 Hello. In this screencast, we're going to do another example of using the graph of a sinusoid to determine an equation for that sinusoid. Just like before, in our previous example, we can choose whether to use a sine or a cosine. And in both cases, we will get the same amplitude, the same period, and the same vertical shift. In other words, we will get the same values of a, b, and d. And the only difference between the two will be in the phase shift. And that, again, will be dependent on whether we use a sine or a cosine. And as before, too, the important things we're going to focus on, to find many of these values, are a high point and the very next low point. We could, if possible, find the two high points because that distance between those two would be equal to one period. And again, recall that basically the difference between the x coordinates of a high point and the next low point is a half a period. And again, we will also use the y coordinates of the high point and the low point to determine the amplitude and the vertical shift. So here's our example. So it might be a good idea for a little practice just to stop the screencast right now, determine the coordinates of the two high points shown and the low point between the two of them, and then see if you can determine the values of a, b, and d from those coordinates. Okay, so here we go. The first high point that we see right there, again, using the grid that we have on there, we can see that the coordinate, coordinates for that point are 0.45. And if we look at the very next high point, we see that that coordinate, that point has coordinates 2.4. From that, we can now quickly get the period is equal to two. Okay, and again, once we have the value of the period, we can determine the value of b. And we can use the formula that 2 pi over b is equal to the period. We get that. Solving that, if we multiply both sides by b, we get 2 pi equals 2b, or b equals pi. One thing we want to work with next is this low point. And although this point here, this low point, doesn't lie right on our grid lines, we can now use the fact that we have a period of 2. And we can say if we start from 0.4 and add half a period, which in this case would be 1, comes out at 1.4. Now we probably could have seen that from the graph, but it's nice to have a little more certainty to it. And we see we get a low point coordinates, 1.4, negative 1. And now we can use that to determine the values of a and d. And again, what we do for that is the amplitude is going to be, actually, two times the amplitude will be the difference of this y-coordinate and this y-coordinate. So it's 5 subtract negative 1, or 6. And so we get an amplitude of 3, which will be our value for a. And again, we now can go from this high point and go down the value of the amplitude to there and get the value for d. And we can see that's 5 minus 3 equal to 2. And so we have d equals 2. So sometimes it's nice to draw this line, try to do it carefully in there, and it helps us see the sine or the cosine curve. And in this case, I think we'll start with getting a cosine curve because this high point, then again, is nice to work with because in some sense that's our basic cosine curve starts at a high point. And we can see then that the phase shift is 0.4. So we get a value of c is 0.4. So for a cosine, we have y equals a, which is 3, cosine b, which is pi, times x minus the phase shift, 0.4 plus 2. And the next thing we'll try to do is determine a sine equation for that. And again, we will use the same values of a, b, and d. So here we have that. Just as a quick review, a is 3, b is pi, and d is equal to 2. And what we're actually going to do here is kind of go back off the graph a little bit. There's our line for the vertical shift of 2. And what we're going to do is kind of extend this curve back because basically we'll consider that to be kind of the starting point for our sine function. And what we're going to do is use this high point, which remember has an x coordinate of 0.4. And to get back to the next intercept, we go back a quarter of a period. So I'll remember again, our period was equal to 2. So a quarter of a period is 2 divided by 4, or since we're working in decimals, 0.5. And with that then, we now has a phase shift of negative 0.5. Oh, I'm sorry. We should, again, maybe go back from 0.4 minus 0.5. And that comes out to be negative 0.1. And that's our phase shift. So the value for c, remember, is going to be negative 0.1. So we have to be a little careful with negative signs when we substitute that in. So our equation becomes y equals 3 sine of pi times x minus a negative 0.1 plus 2. Or again, we would usually write that as 3 times sine of pi times x plus 0.1 plus 2. One thing you might consider if you don't like the use of that negative sign as you could actually go out to this point here and see that that has an x coordinate of 1.9. That would be a whole period from the negative 0.1. And remember, the period is 2. So if I go from negative 0.1 to 2 units, I get at 1.9. And that would just change a phase shift. And what we would end up with is y equals 3 times a sine of pi times x minus 1.9 plus 2. And either one of these should produce that graph for us. The last slide is just the first one. But again, whatever ones we choose to use, we should be able to now reproduce this graph on our graphing calculator. And again, I would set this up pretty much as it's shown there with an x min of 0 and x max of 4 and scale the x axis, perhaps at 0.4. That will produce 10 tick marks on the x axis. And y min of negative 2 of y max of 6. And I might scale that y scale of 1. And it's just so wonderful having the graphing calculator at our disposal to check our work. And I strongly encourage you to do so. That's it for now.