two or more points? Which
send it out of range?
Julia Sets
The Mira mapping is actually an offshoot from some research into the behavior of
elementary atomic particles. The next page shows a Julia set, based on this mapping:
x' = x2 - y2 + a,
y'= 2xy + b
3. Open the Julia Forward page of Chaos2.gsp. The construction is the same as
that on the Mira page, but uses the equations above to define x' and y'.
Q2 The patterns are not generally as striking, but there are similar results. When is
the orbit attracted to a single point? When does it jump between two or more
points? When does it fly off the screen? Again, record settings of parameters a
and b.
4 • Supplemental Activities: Chaos
Exploring Precalculus with The Geometer’s Sketchpad
©2005 Key Curriculum Press
Chaos (continued)
It turns out that this mapping repels a point from a Julia fractal. As before, the fractal
is an attractor for the inverse of the mapping, as defined by the equations below.
Notice that there are two values for x', and a y' corresponding to each step of the
iteration. When performing the iteration backward, the path branches and the
number of iterated points grows exponentially, swarming around the Julia fractal.
x′ = ±
1⎡
x−a+
2 ⎢⎣
( x − a) + ( y − b)
2
2
⎤
⎥⎦
y′ =
y −b
2 x′
4. Open the Julia Back page of Chaos2.gsp.
Experiment by dragging point P and changing the parameters a and b. For certain
settings, the fractal is in a single connected piece. For others, it is broken into discrete
points.
Q3 What is the effect of dragging the starting point P across the screen? Does it
change the overall shape? Does it alter the detail?
Q4 The abbreviation J(a, b) is used to refer to the fractal that comes from some
specific settings for a and b. For each of the following fractals, describe it,
identify its symmetries, and tell whether or not it is connected: J(-1, 0), J(0.3,
0.8), J(0, 0), J(0, 1).
Q5 What general statement can be made about Julia fractals where b = 0?
The Mandelbrot Set
Benoit Mandelbrot divided the Julia fractals into two sets: those that are connected,
and those that are disconnected. The parameters of the fractal can be imagined as
point coordinates. Consider the set of points (a, b) such that J(a, b) is a connected
fractal. That set of points is itself a fractal, the Mandelbrot set. There is a helpful
shortcut for determining which points belong to the Mandelbrot set. Run the Julia
mapping forward, using (a, b) as a starting point. If the orbit diverges, the fractal J(a,
b) is disconnected.
5. Open the Julia Forward page of Chaos2.gsp again. Select parameter a. Choose
Edit Parameter from the Edit menu. Delete the existing value in the
Calculator, and click in the sketch on measurement xP. Use the same procedure
to set parameter b to yP.
Exploring Precalculus with The Geometer’s Sketchpad
©2005 by Key Curriculum Press
Supplemental Activities: Chaos • 5
Chaos (continued)
As you drag point P across the screen, the orbit changes rapidly. When it clearly
converges on one or more points, point P is in the Mandelbrot set. Drag P until the
orbit appears to explode. That point is near the boundary of the set.
6. Select point T and the origin point. Choose Measure | Coordinate Distance.
Hide T and the iterated images. Reduce the depth to about 50.
7. Choose the custom tool Less than 2. Click on point P and then on the
measurement that you created in the previous step.
8. Switch back to the Arrow tool and drag point P.
Point P now leaves a black trail whenever T is within 2 of the origin. Once T has
exceeded that radius, it can never return. If the orbit is still within 2 after 50
iterations, the seed (point P) probably belongs to the Mandelbrot set. Drag point P
across the screen to color a rough approximation of the set.
1
0.5
-2
-1
P
-0.5
EXPLORE MORE
Even such a rough graphical rendering of the Mandelbrot set was not possible until
the development of high-speed computers, but it was possible to draw conclusions
about the orbits of certain individual points. Calculate the images of point (0, 0) after
the first and second iterations. Do the same for the point (-1, 0). Prove that these
points are in the set. Try some other points on the x-axis.
6 • Supplemental Activities: Chaos
Exploring Precalculus with The Geometer’s Sketchpad
©2005 Key Curriculum Press
CHAOS
Objective: Students create various fractal designs
involving strange attractors. In the first part they create
the Sierpiński triangle, and in the second they create
and explore Mira, Julia, and Mandelbrot fractals.
