Extending graphing concepts

 The work of a curious fellow

The plot thickens...
 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.
 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 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.
 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?
 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)
 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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