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Course: Algebra (all content)>Unit 14

Lesson 8: Finding amplitude & midline of sinusoidal functions from their formulas

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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• 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?
• 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.
• 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?
• Based on my understanding. Think of x-axis as "steps required to complete the cycle" instead of angle or time.

Imagine there are 2 best friends, L and S. Mr L has 2 times longer leg than Mr S. At one evening, they decided to walk together from point A to point B and the total distance is 2pi.
Both of them walk one step at a time. As expected, Mr L reached point B with less steps than Mr S.

Now back to unit circle, for y=sin(1X), the total steps required to finish one cycle is shown as below:
total distance / total steps = distance per steps
total steps = total distance / distance per steps
total steps = 2pi / 1
total steps = 2pi
In this case of unit circle, the total distance is the circumference of the circle
So, if he walk ONE step at a time, the total number of step to finish one cycle is 2pi.

for y=sin(2X), the total steps required to finish one cycle is shown as below:
total steps = total distance / distance per steps
total steps = 2pi / 2
total steps = pi
So, if he walk TWO steps at a time, the total number of step to finish one cycle is pi.

Hope it make sense to you ^_^
• what does sinusoidal mean?
• It means to change according to a sine curve or sine wave
• Why is the amplitude always the number the trig function is multiplied with?
• 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.
• What would the amplitude of a tangent function be?
• The amplitude is simply how ample the function is. That is, how much it deviates from zero. Usually we don't talk about the amplitude of the tangent function, since it is infinitely ample.
• 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!
• 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.
• 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.
• 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.
• I don't know about anyone else, but I was initially really confused about why the amplitude was just the absolute value of the coefficient of the sinusoidal function (-1/2 in this example). After thinking about it for a while, I've realized why this is the case. When you're considering the sine or cosine of some angle, you are sort of "bounded" to the unit circle in that the greatest possible value of cos(x) and sin(x) is 1 and the least possible value of the two functions is -1. So when you multiply the value of the function by some coefficient and consider the amplitude, which as Sal explains is 1/2(ymax - ymin)*, you are essentially multiplying 1 (for the max) and -1 (for the min) by that coefficient. In other words, the amplitude of a function is *1/2(+coefficient-(-coefficient))*, which simplifies just to *coefficient. I hope this explanation helps -- even if it was pretty clunky (and I'm aware that this isn't really a question as much as an answer to a question that I imagine some people might have).