Q4 Once an iterated point has left the triangle, no
subsequent iteration can return to the triangle.
Dilating a point outside the triangle about any
vertex results in an image point that is farther from
the triangle than the pre-image point was.
Prerequisites: Students will find this activity easier if
Attracting
they have already encountered some of the concepts
Q5 No pattern is yet visible with only 100 iterations.
and terminology of mathematical iteration (for
instance, pre-image, image, seed, orbit, and mapping).
Sketchpad Proficiency: Intermediate. Students
create iterations, but with the necessary objects
Q6 With 3000 iterations, the shape of the Sierpiński
triangle is filmy but clearly identifiable. The larger
open triangles are quite prominent.
Q7 Dragging P has very little effect on the image. If P
prepared in advance.
is in an area far from the fractal (for instance, in
Class Time: 30–40 minutes for each part. Each part
the middle of the largest midpoint triangle), you
stands on its own. It’s probably best to give students a
can identify the first few iterations, because they
day or two to digest the first part of the activity (the
are near the middle of their respective
Sierpiński triangle) before moving to the second part of
(successively smaller) triangles. Dragging P has no
the activity (Mira, Julia, and Mandelbrot sets).
visible effect beyond these first two or three
Required Sketches: Chaos1.gsp, Chaos2.gsp
images.
Q8 At any particular step, you cannot predict where
Example Sketch: Chaos1 Work.gsp,
the next point will be, because the position of the
Chaos2 Work.gsp
next point is determined by a random process. As a
SIERPIŃSKI’S TRIANGLE AS AN ATTRACTOR
Q1 If the perimeter of the first midpoint triangle is p,
the three triangles of the second iteration have a
combined perimeter of 3p/2, and the triangles of
the third have a combined perimeter of 9p/4. The
terms constitute a geometric sequence with a ratio
of 3/2. The sum of the perimeters in the nth
iteration is p (3/2)n-1. These terms increase
without limit. (Although the activity does not
discuss the concept of fractal dimension, this
ratio corresponds to a fractal dimension of
log 3 / log 2 ≈ 1.58.)
Repelling
Q2 Conjectures will vary. Students use the actual
sketch to investigate these questions in Q3.
Q3 The orbit will stay in or on the triangle only if it
begins from a point on the fractal. Drag point P
slowly to find a “sweet spot.” With a bit of work it is
possible to make as many as 15 iterations stay in
before the orbit leaves. If you could drag P by
result, the iteration has a strange attractor.
Q9 By making the position of R consistent from one
iteration to the next, the process always picks out
the same vertex. The chosen vertex is a point
attractor for the iteration. Dragging point R
changes which of the three vertices of the triangle
serves as the point attractor.
Q10 If you drag point P to a spot that’s clearly not on
the fractal, it’s clear that the first images of P are
also not on the fractal. In fact, none of the image
points lies on the fractal. The fractal is visible only
because the image points are increasingly close to
it, although never actually on it.
Summarize
Q11 Sierpiński’s triangle is self-similar. Dilating it with
respect to one of the vertices, by factor 1/2, maps
the entire fractal onto itself. Hence, any point on
the fractal is mapped to the fractal. Dilating by a
factor of 2 maps the nearest of the three sections
onto the whole fractal.
moving the mouse less than a pixel at a time, you
could get closer to the fractal and keep more
iterations inside the triangle.
Exploring Precalculus with The Geometer’s Sketchpad
©2005 Key Curriculum Press
Activity Note—Chaos • 7
Chaos (continued)
MIRA, JULIA, AND MANDELBROT SETS
J(0, 1)
connected
You may want to introduce students to the shape and
properties of the Mandelbrot set before beginning this
activity. Several programs, books, and video
presentations are available for this.
Q5 When b = 0, the fractal has reflection symmetry on
both axes.
The Mandelbrot Set
Mira Sets
Q1 Answers will vary, with each student trying
different parameter values and different positions
of point P. You may want to have students print
their more interesting images. The orbit diverges
when a or b is greater than one. Experiment with
the settings below for some interesting images.
