 Hello. In pass screencast, we have tried to determine a graph of a sinusoid given its equation. And frequently we try to set that up then using a graphing calculator. In this screencast, we're going to reverse the procedure and use the graph of a sinusoid to determine an equation for that sinusoid. So here's kind of the information we're going to try to work with. General form for a sinusoid equation is either this one or this one. And again, you can see the difference basically is whether you're going to use a sine or a cosine. And we'll talk about that a little bit later. But notice again that the numbers involved, the a, b, c, and d, all have significance with respect to the sinusoid graph. Again, what we're assuming here right now is that a is positive, and so that the amplitude is equal to a. And we also are in this equation using b greater than zero, and then we get the period will be 2 pi over b. The vertical shift, in other words, how much the graph has moved up or down from the x-axis, is determined by the number d. If it's positive, the graph has moved up. If it's negative, the graph has moved down. And the big difference between the sine and the cosine is the phase shift. So all the other constants, the a, b, and d, are the same. But you notice that the phase shifts will be different. And so I've used the c sub 1 and the c sub 2 to indicate the phase shift depending on whether we decide to use a sine or a cosine. And in fact, for most sinusoids, you can use either a sine or a cosine. You just have to make sure that you get the correct phase shift for the situation. So here's what I like to work in thinking about these things. This is a general sinusoid, and it's showing just a little over two periods. But one of the most important things on any of these sinusoid graphs is on this horizontal line, which actually is not necessarily the x-axis. That's what I call kind of the axis of the sinusoid. It's the middle. But these distances between the dots there are all a quarter of a period. And so what I oftentimes like to do is try to find a high point and the next low point on the graph of a sinusoid. There's a couple of reasons for that. If we look at this distance from the high point down to the low point, the vertical distance, this is two times the amplitude. So that distance will be 2a. And so in other words, if we can determine the y-coordinates of those two points, we can determine the amplitude. We'll just subtract the two y-coordinates to get that distance between the high point and the low point. And so that will be very helpful. Once we get the amplitude, the distance from this axis up to the top, the high point, is actually going to help determine the vertical shift. Again, this would be the amplitude. And so what we would be looking at is if we take the coordinate of that, let's just call that point, say t sub 1, y sub 1, and we can then take y sub 1 minus the amplitude and we will get the value of that horizontal, the y-coordinate of that horizontal line. The other great thing about the high point and the low point is if you look horizontally and go from, say, here up to here and look at this distance right here, that's going through two of these quarter periods. So this distance here is actually a half a period. So again, if you know the t-coordinates or x-coordinates, if you're using x, you can look at the difference in those two values and you will have half of the period at which time you can determine the period. So remember what we're actually after are a, b, c, and d. And a will be kind of the amplitude and b will be determined by the period. And d we will get as the vertical shift and c will be the phase shift depending on whether we use a sine or a cosine. So here's the graph. So what I'm going to do here is look for, if at all possible, to read the coordinates of the high point and the low point there. And as I look at this particular graph, I can see the high point has horizontal, or t-coordinate, pi over 3 and y-coordinate, 7. And the low point has coordinates 2 pi over 3 minus 2. I'm sorry, that's minus 3. So we'll take care of that a little bit here. We'll erase that, minus 3. And now the difference between 7 and negative 3, again remember that's a subtraction. So we change that to an addition and we get a value of 10. So right there, that's 2 times the amplitude. So in other words we have 2 times the amplitude equals 10 or the amplitude equals 5. And once we have that, again we can subtract the amplitude and we'll get down to this line here. Now of course we can almost see that line in this case but that's not always the case. And if I look at the upper y coordinate 7 and subtract the value of a, I get a value of 2. So that tells me that d is going to be equal to 2. So you can see we've gone a long ways now to determine an equation for the sinusoid. Next we're going to start working on the horizontal part of this and determine the period. And for the period again, what I'm looking at is this coordinate 2 pi over 3 and the pi over 3. And I want to subtract those two and remember that's the distance between a high point and the next low point. And that is going to give us half of the period. So if I take 2 pi over 3 minus pi over 3, again that's just like subtracting the fractions 2 thirds minus a third and we get pi over 3. That's half of the period. So we have period of 2 pi over 3. But remember that is not the value of b. b is determined by the period and in fact the period 2 pi over 3 is equal to 2 pi over b. Now we solve that equation for b. And it depends on how much detail you want to go through on it. But you can see both numerators are 2 pi. And so we can in this case quickly conclude b is equal to 3. In some more complicated cases we might actually have to kind of solve that equation for b. So we've got three of the values now. And the last thing to do is to determine the phase shift. But that depends on whether we choose to use a sine or a cosine. And in fact we can probably use, probably we can use either one. So the first one I'm going to do is to use a sine function. And what I'm going to focus on now is the horizontal line that is the axis of the sinusoid. And you can see right here that we get that point right there. And that then forms the basis for the sinusoid as we kind of trace this. We can see the sine function being generated. And what we have done is moved over from the origin pi over 6 units to the right. So for the sine function we get a phase shift of pi over 6. And with that now we do have a complete equation for this sinusoid. It would be y equals 5 times the sine. And now I use the value for b, which is 3, times t minus the phase shift, which is pi over 6, plus the vertical shift, which in this case is 2. So there's an equation for this sinusoid. Okay, and as I should point out too, that this is not the only correct answer. There are several different equations that can be used to represent this sinusoid. And in fact in the next slide we're going to take a look at this and determine a cosine function for this. And here, again, we still have a equals 5, b equals 3, and d equals 2. But if we use a cosine function there's a couple things we can do. In particular, one way of looking at this is to notice that if we start here and kind of trace through, we actually get the cosine function turned upside down. This is still kind of the halfway or axis of the sinusoid, and you can see that going from pi over 3 to this would be half a period, which would get us, remember half a period is pi over 3 in this case. Just remember we have a period of 2 pi over 3. So if I go back pi over 3, I'm at this low point. That's a negative cosine. So we could actually write, instead of using a positive value out front, the amplitude is still 5, but we would now have a negative 5. And we have a cosine, and it'll be 3 times t. And there is no phase shift. We've just flipped that cosine graph upside down, and now we have to deal with the vertical shift of plus 2. And so we have a cosine graph that we can, or an equation involving a cosine for that. If you don't like working with a negative value out there, you could start your cosine function at the high point here, and you can see then our phase shift is that distance right there, which is pi over 3. So another possibility is to have this as 5 times the cosine. And now we're going to get a phase shift, so it'll be 3 times t minus pi over 3 plus 2. And again, one thing you should consider doing is using your graphing calculator and checking this out. We'll talk about that a little bit in the next slide. So here's what I have, although you can see right now I'm using x for the independent variable rather than t. No big deal. But you can see I've written the two equations that we have for this, two of the equations that we have for this sinusoid, and we've actually developed a third and we could develop even more. But the great thing now is that we have the technology to check our work. We can enter either of these equations or any other one into our calculator and try to duplicate this graph. And again, if I were doing this, I would use an x min of 0 and x max of pi and try to duplicate this as closely as possible. I might then use an x scale of pi over 6 and y min of minus 4 of y max of 8 and a y scale probably of 2 just so I don't get too many tick marks on my graphing calculator. And of course it'll look a little different than that due to the graphics resolution. But there you have it. That is one way to determine an equation for a sinusoid given its graph. And please remember, the important thing is there's not just one correct answer. There's many, many possibilities for a correct answer here but all you have to do is check your answer on your calculator. Thanks and good luck.