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## Algebra 2

### Course: Algebra 2 > Unit 11

Lesson 7: Transforming sinusoidal graphs- Amplitude & period of sinusoidal functions from equation
- Transforming sinusoidal graphs: vertical stretch & horizontal reflection
- Transforming sinusoidal graphs: vertical & horizontal stretches
- Amplitude of sinusoidal functions from equation
- Midline of sinusoidal functions from equation
- Period of sinusoidal functions from equation

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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.

## Want to join the conversation?

- 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?(67 votes)
- 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.(9 votes)

- 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?(26 votes)- 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 ^_^(14 votes)

- what does sinusoidal mean?(11 votes)
- It means to change according to a sine curve or sine wave(26 votes)

- What would the amplitude of a tangent function be?(6 votes)
- 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.(18 votes)

- Why is the amplitude always the number the trig function is multiplied with?(8 votes)
- 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.(10 votes)

- 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)- 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.(12 votes)

- 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)
- 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)

- 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).(5 votes) - How do I find amplitude and period for a tangent function?(2 votes)
- Tangents do not have amplitude, it would always be infinity. The period is found by dividing pi by the coefficient of x in the equation.(4 votes)

- Suppose I graph the equation on Desmos. Is there a way for Desmos itself to determine the amplitude and period from the equation?(2 votes)
- The things is you should to know what words amplitude and period mean as well as to calculate it.

However you can use following to help you perform calculations though I think it is bad idea.

https://www.desmos.com/calculator/kihjluohq9

Explanation of trig curve sketching

Question: How to sketch Cos(2x-pi/3), why is the phase shift not pi/3.

There two transformations going on, the horizontal stretch and the phase shift.

To stretch a function horizontally by factor of n the transformation is just f(x/n).

So let f(x) = cos(x)

=> f(x/(1/2)) = cos(x /(1/2) ) = cos(2x)

So the horizontal stretch is by factor of 1/2.

Since the horizontal stretch is affecting the phase shift pi/3 the actual phase shift is pi/6 to the right as the horizontal stretch is 1/2.

cos(2x-pi/3) = cos(2(x-pi/6))

Let say you now want to sketch cos(-2x+pi/3). Remember that cos theta is even function. A function is even if f(-x) = f(x). Substitute u = 2x-pi/3 =>

cos(-2x+pi/3) = cos(-u) = cos(u) = cos(2x-pi/3)

Similarly you can sketch sin(2x-pi/3) the same way just beware sin is an odd function not an even function. Note a function is odd if f(-x) = -f(x).

If you wanted to sketch an equation tan theta remember that period of tan theta is pi and tan is an odd function.

Basic notes

Note when equation is in the form asin(bx+c) + d the period is just 2pi/b. The formula for the period might be easier to use than thinking of horizontal stretches.

Vertical shift is just d.

The amplitude is the distance from the midline. i.e. amplitude = |a|. The || (absolute value function) makes everything that is negative become positive. As 'a' can be negative we take the absolute value to make sure it is positive. e.g. |-6| = 6, |-pi| = pi, |-2.78| = 2.78. Note that anything that is positive remains positive i.e. |4| = 4, |6| = 6, |pi| = pi. The reason for the absolute value is we are interested in the distance from the midline which is non-negative.

Note it might to help to learn about curve sketching for more detail on things discussed.(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.