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We're asked to determine the amplitude and the period of y equals negative 1/2 cosine of 3x. So the first thing we have to ask ourselves is, what does amplitude even refer to? Well the amplitude of a periodic function is just half the difference between the minimum and maximum values it takes on. So if I were to draw a periodic function like this, and it would just go back and forth between two-- let me draw it a little bit neater-- it goes back and forth between two values like that. So between that value and that value. You take the difference between the two, and half of that is the amplitude. Another way of thinking about the amplitude is how much does it sway from its middle position. Right over here, we have y equals negative 1/2 cosine of 3x. So what is going to be the amplitude of this? Well, the easy way to think about it is just what is multiplying the cosine function. And you could do the same thing if it was a sine function. We have negative 1/2 multiplying it. So the amplitude in this situation is going to be the absolute value of negative 1/2, which is equal to 1/2. And you might say, well, why do I not care about the sign? Why do I take the absolute value of it? Well, the negative just flips the function around. It's not going to change how much it sways between its minimum and maximum position. The other thing is, well, how is it just simply the absolute value of this thing? And to realize the y, you just have to remember that a cosine function or a sine function varies between positive 1 and negative 1, if it's just a simple function. So this is just multiplying that positive 1 or negative 1. And so if normally the amplitude, if you didn't have any coefficient here, if the coefficient was positive or negative 1, the amplitude would just be 1. Now, you're changing it or you're multiplying it by this amount. So the amplitude is 1/2. Now let's think about the period. So the first thing I want to ask you is, what does the period of a cyclical function-- or periodic function, I should say-- what does the period of a periodic function even refer to? Well let me draw some axes on this function right over here. Let's say that this right over here is the y-axis. That's the y-axis. And let's just say, for the sake of argument, this is the x-axis right over here. So the period of a periodic function is the length of the smallest interval that contains exactly one copy of the repeating pattern of that periodic function. So what do they mean here? Well, what's repeating? So we go down and then up just like that. Then we go down and then we go up. So in this case, the length of the smallest interval that contains exactly one copy of the repeating pattern. This could be one of the smallest repeating patterns. And so this length between here and here would be one period. Then we could go between here and here is another period. And there's multiple-- this isn't the only pattern that you could pick. You could say, well, I'm going to define my pattern starting here going up and then going down like that. So you could say that's my smallest length. And then you would see that, OK, well, if you go in the negative direction, the next repeating version of that pattern is right over there. But either way you're going to get the same length that it takes to repeat that pattern. So given that, what is the period of this function right over here? Well, to figure out the period, we just take 2 pi and divide it by the absolute value of the coefficient right over here. So we divide it by the absolute value of 3, which is just 3. So we get 2 pi over 3. Now we need to think about why does this work? Well, if you think about just a traditional cosine function, a traditional cosine function or a traditional sine function, it has a period of 2 pi. If you think about the unit circle, 2 pi, if you start at 0, 2 pi radians later, you're back to where you started. 2 pi radians, another 2 pi, you're back to where you started. If you go in the negative direction, you go negative 2 pi, you're back to where you started. For any angle here, if you go 2 pi, you're back to where you were before. You go negative 2 pi, you're back to where you were before. So the periods for these are all 2 pi. And the reason why this makes sense is that this coefficient makes you get to 2 pi or negative-- in this case 2 pi-- it's going to make you get to 2 pi all that much faster. And so it gets-- your period is going to be a lower number. It takes less length. You're going to get to 2 pi three times as fast. Now you might say, well, why are you taking the absolute value here? Well, if this was a negative number, it would get you to negative 2 pi all that much faster. But either way, you're going to be completing one cycle. So with that out of the way, let's visualize these two things. Let's actually draw negative 1/2 cosine of 3x. So let me draw my axes here. My best attempt. So this is my y-axis. This is my x-axis. And then let me draw some-- So this is 0 right over here. x is equal to 0. And let me draw x is equal to positive 1/2. I'll draw it right over here. So x is equal to positive 1/2. And we haven't shifted this function up or down any. Then, if we wanted to, we could add a constant out here, outside of the cosine function. But this is positive 1/2, or we could just write that as 1/2. And then down here, let's say that this is negative 1/2. And so let me draw that bound. I'm just drawing these dotted lines so it'll become easy for me to draw. And what happens when this is 0? Well cosine of 0 is 1. But we're going to multiply it by negative 1/2. So it's going to be negative 1/2 right over here. And then it's going to start going up. It can only go in that direction. It's bounded. It's going to start going up, then it'll come back down and then it will get back to that original point right over here. And the question is, what is this distance? What is this length? What is this length going to be? Well, we know what its period is. It's 2 pi over 3. It's going to get to this point three times as fast as a traditional cosine function. So this is going to be 2 pi over 3. And then if you give it another 2 pi over 3, it's going to get back to that same point again. So if you go another 2 pi over 3, so in this case, you've now gone 4 pi over 3, you've completed another cycle. So that length right over there is a period. And then you could also do the same thing in the negative direction. So this right over here would be negative, negative 2 pi over 3. And to visualize the amplitude, you see that it can go 1/2. Well, there's two ways to think about it. The difference between the maximum and the minimum point is 1. Half of that is 1/2. Or you could say that it's going 1/2 in magnitude, or it's swaying 1/2 away from its middle position in the positive or the negative direction.