With some settings, the position of point P has
little effect on the overall image. With others, the
effect is significant.
The rendering of the Mandelbrot set created in this
activity is rather crude, but so was the first version
created by Benoit Mandelbrot himself. Students can
trace the figure more easily if they first see a finished
picture. However, if you give them time to find it
themselves, they may share a certain sense of discovery.
EXPLORE MORE
pre-image
1st image
2nd image
a
b
(0, 0)
(0, 0)
(0, 0)
-0.05
1.00
(-1, 0)
(0, 0)
(-1, 0)
-0.01
0.99
-0.75
0.92
0.99
0.99
0.94
0.91
Q2 Try the settings below and experiment with others.
The point (0, 0) continues to be mapped onto itself no
matter how many iterations. The point (-1, 0) is
mapped to (0, 0), then back, then continues to jump
between these two points.
RELATED ACTIVITY AND SKETCH
The related activity The Mandelbrot Set enables
The position of point P has a great deal of influence
students to create and explore the Mandelbrot set.
on whether the orbit converges, but it does not
The sketch that accompanies that activity
appear to change the attractor point(s).
(Mandelbrot.gsp) shows for a more detailed
a
b
0.32
0.39
0.25
-0.44
-0.97
0.23
rendering of the Mandelbrot set and allows students
to zoom in on any portion of the rendering to see the
fractal boundary in more detail.
Q3 When you drag point P, the overall shape does not
change, but the detail does.
Q4 All Julia fractals have 180° rotational symmetry on
the origin.
J(-1, 0)
reflection symmetry on both axes,
connected
J(0.3, 0.8)
disconnected
J(0, 0)
a circle centered on origin,
unlimited number of rotation and
reflection symmetries, connected
8 • Activity Note—Chaos
Exploring Precalculus with The Geometer’s Sketchpad
©2005 Key Curriculum Press
Barnsley’s Fern
Barnsley’s fern is a fractal created by an iterated function
system, in which a point (the seed or pre-image) is
repeatedly transformed by using one of four transformation
functions. A random process determines which
transformation function is used at each step. The final
image emerges as the iterations continue.
Affine transformations
preserve collinearity and
ratios of distances.
The transformations are affine transformations of form
x' = ax + cy + e
y' = bx + dy + f
Thus each transformation can be specified by six constants
(a, b, c, d, e, and f in the example).
SKETCH
1. Open the sketch Fractal Fern.gsp in the folder Supplemental Activities |
Fractal Fern. This sketch contains the constants to be used in the four
transformations and tools to simplify the construction process.
A random process controls which transformation is chosen for each iteration. For the
Barnsley fern, the first transformation is chosen 1% of the time, each of the next two is
chosen 7% of the time, and the fourth transformation is chosen 85% of the time.
You’ll start out by constructing a point that can be used to make the random choices.
To measure the ratio,
select A, B and C in
order. Then choose
Measure | Ratio.
The Between tool
performs a calculation
whose result is 1 when a
test value is between
two reference values,
and whose result is 0
otherwise. To choose
this custom tool, click
and hold the Custom
Tool icon, and choose
Between from the menu
that appears.
2. Construct point C on horizontal segment AB and measure the three-point ratio
AC/AB. Label the ratio r. Drag C to be sure that r stays between 0 and 1.
3. To assign probabilities to each of the transformations, create five parameters t1
through t5. These parameters define the range of values associated with each
transformation. To achieve probabilities of 1%, 7%, 7% and 85%, assign the
parameters values of 0.00, 0.01, 0.08, 0.15, and 1.00. Each successive pair of
parameters defines the range of values for one of the transformations.
4. Choose the Between custom tool and use it to generate results m1 through m4,
using r as the test value and pairs of parameters t1 through t5 as the reference
values. (In other words, m1 should produce 1 when r is between t1 and t2, and
should produce 0 otherwise. Similarly, m2 should produce 1 only when r is
between t2 and t3.) Drag C to make sure that, no matter where C is, exactly one of
the values m1 through m4 produces 1, and the other three produce 0.
