 Greetings. Today we're going to discuss how to graph functions of the form y equals a times sine of bx or y equals a times cosine of bx. And although some comments will be valid for all values of a and b, our examples today will only use situations where a and b are both positive. The starting point for many of these graphs are the standard graphs for y equals sine of x and y equals cosine of x. The textbook seems to call them the pure sinusoids. I just prefer to call them the standard graphs for sine of x or cosine of x, or if you want standard sinusoids. And again, although it may not be quite easy to see on this screen, things like the tick marks on the x axis are in terms of pi. So that one there is 3 pi over 2, that's 2 pi, this is pi over 2, and so forth. This basically represents one complete cycle of the sine graph and of the cosine graph. And as you can see, both sine and cosine satisfy the inequalities minus 1 less than or equal to the value at x less than or equal to 1. So, some of the things that we're going to assume you're familiar with at this point are the kind of standard meanings of the values of a and b. Basically, a controls the amplitude of the function and b controls the period. And basically, if a technically absolute value of a is greater than 1, it stretches the graph vertically so that the high point of the graph will be at y equal to a and the low point will be at y equal to negative a. And if the absolute value of a is less than 1, it's just a compression or contraction of the so-called pure sinusoidal wave or the standard sinusoidal wave. The period of the function is 2 pi divided by the absolute value of b or, in our situations, 2 pi over b. If you kind of recall those sine and cosine curves, both of them more or less look something like this. I'm going to set the vertical or y-axis has not been drawn in here. And to some extent, that's about the only distinction between the sine curve and the cosine curve. So, for example, if we want to draw a sine curve, kind of bear with me as I tried to draw a relatively straight line, we could, for example, draw our vertical axis right there. And that basically gives us just a little bit more than two complete cycles of the sine curve. On the other hand, if we have a cosine curve, then we're just going to draw the vertical axis, the y-axis, in a slightly different position. And it might go somewhere like that. And we now have a cosine curve. And this is what we're going to be doing on the next examples is kind of determining where the axis goes, whether it's a sine or cosine, determining the amplitude, and then determining the period. And with that, we should be able to get a complete sketch of the graph. So, our first example is y equals 4.8 times cosine of one-third x. And so, because this is a cosine curve, I'm going to draw the vertical axis, where we did on the last one, about right through there. And the coefficient of 4.8 tells me that I have an amplitude of 4.8. Or in other words, the y-value is going to be strictly between minus 4.8 and 4.8. So, although we might not completely scale the y-axis, the one thing we do know is that the high point here is at 4.8. And the low point will extend that axis a little bit is at minus 4.8. And for many purposes, that's good enough for the graph. The question right now then is, what is the period so that we can mark off the scale on the x-axis? The period is going to be, in this case, 2 pi divided by one-third. So, you have to be a little careful with your division of fractions. But that comes out equal to 6 pi. So, what that tells me is if I do one complete period of the curve, we're going to be out to x equal to 6 pi. So, I can mark my x-axis here at 6 pi right there. And now what I try to focus on are what I sometimes call the quarter points and the half points. Half a period gets me roughly to there. So, that means I'll have a 3 pi on the x-axis right there. And a quarter of a period puts me right there. And that's going to be 6 pi over 4. Or that value here, going to write it on the side here, will be 3 pi over 2. And then if I wanted to complete one complete cycle, I would scale that point there. And again, I'm going to take from 3 pi, we go over one-quarter of a cycle. So that is actually 3 pi over 2 plus 3 pi. And that basically comes out 4.5 pi. Or if we wanted to, we could write it as 9 pi over 2. And so, that tick mark is at 9 pi over 2. And just because I didn't have enough room maybe to squeeze all that in on this graph, it comes out with those errors on it. But basically, that gets us a graph of the function y equals 4.8 times cosine of 1 third x. If we wanted to, we could extend the tick marks out further on the x-axis using the periodic properties of the function. For our second example, we're going to sketch the graph of the sinusoidal function y equals 30 times sine of 10 pi x. Our main task will be first to draw in the y-axis, then scale the y-axis, and finally, scale the x-axis. Since this is a sine function, what we will do is say, okay, the basic sine function starts at the origin, so the y-axis can be drawn in like that. And what we need to know in order to scale the y-axis is the amplitude. In this case, the amplitude is this number right here, so the amplitude is 30. So that tells us on the y-axis we're at 30 at the high point and minus 30 at the low point. And that's basically all we have to do for the y-axis right now. That's going to be good enough to get a reasonable sketch of this sinusoidal function. The next task is to determine or scale the x-axis, and for that we need to know the period. Remember that the period is determined by this number right here, this 10 pi. And in fact, we know that the period will be 2 pi divided by 10 pi, and as we can see in that setup the pi's cancel in 2 tenths is also equal to 1 fifth. So the period is 1 fifth. That immediately allows us to start scaling the x-axis. If we start at the origin, say right there, and go one complete cycle, we end up at this point right here. And the x-coordinate, that is determined by the period, so the x-coordinate to that point is 1 fifth. Now if we continue on and go two complete cycles, we end up at this point right here. And again, that's two periods, or the x-coordinate is 2 fifths. And now what we try to do is make basic use of one half periods and quarter periods. Focusing on the x-intercepts, if we look at this x-intercept here, we can see it's exactly half way between zero and one fifth. In other words, one half of a period. So what we do is take one half times one fifth, and we get one tenth. So the x-coordinate to that point is one tenth. And I'm going to write that as a decimal to save a little room. So that's 0.1. And again, we can now go from one fifth, which is 0.2. Another half a period out to here, half a period is 0.1. So this x-coordinate is 0.3. And now we focus on the high point. And again, we can see if we look at this and we use the fact that this high point basically will occur one quarter of a period from the origin in this case. And all we have to do is a little arithmetic with fractions, and we get that the coordinate of that is one twentieth. And again, maybe shoot it over here, convert it to a decimal, we can see that that coordinate is 0.05. That gives us the coordinates of the high point, this high point on the graph. The x-coordinate is 0.05, and the y-coordinate is 30. Now we can do something very similar for the low point. Again, if we draw that in there, this low point is going to be a quarter of a period from this x-intercept here. So it's basically 0.1 plus a quarter of the period, which is 0.05. So that comes out as 0.15. And that gives us then the coordinates of the low point. The x-coordinates will be 0.15, and the y-coordinate will be minus 30. That gives us a nice complete graph of this sinusoidal function. And you can see the basic strategy was to first draw in the y-axis, use the amplitude to scale the y-axis, and then use the period to scale the x-axis. And that completes this example, and I'll see you later so long.