Extending graphing concepts

The work of a curious fellow
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 fixed amount.

On the Next x by Multiplying display we repeat the graph of

except that we use the multiplication rule for picking the next x value.
next x by multiplying
next x by division

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.

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 will emerge.
next x by chance
next x by 1/(x+.1)

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.
devilish hardDividing 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
next x by feedback
new y by feedback

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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