 This screencast will be another another one dealing with a sinusoid. In particular, we're going to be discussing the phase shift for a sinusoid. In the process of doing this, we will, of course, mention things like amplitude and period, but the main focus will be on the phase shift for a sinusoid and to some extent how to determine the phase shift given the equation of a sinusoid. Here's what we what we already know and this is called general form for a sinusoid. It might be just as appropriate to call it the standard form for a sinusoid. That really doesn't matter. The key here is that when we have this expression written in exactly this form, notice it's really the same for both sine and cosine. We will be able to read the amplitude, the period, the phase shift, and the vertical shift. This is assuming, of course, that A is positive and B is positive. C or D could be negative or positive. But when we have that, we can see we can read the various values. The period we have to calculate, but the amplitude, the phase shift, and the vertical shift can be read directly from those forms. The problem sometimes is our calculator or computer software, if we ask it for a sinusoid function in a certain situation, it may not give us the sinusoid in this form. And we need it in exactly this form in order to read these quantities from the expression. So if it's not in that form, we have to rewrite it so that it is. So for example, here's an example we're going to look at. This is, of course, a sine, but the same principle will be true, will be a whole if we also would have a cosine. And again, you can see that the difficulty here is this part inside the parentheses. It's not in this form. Okay, we can we can still look at this and say, okay, in this example, A equals 5, B, what's multiplying x will give us the B, that's 2. In this case, the vertical shift or D is equal to 0. What we don't see immediately is what the value of C is. And to get that, we need to write it in one of our standard forms. The key here is the expression 2x plus 3. What we want to do is rewrite that expression in this form here. And what we want to really do is factor out the 2. Well, that can look a little funny because how do we factor 2 out of pi over 3? Well, the idea, if you want to look at it, is you can always rewrite this fraction by multiplying both numerator and denominator by 2 and you can write pi over 3 as 2 times pi over 6. So if we do that, we can factor out a 2 and we'd get a result something like that. And again, we see one other issue that's come up here is this this has got an addition sign where our general form here requires a subtraction sign. So what we have to do is rewrite this at pi over 6 in a form of a subtraction. In order to do that, we use a property that says, okay, if we have a number q, we can always rewrite it in that form. And so what we do is take that x plus pi over 6 and write it as a subtraction and what we end up subtracting is minus pi over 6. And now we have this written in exactly the form b times x minus c. And this will allow us to determine the phase shift for this sine-us weight. So what we have right now then is y equals 5 times sine of 2 times x minus subtract minus pi over 6. And now we have exactly the right form of a times sine of b times x minus c. And we can see the amplitude is 5. The period is 2 pi over 2, or is equal to pi. And the phase shift is negative pi over 6. So we can see that what would happen is the sine curve would shift to the left, pi over 6 units. So there we've determined these that we want. Here's the summary of the work that we have done so far. We've got the original form of the expression. And this third form down here is the one that we really used to determine amplitude period phase shift and vertical shift. What we want to do now, and what we can do and also is check this work using our calculator. And it's nice to be able to then sketch a complete cycle of this sine-us weight. In other words, at least one period. In doing this, what I would really like to be able to do is show the phase shift. So notice that minus pi over 6, the phase shift, the period is pi. So if I add those two things, and we'll put some of the steps in here, we can see we write pi as 6 pi over 6 to get a common denominator. And so we get 5 pi over 6. So minimally, what I would like to do is have my x axis go from minus pi over 6 to 5 pi over 6. But it's also nice to see a little room on the sides. So what we're going to do is use x greater than or equal to minus pi over 3, which is basically negative 2 times pi over 6. And instead of going up to 5 pi over 6, we'll run it out to pi. And of course, the y axis, we will scale because of the amplitude of 6. And again, using that principle, we like to see a little extra room on the top and the bottom. So we'll use minus 6 less than or equal to y less than or equal to 6. So if you got your graphing calculator handy, you might set this up and see if you can get that graph using this viewing window. Okay, if you do so, you should see something like this. Now this was generated using geojabra. And although the x axis is not scaled for everything here, remember we went minus pi over 3, this here is x equal to minus pi over 6. So that's showing the phase shift of pi over 6 units to the left. And as we look at this graph and kind of sketch that out, we see one complete cycle out to here. And of course, we've already calculated this on the previous slide. That would be 5 pi over 6. Very nice way to be able to check your work on this. And I encourage you strongly to do that any time you're confronted with a problem such as this. That's it for now.