>> Let's get away from this idea of adding the functions, and let's consider what happens if we multiply functions. It turns out that we can define the borders of a graph, and make a sine or cosine pattern go into a certain border. Let me show you what I'm talking about here. Suppose I enter 3 and negative 3, something very basic, and then we know what the sine curve looks like but I'll enter it anyway, and we'll take a look at these. And I think I'd like to have a look at these in a different window setting. This is okay for X, but I want plus and minus 4 I think for the Y's. And the graphs then look like this. Okay, these are all the expected outcomes, but in order for me to cause this sine curve to go up to this mark and down to this mark, go to the, make this the top of the amplitude and this the bottom of the amplitude, all I have to do is multiply them together. And we've seen it a number of times before, this would then be 3 sine X, and let me take off the regular sine curve here so we can see this more clearly, but here is the graph. So by multiplying the two functions together, we are defining, or describing a border sort of for our sine curve, and then we're placing that sine curve within a particular border. Now we can make that border look as exotic as we would like for it to look. Let's clear this one off, and come up with a rather exotic looking border. How about a border that is something like oh 1 over X, and the negative of 1 over X. And so here's the border, and I'm going to change window settings. We'll look at the border in just a second, I'm going to make up my window settings, X min negative 12, and the max 12. And then I'll come down for the Y's and set the Y at negative oops, negative .5, and the max at 1.25. Kind of tight on the Y's, but you'll see why in a minute. And these two graphs are here. Now what I'm going to try to do is get a sine curve to wiggle in between these borders, and I can do that by doing this, just enter 1 over X times sine X. Now I would like to make this a distinct coefficient. So I'm going to go back here and I'm going to put a parenthesis around the 1 over X. I'm parenthesing second and insert, because I want to insert an open parenthesis here, and I'll close that parenthesis on the other side. And now sine X. So all I'm doing is multiplying the 1 over X times sine X, and I am effectively putting that sine curve inside this border. This is called a damping effect, a damping effect. Now obviously I could take the borderlines off, but it kind of looks neat to see those border lines. Now remember, all I'm doing by, in multiplying here, remember that a coefficient of sine X is sort of defining the amplitude, and this then, this is the amplitude of that function. And so it goes up to that mark and down to the bottom of that, the reflection of that mark and so on, and just back and forth, just bounces back and forth between that kind of graph, plus and minus that amount.