5. Make sure the origin is near the bottom center of the screen, and that the scale of
the axes leaves 10 on the y-axis slightly below the top of the screen.
Exploring Precalculus with The Geometer’s Sketchpad
©2005 Key Curriculum Press
Supplemental Activities: Barnsley’s Fern • 1
Barnsley’s Fern (continued)
Make sure these labels
are correct, or the
Affine Transformation
tool won’t work correctly.
This calculation will
compute the x value
corresponding to the
current value of r.
6. Construct an independent point P to be the pre-image for the iteration, and
measure its x- and y-coordinates. Label the coordinates x and y (no subscripts).
7. Choose the Affine Transformation tool, and use it to generate the four
transformation functions. For each function, click on the appropriate values for
a, b, c, d, e, and f.
8. To calculate a transformed x value from the four functions, create a calculation
that adds the x values of all four transformations, multiplying each x value by the
corresponding value from m1 through m4.
m1 (a1x + c1y + e1) + m2 ( a2x + c2y + e2) + m3 ( a3x + c3y + e3) + m4 ( a4x + c4y + e4)
Point Q is the
transformed image of P.
As you drag C, you
should observe Q
switching among four
possible transformed
images.
9. Calculate a y value in the same way. Plot the point determined by these two
calculations, and label it Q.
10. The value of r determines which transformation is used. Drag C to change the
value of r. Observe that Q moves as each of the transformations is used in turn.
You’ll iterate the construction by iterating P to Q and by moving images of C to new
random positions on the segment.
11. Iterate the function system: select points P and C. Choose Transform |
Iterate, and iterate P to Q and C to itself. Click the Iterate button.
12. Delete the table of iterated values that appears. Choose Edit | Properties |
Iteration for the iterated image of P, and set the number of iterations to 20.
Choose Display | Line Width | Dashed to show the image with small points.
Q1 Observe that the orbit of the iterated images tends toward a fixed point. Drag
point C to change the transformation used. Does each transformation function
appear to be associated with a single fixed point? Give the approximate
coordinates of the fixed point corresponding to each function.
13. The initial orbit that appears uses only a single transformation, because the
images of C are not yet taking on new random positions. Select the iterated
image of P and choose Edit | Properties | Iteration. Change the number of
iterations to 1000, and click the radio button labeled To Random Locations On
Iterated Paths.
The 1000 iterations that appear produce an outline of the final shape, but you’ll need
many more iterations to develop it in detail. One way to generate more detail is to
trace the iterated image of P while rerandomizing the positions of images of C.
14. Select point C and choose Display | Animate.
To trace, choose
Display | Trace Iterated
Point.
15. Select the iterated image of P and turn on tracing. Then open Iteration
Properties, and click the Randomize Locations button. Each time you click the
button, the images of C are assigned new random locations, and new images of P
leave traces on the screen. Continue pressing the Randomize Locations button
2 • Supplemental Activities: Barnsley’s Fern
Exploring Precalculus with The Geometer’s Sketchpad
©2005 Key Curriculum Press
Barnsley’s Fern (continued)
until you have a fair amount of detail in the image of the fern. Once you have
enough detail to see the fern clearly, click OK to dismiss the dialog box.
16. Stop the animation of point C, and then drag C so the fourth function is used to
generate point Q from P.
Q2 Move P to different locations on the fern, and describe the effect of this
transformation function in terms of the shape of the fern.
Q3 Similarly, activate each of the other transformation functions, and describe the
effect of these functions.
To find a fixed point of a
transformation, drag P
until P and Q coincide.
Q4 Where are the fixed points of the four transformations in terms of the shape of
the fern?
Q5 Change the t parameters so the first transformation is never used. Erase the
traces and regenerate the fern using only transformations 2, 3, and 4. What’s the
difference in the new shape? Describe the role of the first transformation.
Q6 Determine and describe the role of each of the other transformations in
generating the image.
EXPLORE MORE
Q7 Determine the effect of the various constants in specifying the fern’s shape.
Q8 Modify the fern transformations so that the leaves of the fern are opposite each
other, instead of alternating.
Q9 Make other minor changes to the function system to produce a modified shape.
(Be prepared to undo your changes – not all function systems generate
interesting results.)
