Now let's talk about graphing trigonometric functions. Let's start with the sine function, sine x. Sine x is basically a sinusoidal function.
It's a sine wave. And that's how it looks like. At least that's one period. This ends at 2 pi.
That's one cycle of the wave. Now let's say if you put a negative in front of the sine function. it's going to flip over the x-axis.
So instead of going up initially, it's going to start from the origin, it's going to go down, and then back up, and then back down. So that's the shape of sine and negative sine. Now keep in mind, this wave keeps on going forever in both directions.
But for the course of this lesson, I'm going to focus on graphing one period, which is basically one cycle of the wave. Now what about the graphs of cosine x and negative cosine x? Cosine starts at the top, whereas sine starts at the center.
So that's one period of the cosine wave. Let me do that a little bit better. But it can continue going on forever.
Negative cosine starts at the bottom, it goes up to the middle, and then goes up, and then back down. So that's the graph of one period of negative cosine. So this is one cycle.
Now let's go back to the sine graph. Let's draw two cycles of this graph. So one cycle, you need to break it up into four useful points. One cycle is 2 pi. You want to break that up into four points, such as pi over 2, pi, and 3 pi over 2. Now, if we want another period, let's add 2 pi to it.
So, we want to go to 4 pi. In between 2 pi and 4 pi is 3 pi. In between 2 pi and 3 pi, it's 5 pi over 2. You add these two, then divide by 2. If we add 3 pi and 4 pi, that's 7 pi, and then divide it by 2, we get 7 over 2. Now sine starts at the center, and then it's going to go up, back to the middle, down, back to the middle.
middle. So that's one cycle of the sine wave. Then it's going to go back up, back to the middle, down, and then back to the middle. That's why it's helpful to plot the points first before put in everything else.
If you break up each cycle into five key points, which equates to four intervals, it's going to be easier to graph the sine wave. Let's do the same for cosine. Let's graph two periods of the cosine wave. So one period is going to be 2 pi, two periods, 4 pi.
For each period or each cycle, break it up into five points, which is four intervals. The first point, by the way, is the origin. It's 0. So these points will be the same as the graph assigned. Now we know that cosine starts at the top, it's going to go back to the middle, and then to the bottom, back to the middle, and then to the top, and it's going to alternate.
So it's going to look something like this. So that's one cycle. And here's the second cycle. So that's two cycles of the cosine wave. Now let's talk about the amplitude of the sine wave.
The generic formula is A sine Bx plus C plus D. Now we're going to focus on A. A, the number in front of sine, is the amplitude.
So in this case, the amplitude is equal to 1. So when you graph a sine wave, and you plot your four points of interest for one full cycle, The amplitude is going to be 1, so it's going to vary from 1 to negative 1. So we know sine starts at the center, it's going to go to the top, back to the middle, and then to the bottom, and then back to the middle. So it's going to look like this. And we know the period is 2 pi.
Now what if we wanted to graph 2 sine x? So if we increase the amplitude, This graph is going to stretch vertically. So it's going to vary from 2 to negative 2. By the way, this is the amplitude. It's the distance between the midline of the sine wave and the highest point.
Now let's plot one period. So this is going to be 2 pi. So once again, sine is going to start at the middle, then it's going to go up, back to the middle, and then down, and then back to the middle. So it's going to look like that.
And so the amplitude tells you how much it's going to stretch or compress vertically. Consider the equation y is equal to negative 3 cosine x. What is the amplitude of this function? The amplitude is always a positive number, so you ignore the negative sign, and it's going to be 3. The amplitude is the absolute value of a, the number in front of cosine. Now, let's go ahead and graph it.
Let's plot one period. So let's break it up into five points or four intervals Now, the amplitude is 3, so we need to vary the sine graph, or rather the cosine graph, from negative 3 to 3. So, cosine typically starts at the top, but we have negative cosine, so it's going to start from the bottom. Then it's going to go to the middle, to the top, back to the middle, and then back to the bottom. So that's how we can graph one cycle of negative 3 cosine x. Now keep in mind, this graph can keep on going forever in both directions.
