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Amplitude & period of sinusoidal functions from equation

Sal finds the amplitude and the period of y=-0.5cos(3x). Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user garyluo247
    what does speeding up the rotation by 3 times mean? if one period is 2pi how can it speed up the rotation without completing 3 circles? I'm like stuck in this thought, can anyone help me understand what is a period? Every time when the period is 2 or more, I think of it completing 2pi twice, but apparently it does not. How can you complete 2pi 3 times faster without going around 2pi 3 times? I don't know if I'm explaining my struggle right, but that is pretty much it, can anyone help me? Does going 3 times faster literally means condensing the graph into three 2pis? Which means like doing the same length of 2pi 3 times in one circle?
    (63 votes)
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    • piceratops ultimate style avatar for user Ed Miller
      The x-axis shows the measure of an angle. We know y=cos(x) completes a full cycle or period for every change of 2π radians along the x-axis, and as a consequence cos(2π) = cos(0). y=cos(2x) completes a full cycle for every change of π radians along the x-axis, and when x = π, cos(2x) = cos(2 * π) = cos(0). So, for a given change in x, cos(2x) completes more cycles than cos(x).

      I would say you are right to think of this as meaning that cos(2x) completes 2π twice over the interval that cos(x) completes only one cycle, but wrong to say that "one period is 2π". One period of cos(x) is 2π, but one period of cos(2x) is π. In other words, "period" is descriptive of a specific function, not of whatever function you perceive to be the "underlying function". In some sense, cos(2x) does behave very like cos(x), but its period is different, judging by our definition of period, which is the change in x over one complete cycle of a function.
      (8 votes)
  • leaf green style avatar for user QS
    Can someone please illustrate what is happening with the unit circle for the equation in this video?

    I think Sal and the people below did an excellent job explaining the "periodic" concept verbally and mathematically, but what is happening on the unite circle/cartesian graph? It makes sense visually on a sinusoid graph (x axis in units of Pi), but not the unit circle/Cartesian graph as theta goes in circles.

    The cos(x) graph repeats because it completes a cycle(2pi). How can a value repeat it'self 1/3 of the rotation, without completing a cycle?

    I've looked online at other sources, and everyone is more than happy describing what's happening on the unit circle when y=sin(x), or cos(x) with a cycle of 2pi. But when looking at an equation of y=sin (3x) they completely avoid describing it in terms of the unit circle graph, and only allude to the sinusoid graph. Can someone please illustrate what is happening with the unit circle for the equation in this video?
    (25 votes)
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    • leaf green style avatar for user QS
      I've done a lot of work and understand the math and sinusoid graph throughly, but there's something fundamental/visual that's off. I spoke to a friend, and he says the issue is, I'm thinking of the unit circle as a function, and I should not interpret something like y=cos(1/3x), which has a period of 6pi back into the unit circle. I have to wrap my head around this a little more, sometimes the x axis sounds like a function of time, and some times it's a representation of angles. That's the best way I can explain it right now,but feel free to give me more input if you can understand this issue better..
      (8 votes)
  • spunky sam blue style avatar for user dlillard07
    what does sinusoidal mean?
    (10 votes)
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  • mr pants teal style avatar for user Richard
    What would the amplitude of a tangent function be?
    (6 votes)
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  • primosaur ultimate style avatar for user thaingocnguyendinh
    Why is the amplitude always the number the trig function is multiplied with?
    (8 votes)
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    • leaf orange style avatar for user Z.K.
      Both the normal sine and cosine functions sway between 1 and -1. When you add a coefficient, you are multiplying that positive one or negative one by the coefficient, giving you a new amplitude equal to the absolute value of your coefficient.

      Example: y= 2 sin (x)
      The normal sine function is bound between 1 and -1, so the 2 coefficient multiplies those values by two, giving us a function with an amplitude of 2.
      (7 votes)
  • piceratops ultimate style avatar for user jerryma2004
    Could someone please tell me how does the number before x make the rotation faster?
    E.g. in
    y = -3sine(4x)
    How does the 4 make x rotate around the unit circle four times faster? What I thought was that surely the 4 only increases x by four times, then it will be a different angle so the period is still going to be 2pi.

