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The plot thickens...
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In the graphing example which we first did where
y=x*(10-x)
we made an assumption that is so basic that it is almost unconscious.
That assumption is that we get our next choice for x by adding a fixed
amount to the current value of x. Starting at x=0, the next point was
at x=1, then x=2 and so on until x=10. Let's spend a few minutes
thinking about other possible rules for plotting the graph of a function.
Suppose for example that the way to get the next value of x was
to multiply the current x by a fixed amount rather than to add a
fixed amount.
On the
Next x by Multiplying
display we repeat the graph of
y=x*(10-x)
except that we use the multiplication rule for picking the next x value.
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Now let's consider what happens if we divide the current value
of x by a number greater than 1 to get the next x. Take 1.25 for
example. The
Next x by Dividing
display illustrates that situation.
It turns out that multiplying by a constant to select the next
x point for a graph and dividing by a constant amount to the same
thing. This follows from the fact that multiplying by a number
less than 1 is equivalent to dividing, and dividing by a number
less than 1 is equivalent to multiplying. Think about it.
Likewise adding and subtracting a constant to get the next x are
the same since subtracting is just adding a negative number. So
in our examples we have exhausted the ordinary arithmetic
operations as means of selecting the next x to plot.
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How about some other innovative schemes for deciding which point to plot next on a graph? What if instead
of applying some constant to the old x to get the next one we
depend on chance to fill in the graph. On our graph of
y=x*(10-x)
let us just roll a ten-sided die,
the singular of dice, (as opposed to "douse") to select our next x. This as you will see on the
Next x by Chance
display is not particularly efficient, what with
landing on the same x repeatedly. Still, after a while the graph
will emerge.
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Consider the effect of selecting the next x in graphing a
function by applying x itself in some way rather than some
constant. Take for example division by x. Try to predict what
will happen if we pick some x to start with and then take 1/x to
be the next point on our graph. The
Next x by 1/x
display will demonstrate that technique.
Let's modify the ineffectual approach on the previous
example in a very simple way. As before we will pick some value
of x to start with. Then take the next x to be 1/(x+0.1) instead
of just 1/x. Run the
Next x by 1/(x+.1)
display.
 Dividing by something more than x will evidently cause the next point to fall short of of the reciprocal. Would replacing the 0.1 in the next x selection rule with 0.2 converge to a different location or just converge faster?
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As you have seen, even minor changes
in the "next x selection" rule can make quite a
difference in how a graph gets filled in, or not filled in as the
case may be. We could come up with all sorts of functions of x to
select the next x for plotting. There is one particular function
of x though which leads to some very interesting results. That is
just the function being plotted itself. Look at the example.
y=g*x*(1-x)
We could just pick some x to
start with, then select the next x equal to g*x*(1-x). Or more
simply stated, make the new x equal to the old y. In the
Next x by Feedback
display you will be
able to feed the output back into the input of the function
y=g*x*(1-x)
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An alternative way to display this
iteration process might be to plot successive values of y over
the initial value of x. This would allow us to readily see the
effect of starting with different initial x values on the process
of iteration. Run the
New y by Feedback
display.
This is the last perversion of normal graphing which we will
undertake for now. All of the ideas introduced here are intended to
extend your vision of what a graph might be. The next lesson in
the sample is called Iteration and Attractors.
Are there any questions?
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