 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 3pi over 2, that's 2pi, 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 one less than or equal to the value at x less than or equal to one. 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 one, 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 one, it's just a compression or a contraction of the so-called pure sinusoidal wave or the standard sinusoidal wave. The period of the function is 2pi divided by the absolute value of b or, in our situations, 2pi over b. If you kind of recall those sine and cosine curves, both of them more or less look something like this. And notice that 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 try 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 a 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 1 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 1 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 1 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 in 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. So for our second example, we're going to look at this one. And as you can see, this is a sine curve. So right now I will draw the axis in about right there. And again, we can see the amplitude is the value of a, as we've called it. So in this case, it will be 30. So our high point occurs at 30 and our low point at minus 30. Now here's one where the b value, if you want, is actually 10 pi. So that's actually going to affect the period. And the period for this will then be 2 pi divided by 10 pi. And you can see the pi is canceled here and we get a period of 1 fifth. So that may seem a little strange. You expect to see multiples of pi on that. But if we go one complete period here, that's one fifth. And if we want two complete periods, if you want, we put a point out there. It's at two fifths. And then again, what we would like to do is fill in some of the other points. The x coordinate where you get the high point, the other x intercepts and so forth. And remember, this point here is one half of a period from zero. And so that will be one fifth divided by two or one tenth. And maybe just to save a little space, I'll go to a decimal there and put that in as 0.1. The high point now, which would be about there, this time I'm only going to go a quarter of a period. And if you look at a quarter of a period, it basically comes out to be one twentieth. Or in decimal form, 0.05. So this point on the x-axis is at 0.05. And now we can basically, every time we kind of move a quarter of a period, we know we're going to add 0.05. So if I go to this point, which is where the low point is, we will be at 0.15. And now I can, yeah, it's pretty easy now to continue on. That one fifth is 0.2. So for example, if I wanted that point there, that's half a period beyond that. So it's 0.2 plus 0.1. I'm sorry, that's 0.2 plus 0.05. And so we get 0.25. So there you have it. This gives us a nice convenient way to graph any function of the form y equals a times sine of b times x or y equals cosine of bx. What I focus on, of course, is you can just have one of these sinusoid graphs at your disposal and then put in the axes and then scale as appropriate, whether it's a sine or a cosine. And then scale the y-axis according to the amplitude and scale the x-axis according to the period. And you'll have a nice sketch of the graph. That's it for now. It's along.