The plot thickens...
In the graphing example which we first did where
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
Next x by Multiplying
display we repeat the graph of
except that we use the multiplication rule for picking the next x value.
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
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.
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
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
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)
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?
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.
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
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
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?