So let's say if we want to write the domain and range of this cosine graph. The domain for sine and cosine graphs will always be the same. are real numbers. The range is based on the amplitude. The lowest y value is negative 3, the highest y value is 3. So that's how you can write the domain and range of this particular cosine graph.
Now let's talk about finding the period. So given this sine function, a sine bx, we know a represents the amplitude. Now b is not the period itself, but it's used to find the period.
The period is 2 pi divided by b. So in the case of sin x, b was equal to 1. So the period was 2 pi divided by 1. Now let's go ahead and graph these two functions. Sin x and sin 2x. Let's see what effect b has on the graph.
Now we know the general shape of sine x. It has a period of 2 pi, and for the most part, it looks like this. Now, if b is equal to 2, in this example, the period is going to be 2 pi divided by b, so the period is pi. So therefore, it's going to do one full cycle in less time, so to speak. So what happens is the graph, it shrinks horizontally.
So one full cycle occurs in 1 pi. Two cycles occur in 2 pi. Here's another example. Go ahead and graph this function, 2 sine 1 half x. So first we need to find the amplitude.
The amplitude is the number in front of sine, that's 2. The period is 2 pi over b, where b is the number in front of x. So in this case it's 1 half. 2 pi divided by 1 half is 4 pi. So this one is going to stretch horizontally. The amplitude is 2, and the period is 4 pi.
But we need to break it up into 4 intervals. So that's 1 pi, 2 pi, 3 pi, and 4 pi. Sine starts at the center, then it goes up, back to the middle, down, and then back to the middle. So we're going to have a graph that looks like that. So if you have a fraction, what's going to happen is it's going to stretch horizontally.
Let's try another example. Let's graph 4 cosine pi x. So first, identify the amplitude and the period. The amplitude is simply 4 in this example, and the period is 2 pi over b. In this case, b is the number in front of x, so b is pi.
2 pi divided by pi is 2. So that's the period in this example. So let's go ahead and make a graph. So the amplitude is 4. So it's going to vary from 4 and negative 4. The period is 2. So 2 should be about here.
And we need to break it into four parts. So this is 1, 1 half, and then between 1 and 2, you add them up. 1 plus 2 is 3. Then you average it, or you divide it by 2. So it's 3 over 2. So those are the four points of interest.
Cosine starts at the top. Then it's going to go to the middle, and then back to the bottom, to the middle, and to the top. So we're going to have a graph that looks like this. That's one cycle.
And if we wish to extend it to draw another cycle, this is going to be 3. Next one is 2.5, or 5 over 2. And then 3 plus 4 is 7, but then divided by 2. So 3.5 is 7 over 2. The next point is going to be at the middle, and then back to the bottom, back to the middle, and then to the top. And that's it. So that's how you can graph 4 cosine pi x. So when you find your period, make sure you put that first on the x-axis, and then break it into 4 intervals.
Now what is the domain and range of this function? As you recall, the domain for any sine or cosine wave is R-row numbers. The range is from negative 4 to 4. It's from the lowest y-value to the highest y-value. Now let's talk about what to do when there's a vertical shift. Let's say if we wish to graph sine x plus 3. So the vertical shift is 3. The amplitude is 1. So what you want to do first is you want to plot the vertical shift.
So at 3, I'm going to draw a horizontal line. That's going to be the new center of the graph. The amplitude is 1, so sine is going to vary 1 unit higher than the midline and 1 unit lower than it.
So it's going to vary between 2 and 4. Now we're still going to plot just one period. So let's write our four key points. Sine starts at the top, and then it goes to the middle. Actually, I take that back.
Sine starts at the middle, and then it goes to the top, and then back to the middle, to the bottom, and then back to the middle. So this would be one sine wave. So that's how you can graph sine x plus 3. Let's try another example. Let's graph two periods of 2 cosine x.
Minus 1. So this is going to be 1 cycle. And 2 cycle. But let's start with the first cycle.
So the midline is at negative 1. Now the amplitude is 2. So we got to go up two units and down two units. Now cosine will start at the top. And then it's going to go to the middle, back to the bottom, and vice versa.
Now we need to plot one more cycle. So this is pi, and this is 3pi. So it's going to go back to the middle, and then to the bottom, back to the middle, and to the top.
