In this video, we're going to focus on how to graph sine and cosine functions, particularly with horizontal phase shifts. But we're going to start with some basic structures, some basic equations, and then we're going to make it progressively harder. So let's say if we want to graph, let's start with the very basic.
y equals sine x. The amplitude is 1 and for this graph the period is 2 pi. Sine starts at the center and goes up, back to the middle, down, and then back to the middle. And here's the period, 2 pi. The amplitude varies from 1 and negative 1. Now let's go over the four types.
So let's say if we want to graph negative sine x. Negative sine x is going to look like this. Instead of going up, it goes down first. So negative sine goes down, positive sine goes up. Now let's say if we want to graph, and by the way this is going to be 2 pi, and the amplitude is 1. Let's say if we wish to graph cosine x.
Cosine doesn't start at the center. Positive cosine starts at the top. Then it goes down, and then back up. And then you can keep continuing the graph if you want.
Same thing for this one, you can continue it. Cosine also varies between 1 and negative 1. And a period is from here, peak to peak. So this would be 2 pi at that point.
That's one full cycle. Negative cosine, if you wish to graph it, it starts over here instead and goes like this. That's negative cosine. It starts at the bottom instead of the top. So those are just some generic graphs that you just have to know.
So now let's say if we want to graph y equals 2 sine, let's say, 2 sine x. Let's just keep it simple. So we still have the regular sine wave, the period is 2 pi, but the amplitude now is 2. So it's going to vary from 2 and negative 2 instead of 1 and negative 1. But now let's say if we want to graph y equals 2, actually, just sine 2x.
By the way, this is known as the vertical stretch. The graph will stretch two units. It doubled in the y direction. If we have 2x, it's going to be a horizontal shrink. It's going to vary between negative 1 and 1, but it's going to be much shorter.
The period is going to be pi. And the way you calculate that... Period is equal to 2 pi over b. b is the number in front of x, so it's 2 pi over 2. And so we get pi for the period. So it shrinks from 2 pi to pi.
The generic equation for a sine or cosine wave is y is equal to a sine bx plus c plus d. As we've talked about already, a is the amplitude. b helps you define the period. c tells you the presence of any horizontal or phase shift. And d is the vertical shift.
so let's try another example let's say if we wish to graph y is equal to negative sine of 1 half x so because it's a negative sign it's going to instead of going up it's going to go down Now, because we have a half, it's going to be a horizontal stretch by a factor of 2. If we calculate the period, it's 2 pi over b, and b is 1 half. 2 pi divided by 1 half is the same as 2 pi times 2, so the period has doubled into 4 pi. The amplitude is still 1, so that's going to be the same. so for the sake of practice let's try another example let's say if we want to graph 3 cosine 1 3rd X so the amplitude is 3 the period is 3 pi divided by 1 3rd so that's 3 pi times 3 that's 9 pi so we have a vertical stretch of 3 and a horizontal stretch of 3 as well at the same time so we know cosine starts at the top so let's say this is 3 and that's negative 3 and that's one cycle if we wish to continue it that's another cycle So from here to here is one cycle, so this is 9 pi.
And that's how you would graph 3 cosine 1 third x. So now let's introduce the vertical shift, and then we're going to introduce the phase shift. So now let's say if we have y equals sine x plus 2. The 2 is the vertical shift. So, what you should do is first plot 2, which is going to be your center line.
Your amplitude is 1, so you need to go up 1 from 2, which is 3, and down 1, which is 1. So, sine starts at the center. It's going to go up, back to the center, and then down, back to the center. And that's how you graph it.
So, let's say if you want to graph... 3 sine X plus 4 so the vertical shift is 4 so let's plot that first and the amplitude is 3 so 4 plus 3 is 7 and 4 minus 3 is 7 is 1 So we need to go up to 7 and stop at 1. Or the lowest we can go is 1. So plot the midline, which is where the vertical shift is. And let's make this negative 3 sine plus 4x.
It won't change these numbers, so that's going to be the same. But instead of going up, we're going to go down, back to the center, and then up. Okay, so let's say if we want to graph y is equal to 2 sine 4x minus 3. so let's start with a midline the vertical shift is negative 3 and amplitude is 2 so negative 3 plus 2 is negative 1 that's our maximum and negative 3 minus 2 is negative 5 that is our minimum And we have a positive sign graph. The period is 2 pi over b, or 2 pi over 4, which is pi over 2. So we're going to start here. And let's plot the period.
So let's say pi over 2 is somewhere over here. What we need is 4 points. Half of that is pi over 4, and half of that is pi over 8. And 3 times pi over 8 is 3 pi over 8. So those are the four important points that we need. So here's the first one. The positive side is going to go up, back to the middle, and then down, back to the center.
And so that's our graph. That's one cycle for it. Okay, let's try one with cosine.
So let's say y is negative 3 cosine 1 half x plus 5. So let's plot the midline first let's go up three to eight and down three to two Let's calculate the period. It's 2 pi over b, or 2 pi divided by half, so it's 4 pi. So let's say 4 pi is over here. Half of that is 2 pi, half of that is 1 pi. Times that by 3, you get 3 pi.
So those are the four important points that we need. Cosine starts at the bottom, which is at 2. Then it's going to go back to the center, and then to the top, back to the center, and at the bottom. So it's going to look like this.