Similar iterated function systems can generate a wide variety of fractals. For instance,
consider a system of three functions that have the following effects:
•
Dilate the pre-image point by a scale factor of one-half toward the point (6, 0).
•
Dilate the pre-image point by a scale factor of one-half toward the point (-6, 0).
•
Dilate the pre-image point by a scale factor of one-half toward the point (0, 10).
Q10 Write down the transformation functions for these three transformations.
Q11 Modify a copy of the fern sketch to use these transformations, with equal
probabilities for all three transformations. What figure results?
Q12 Build a similar iterated function system with four transformations, dilating the
pre-image by a factor of one-third toward one of the points (0, 0), (1, 0), (0, 1),
(1, 1) with equal probability. Describe the resulting figure.
Q13 Research Barnsley’s method and fractal compression. Report your findings.
Exploring Precalculus with The Geometer’s Sketchpad
©2005 Key Curriculum Press
Supplemental Activities: Barnsley’s Fern • 3
BARNSLEY’S FERN
transformation is responsible for generating the
Objective: Students plot points using four pairs of
iterated functions. By choosing randomly among the
function pairs, they create a fractal Barnsley’s Fern.
right-hand leaves. Finally, the 85% transformation
generates higher leaves from lower leaves, moving
toward the tip of the fern.
Prerequisites: Students should be familiar with the
EXPLORE MORE
use of mathematical iteration to produce fractals.
Q7 The 0.85 in function 4 makes each succeeding leaf
Sketchpad Proficiency: Advanced. Students should
be familiar with iteration and with the use of custom
tools.
85% of the size of the previous leaf. The .44 in
function 3 and the 1.6 in function 2 determine the
height at which the leaves begin on the two sides.
The numbers clustering around .20 in functions 2
Class Time: 40 minutes
and 3 make the side leaves about 1/5 the size of the
Required Sketch: Fractal Fern.gsp
main leaf.
Q1 Each of the four functions is associated with a fixed
point:
r values
Fixed point
0.00–0.01
(0, 0)
0.01–0.08
(-0.61, 1.87)
0.08–0.15
(0.15, 0.63)
0.15–1.00
(2.66, 9.96)
Q2 The fourth function transforms a point on one leaf
to the corresponding point on the next higher leaf.
Q3 The first function transforms a point anywhere on
the fern into a point on the lowest portion of the
stem. The second function transforms a point
anywhere on the fern to the corresponding point
on the lowest left leaf. The third function
Q8 To make the leaves opposite instead of alternating,
use the same value for f in the second and third
functions.
Q9 Answers, and resulting shapes, will vary. You may
want to ask students to print their most interesting
shapes to share with the class.
Q10 The three functions use these coefficients:
0.50
0.00
3.00
0.00
0.50
0.00
0.50
0.00
-3.00
0.00
0.50
0.00
0.50
0.00
0.00
0.00
0.50
5.00
Q11 These functions produce the Sierpiński triangle.
Q12 The four functions use the coefficients
transforms a point anywhere on the fern to the
0.33
corresponding point on the lowest right leaf.
0.00
0.33
0.00
0.33
0.00
0.67
Q4 The first fixed point is the base of the stem, and the
0.00
0.00
fourth is the very tip of the fern. The second is the
0.00
0.33
0.00
point on the bottom left leaf that’s in the same
0.33
0.00
0.00
relation to both that leaf and the entire fern, and
0.00
0.33
0.67
the third is the point on the bottom right leaf that’s
0.33
0.00
0.67
in the same relation to that leaf and the entire fern.
0.00
0.33
0.67
Q5 If the first transformation is never used, the fern
The resulting figure is a Sierpiński square:
appears without a stem, and each leaf appears
without its stem. The first transformation takes the
point to the lowest part of the stem, where the 85%
transformation can move it up the stem. Thus the
entire stem is produced.
Q6 The second transformation takes a point to the
corresponding part of the lowest left leaf.
Subsequent 85% transformations generate the
other left-hand leaves. Similarly, the third
Exploring Precalculus with The Geometer’s Sketchpad
©2005 Key Curriculum Press
Activity Notes—Barnsley’s Fern • 4