Basins of Attraction for Real Numbers
If we already know the zeros of a function, then Newton's method suggests another problem: which initial values
iterate to which zeros? If the iterates 𝑥1 , 𝑥2 , 𝑥3 , … , 𝑥𝑛 , … obtained from Newton’s method approach a certain zero,
𝑧, as 𝑛 gets larger, then we say that the iterates converge to 𝑧 and that 𝑥0 , the initial value, is in the basin of
attraction of 𝑧. In this investigation, we will consider the following question:
For a given function, where are the basins of attraction of each zero?
I. Consider the function 𝑓(𝑥) = 𝑥 3 − 4𝑥.
1. Find the zeros of this function using algebra.
2. Write the particular recursive equation that would lead an initial value 𝑥0 to one of these zeros using
Newton’s Method.
3. Verify that 𝑥0 = −3 is in the basin of attraction of the zero −2. On a graph of 𝑓(𝑥) = 𝑥 3 − 4𝑥, show
graphically what happened when you applied Newton’s Method starting at 𝑥0 = −3. Your drawing should
include tangent lines and locations of the intermediate iterates 𝑥1 and 𝑥2 .
4. Explain why there is no number less than −2 that is not in the basin of attraction of −2.
5. Verify that 𝑥0 = 0 is in the basin of attraction of 0 but 𝑥0 = 1 is in the basin of attraction of −2.
6. Somewhere between 0 and 1 there appears to be a transition between the basins of attraction of 0 and of
−2. To determine where the transition takes place, divide the interval between 0 and 1 into tenths.
Underneath each number, write down the basin of attraction to which it belongs.
𝑥0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Basin of attraction 0
1
−2
After completing this, you should know an interval 0.1 units long that contains the boundary between the
basins of attraction of 0 and of −2. Subdivide this interval into 10 equal parts each of length 0.01.
Beneath each number, write down the basin to which it belongs. You can now better estimate the location
of the boundary between the basins of attraction of 0 and of -2.
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II. Consider the function 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 2)(𝑥 + 5).
1. Use Calculus to find the coordinates of the turning points on the graph.
2. Implement Newton’s method with 𝑥0 = −4. This initial value is in which basin of attraction? Repeat this for
each value of 𝑥0 in the table below, and record your results.
𝑥0
−4
−3.8
−3.7
−3.698
−3.695
−3.6945
−3.6942
−3.68
−3.6
−3.55
−3.5
−3.4
−3.3
−3.2
−3
−2.5
−1
−0.5
Basin of Attraction
𝑥0
Basin of Attraction
−0.2
−0.185
−0.16
−0.1515
−0.1512
−0.1511
−0.15
−0.1
0
0.1
0.2
0.3
0.36
0.364
0.37
0.4
0.5
0.8
3. Use a tangent line argument to explain why 𝑥0 = −3.6 is in the basin of attraction of the zero 𝑧 = 2 even
though 𝑥0 is between the other two zeros.
4. Use a tangent line argument to explain why 𝑥0 = 0.36 is in the basin of attraction of the zero 𝑧 = −5 even
though 𝑥0 is between the other two zeros.
5. Make a large, accurate graph of 𝑓, filling an 8.5 x 11 piece of paper. On the x-axis of your graph, make a
colored dot on each value of 𝑥0 based on which basin of attraction that x-value belongs to. Color code as
much of the x-axis as you can based on which 𝑥0 values are in each basin of attraction. If you could
successfully color code all of the x-axis, the image you would produce would be a one-dimensional fractal.
6. Find all initial values 𝑥0 for which Newton’s Method will fail at the first step. If you start with one of these
initial values when you execute Newton’s Method, does the method actually fail? Why or why not?
7. Find all initial values 𝑥0 for which Newton’s Method will fail at the second step. That is, find 𝑥0 so that 𝑥1
exists but 𝑥2 is undefined. If you start with one of these initial values when you execute Newton’s Method,
does the method actually fail? Why or why not?
8. How many initial values exist that would cause Newton’s Method to fail at some stage in the iterative
process? If you use technology to execute Newton’s method, will you observe these failures?
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