So that's how we can graph two cosine periods. Now what is the range for this graph? Notice the lowest y-value is at negative 3, but the highest is at 1. So the range is from negative 3 to 1. Let's go ahead and graph this one.
Negative 3 sine x plus 4. So feel free to pause the video. Actually, let's also, let's change it a bit. Let's make it 1 third x plus 4. The majority of the graph will be above the x-axis.
So let's draw the center line at 4 first. The amplitude is 3, so we're going to have to go up 3, 4 plus 3 is 7, and then down 3, starting from 4, 4 minus 3 is 1. So the range is going to be from 1 to 7. Now let's find the period. We know the period is 2 pi divided by b, and b is 1 third.
So it's 2 pi divided by 1 third, so it's equal to 6 pi. And let's break it into four points. Half of 6 pi is 3 pi, half of 3 pi is 3 pi over 2. And if you multiply this number by 3, it will give us this point, which is 9 pi over 2. Now we know that sine starts at the center.
Positive sine will go up initially, but negative sine will go down. And then it's going to go back to the middle, and then to the top at 7, and then back to the middle. So that's how you can plot negative 3 sine 1 third x plus 4. Now let's talk about how to graph this function.
Sine x minus pi divided by 2. How can we do so? So considering the generic formula, a sine bx plus c plus d. Anytime there's a c value, there's a phase shift, which means that the graph is going to shift either to the right or to the left.
And so you want to find the phase shift because sine won't start at the origin in this case. So to find the phase shift, shift set the inside equal to 0 and solve for X so when you set B X plus C equals 0 X is going to equal negative C divided by B and this is your face if that's where it starts on the x-axis So let's set x minus pi over 2 equal to 0. So we can see x is at pi over 2. So that's where the sine wave is going to start. Now let's go ahead and graph it.
The amplitude is 1, and the period is 2 pi over 1, so it's 2 pi. But first, plot pi over 2, because that's where the phase shift is. And then what you want to do is add one period to the phase shift.
So you're adding 2 pi to pi over 2. 2 pi is the same as 4 pi over 2. So this will give you 5 pi over 2. So this is going to be 3 pi over 2, and you want to break it into 5 key points. This is 1 pi over 2, and between 1 and 3 is 2. 2 pi over 2 is pi. And between 3 pi over 2 and 5 pi over 2, we have 4 pi over 2, which reduces to 2 pi.
Now the amplitude is 1, so it's going to vary from 1 and negative 1. Now sine starts at the middle, but we're not going to start at the origin. In this example, we're going to start at the phase shift, which is pi over 2. Positive sine. it's going to go up.
The negative sign is going to go down first. So negative sign will be like this. Positive sign will have that shape.
And then at 2 pi, it's going to have a y value of negative 1. And at 5 pi over 2, it's going to be back on the x-axis. So that's how you can plot this particular sine wave with a phase shift. Now let's try another example.
Let's say if we want to plot 2, sine x minus pi over 4 plus 3. So we have a vertical shift of 3, an amplitude of 2. The number in front of x is 1, so 2 pi over 1 is 2 pi. The period is still 2 pi. But we do have a phase shift.
So if we set the inside equal to 0, the phase shift is positive pi over 4. The majority of the graph will be above the x-axis. So we're going to plot it up there. So let's plot the midline first, or the centerline, which is at 3. The amplitude is 2, so we need to travel 2 units above the centerline, which will take us to 5. 3 plus 2 is 5, and then 2 units down.
3 minus 2 is 1. So the graph is going to vary from 1 to 5, and that's the range of this sine function. Now the phase shift is going to start at pi over 4. That's where the sine wave is going to start. And if we add one period to that, the period is 2 pi. 2 pi over 1 is the same as 8 pi over 4. We need to get common denominators. So if we add these two numbers, this will give us 9 pi over 4. So that's where the first period will end.
The midpoint between 1 and 9 is 5. And the midpoint between 1 and 5 is 3. And between 5 and 9 is 7. Now we can graph it. So let's start with the phase shift. The sine is going to start at the middle. And then it's going to go up, back to the middle, and then down, and then back to the middle.
So this is the graph of just one period.