    Please could someone help me. I'm really confused.

    Thank you very much!
    (4 votes)
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    • starky ultimate style avatar for user Andras Viczian
      Normally the period of sin(x) would be 2pi long. Lets compare sin(4x) and sin(x):
      At x=1
      sin(x) = sin(1) , sin(4x) = sin(4)
      At x=2
      sin(x) = sin(2) , sin(4x) = sin(8)

      As you can see in sin(x) x goes up by plus 1 every time (0, 1, 2, 3, 4,...).
      On the other hand sin(4x) x goes up by plus 4 every time( 0, 4, 8, 12, 16,...).

      In sin(4x) the gap between the inputs are 4 times as large (4/1=4, 8/2=4,...) so it technically jumps over 1/4 of sin(x) (1/4 = 2/8 = 3/12,...).

      I'm sorry if my answer is confusing, but I hope it helps.
      (10 votes)
  • leafers ultimate style avatar for user fisherlam→ΣβФ
    So amplitude refers to the highest point the graph of the sine/cosine function reaches on the y axis while period is the length on the x axis in one cycle, am I right? Thanks.
    (5 votes)
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    • hopper cool style avatar for user Christi
      You are partially correct: the period is the length on the x axis in one cycle. However, the amplitude does not refer to the highest point on the graph, or the distance from the highest point to the x axis. The amplitude is 1/2 the distance from the lowest point to the highest point, or the distance from the midline to either the highest or lowest point. This is an important distinction when the trig function is shifted up or down.
      (5 votes)
  • aqualine ultimate style avatar for user Rebeca Flores
    How do I find amplitude and period for a tangent function?
    (2 votes)
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  • primosaur ultimate style avatar for user JeremiahJTReed
    Is there a "proof of sinusoidal functions" video? I see the one where we graph from a unit circle, but is there a video explaining how each of these parts of the sinusoidal function cause the manipulations the way they do?
    (2 votes)
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    • male robot hal style avatar for user Madd Sam
      Everything Vader said is correct. Just want to add that I had an insight as to what's really happening when I realized what b is doing. If you write a function for temperature over a 24 hr period, then 2π/24 really just breaks the unit circle into 24 equal parts, and t chooses one of those parts. Once I figured that out, the rest of it just began to fall into place. Hope that helps.
      (2 votes)
  • piceratops ultimate style avatar for user meazza718
    At , I don't understand why we start at y=-1/2 when x=0. When I read Y=-1/2cos3X, I get Y= 1/2(times)cos3(times)X, wich should be Y=0 when X=0. Where do I get it wrong ?
    (1 vote)
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    • aqualine ultimate style avatar for user Oreo
      A point on the Y axis is given by running a value for the variable x through the equation Y=1/2 * cos(3*x)

      A careful point to note here. Lets look at a cosine function all by itself. it is written as cos(x). whatever is in the parenthesis is called the argument of the function. Now recall the order of operations. Parenthesis, then exponents, then multiplication and division, then addition and subtraction. You must clear parenthesis first. So for a basic cosine function of cos(x) you would do whatever operations exist inside of the parentheses.

      Returning to our original equation and just looking at the cosine function for now, we see that we have cos(3*x). The variable x here can represent any point on the x axis, lets choose 0.
      We begin by substituting 0 for x in the cosine function.
      We now have cos(3*0). This evaluates to cos(0) because 3*0 =0.
      Now we have the equation Y=-1/2 * cos(0).
      Lets continue evaluating our cosine function, we aren't done yet!What is the cosine of 0? it is 1.
      So now we have Y= -1/2 * 1.
      Continuing to solve and simplify and you see that -1/2 * 1 = -1/2.
      So we now have Y=-1/2.
      Thus for the coordinate 0 on the x-axis, the y coordinate is -1/2.

      Its important to do the order of operations, especially with trig functions!

      Good luck!
      (5 votes)

Video transcript

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.