So that's how we can graph that function. So now let's introduce the vertical shift. If y is equal to 2 sine x minus pi over 2 plus 3. Okay, so let's make the graph. We can see that the vertical shift is up 3, and the amplitude is 2. So 3 plus 2 is 5. and 3 minus 2 is 1 so let's draw the horizontal line here now our phase shift to find it set the inside equal to 0 and solve for X so it's going to be PI over 2 next let's calculate the period the period is 2 PI over B and the number in front of X. which there is no number, and it's an invisible 1, so it's 2 pi over 1, which is just 2 pi.
Now what you need to do is add 2 pi to where you start from, to get to your end point. So pi over 2 plus 2 pi. 2 pi is the same as 4 pi over 2. And 4 pi over 2 plus pi over 2 gives you 5 pi over 2. So this graph is going to end.
Let's extend it. Thank you. It's going to end at 5 pi over 2. Now what you need is 5 points.
You want to break it into 4 intervals. The midpoint between 1 pi over 2 and 5 pi over 2. Between 1 and 5 is 3. And between 1 and 3 is 2. 2 pi over 2 is the same as pi. Between 3 and 5 is 4. 4 pi over 2 is the same as 2 pi. If you reduce it. So now we have our 5 points.
So now you want to plot it. Sine starts at the center. Yeah.
And because it's positive 2, it's going to go up to 5. Then it's going to go back to the middle, down to 1, and then back to the middle. And so that's how you can graph one cycle. And if you want to, you can continue it if you want.
Let's try another example. So let's say if we wish to graph... y equals 3 cosine 1 half x plus pi minus 5. So I'm going to draw the graph like this. So the vertical shift is at negative 5. And the amplitude is 3. So, negative 5 plus 3, that's negative 2, that's my maxima. Negative 5 minus 3 is negative 8, that's my minimum.
By the way, if you want to find the range of this graph, as you can see, it's from low to high, negative 8. negative 2 for the last one that we did it varied from 1 to 5 so the range will be 1 to 5 the domain for sine graphs and cosine graphs is negative infinity to infinity because they can keep going on forever unless you restrict it So I just want to mention that in case you had questions on it. So we talked about the amplitude already. It's 3. We considered the vertical shift.
Now let's calculate the phase shift, where it starts. To find the phase shift, set the interval. side equal to 0 and solve for X so if we move pi to the other side a half X is equal to negative pi and if we multiply both sides by 2 X is equal to negative 2 pi which I should have extended the graph here so let me redraw case is negative 5 negative 2 negative 8 so here's negative 2 pi and here is our midline and I'm just going to rewrite the equation So now that we have the phase shift, this is where the graph is going to start. Let's calculate the period. The period is 2 pi over b, or just 2 pi divided by half.
And we're going to get 4 pi. Now what you should do is add your... starting point your phase shift to your period negative 2 pi plus 4 pi is Equal to 2 pi So that's where it's going to stop so over here so that's going to be just one cycle and we can continue it if we choose to. So, cosine starts at the top.
So, at negative 2 pi, negative 2, which is right there, and it's going to go back to the middle. Well, first, we got to break it into four intervals. The midpoint between negative 2 pi and 2 pi is 0, and between negative 2 pi and 0, it's negative pi, and between 0 and 2 pi, it's pi. So, then cosine is going to go back to the middle, which is going to be here, at negative pi, and at 0, it's going to go to all the... all the way to the bottom at negative 8, and that pi back to the center, and that 2 pi over it's going to be right there.
It's going to look like that. Now be careful, don't draw a V shape. Now let's try one more example. So let's say if y is negative 2 sine 1 third x minus pi over 2 plus 3. Okay, so most of the graph will be on the first quadrant, so I'm going to draw it like this. So let's start with the vertical shift.
It's at 3, and the amplitude is 2. So 3 plus 2 is 5, 3 minus 2 is 1. That means that the range is from 1 to 5. And the domain goes on from negative infinity to infinity. so this is the estimate line and next we need to find the face shift so set the inside equal to 0 and solve for X so 1 3rd X is equal to pi over 2 and if we multiply both sides by 3 X is 3 pi over 2 so that's where the graph is going to start so now we need to calculate the period the period is 2 pi divided by 1 3 which is the same as 2 pi times 3 so that's 6 pi so we need to add 3 pi over 2 the phase shift plus the period which is 6 pi 6 pi if we get common denominators it's the same as 12 pi over 2 so 3 plus 12 was 15 so we get 15 pi over 2 I'm going to extend this graph so here's 15 pi over 2 the midpoint between 3 and 15 is let's see 3 plus 15 is 18 if you divide it by 2 you get 9 so it's 9 pi over 2 the midpoint between 3 and 9 is 6. 3 plus 9 is 12, divided by 2 is 6. So we get 6 pi over 2, which reduces to 3 pi. And the midpoint between 9 and 15. And this doesn't look like a 15, but it's supposed to be 15. The midpoint between 9 and 15 is 12. So 12 pi over 2 reduces to 6 pi. So now we have our 5 points. And so now we can graph it.
Sine starts at the center. And because it's a negative sine, it's going to go down before it goes up. then it's back at the center then it goes to the top at five and then back to the middle now let's say if you only want to draw one cycle notice that your domain is restricted from 3 pi over 2 to 15 pi over 2 based on the graph that we drew because it starts here and it stops here. However, if you wish to continue the graph, it can go on forever.
And the domain, therefore, can be negative infinity to infinity. So it really depends on whether if you choose to restrict your graph or if you choose to expand it. So just know that for sine waves, typically the domain is this, negative infinity to infinity, unless...
you pretty strict and his range so that's it for this video that's how you can graph sine and cosine functions I'm when you get an aptitude vertical shift I face shift and features give me a question you not find everything so that's it and I thanks for watching