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### Course: Algebra 2 (Eureka Math/EngageNY) > Unit 2

Lesson 4: Topic B: Lesson 11: Features of sinusoidal graphs- Features of sinusoidal functions
- Midline of sinusoidal functions from graph
- Amplitude of sinusoidal functions from graph
- Midline of sinusoidal functions from equation
- Amplitude of sinusoidal functions from equation
- Period of sinusoidal functions from graph
- Amplitude & period of sinusoidal functions from equation
- Period of sinusoidal functions from equation
- Midline, amplitude, and period review

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# Features of sinusoidal functions

Sal introduces the main features of sinusoidal functions:

**midline, amplitude, & period**. He shows how these can be found from a sinusoidal function's graph. Created by Sal Khan.## Want to join the conversation?

- Hello! How do I determine if a function has a period algebraically? Thanks!(46 votes)
- There is a way to do this, but to be honest it is much easier to do graphically. Also, the math involved can get fairly advanced and rather hard to avoid making errors with.

But here is how you would do it:

The function f(x) is periodic if and only if:

f(x+nL) - f(x) = 0, where n is any integer and L is some constant other than 0.

If the only solution for L is 0, then the function is NOT periodic.

Thus, set n=1 and solve for L.

After doing so, demonstrate that

f(x+nL) - f(x) = 0, for integer values of n.

So, that is how you would determine this mathematically. I don't recommend attempting it because it is quite difficult and often involves nonreal complex exponents or complex logarithms.

Note: there are some functions that have more than one period, but these are really advanced level math and you probably won't encounter them at this level of study.(70 votes)

- How do I know whether I must use midline = (max val + min val) / 2 or (max val - min val) / 2? I'm really confused(16 votes)
- Instead of relying on formulas that are so alike that they're confusing (to me, too!), maybe try to think it through each time (at least in the beginning) until it gets more familiar). I had a LOT of difficulty with this type of problem and I found that I had to go slowly and think things through each step EVERY time I did a problem.

Here's a method I found helpful. Maybe it will be of use to you.

I know that the midline lies halfway between the max and the min. So I need to get the total height (by subtracting the min from the max). (That gives me ( 4 - (-2) ). Now I divide by 2. (That gives me 3.) By definition that is the AMPLITUDE. Now I can either add that to the min (or subtract it from the max), and where I end up is the MIDLINE ( at 1 ).

Good luck!(31 votes)

- Can someone please explain how to find the midline of a sinusoidal function from its equation, instead of the graph?(10 votes)
- y = A sin (B(x - C)) + D is a general format for a sinusoidal function. The number in the D spot represents the midline. The equation of the midline is always 'y = D'.

Example:

y = 3 sin(2(x - π)) - 5 has a midline at y = -5(20 votes)

- What are sinusoidal functions?(6 votes)
- A sinusoidal function is one with a smooth, repetitive oscillation. "Sinusoidal" comes from "sine", because the sine function is a smooth, repetitive oscillation. Examples of everyday things which can be represented by sinusoidal functions are a swinging pendulum, a bouncing spring, or a vibrating guitar string.

I hope this helps. Good luck!(23 votes)

- Hello, I'm just wondering why Sal choice to use the Midline to find the period: is this always the case? or is it just easier to use the Midlines y value instead?

Thank you!(5 votes)- Hi Daniel,

No, you do not have to use the midline to find the period. You can find the period by going from peak to peak, or trough to trough, or midline to midline. If you use midline of course you will need to keep in mind that you will need to skip a midline (because the midlines you measure from must be going the same direction).

Hope this helps,

- Convenient Colleague(15 votes)

- Do you have any videos that actually talk about the graphs of trig functions? If so please post as soon as possible. This title is very misleading. I assumed you would teach what the trig functions looked like but it seemed more like you expected us to know it already :(. On the next video I was so frustrated because I did not even know what -1/2 cos(3x) meant. I didn't even know these things could be graphed. I thought you only used for triangles or something. What is all this graphing stuff? SO frustrated :/(7 votes)
- If you watch the videos in the preceding section headed "Unit circle definition of trig functions", you will appreciate that the cosine and sine functions take an angle as the input value, and give output values that repeat every so often, and that always remain within the values -1 and 1. If, instead of thinking about the x and y coordinates of points on the unit circle, you decide to plot a graph with angle on the x-axis, with the y axis being the cosine or sine of the variable x, you will obtain a pattern like the one in this video.

So, this*is*the video where Sal is showing you what the trig functions look like. If the maximum value of the cosine or sine of any angle is 1, and the minimum value is -1, then the amplitude of these functions is 1, and any function that is a multiple of one of these functions will have an amplitude of 1 times that multiple, or -1/2 in the case of cos(3x).

Edit: Actually, all this is made more explicit in this video: https://www.khanacademy.org/math/trigonometry/trig-function-graphs/trig_graphs_tutorial/v/we-graph-domain-and-range-of-sine-function(6 votes)

- Can the "midline" also be called the "sinusoidal axis"?(4 votes)
- Yes, I think that it can be referred to by both names.(7 votes)

- How is the wavelength of a wave(distance from crest to crest or trough to trough) related to the period? Is the wavelength of a sinusoidal function equivalent to the period? Sorry for including a bit of science here, I'm just confused.(4 votes)
- They aren't the same. Wavelength refers to the distance between two troughs/crests while period refers to the
**time**taken to complete one wavelength(7 votes)

- I know amplitude can't be negative, but can the period?(3 votes)
- In sinusoidal waves, the period represents the duration it takes for one complete cycle of the wave to occur. The period cannot be negative because it is a measure of time, and time cannot be negative in the physical sense. However, the period can be expressed as a negative value in mathematical representations or equations to indicate a phase shift or a reversal in the direction of the wave. For example, a negative period value would indicate that the wave is shifted to the right or has undergone a phase shift. But in practical terms, the period is always considered as a positive value.(5 votes)

- Hey Sal,

I think there was a mistake at2:37. Isn't 1-3=-2? Just wanted to point out.(2 votes)- If you don't watch the video at full screen you can see the mistake was pointed and corrected at the bottom right corner of the video in a small box(7 votes)

## Video transcript

We have a periodic
function depicted here and what I want
you to do is think about what the midline
of this function is. The midline is a line,
a horizontal line, where half of the
function is above it, and half of the
function is below it. And then I want you to
think about the amplitude. How far does this function
vary from that midline-- either how far above does it go or
how far does it go below it? It should be the same amount
because the midline should be between the highest
and the lowest points. And then finally,
think about what the period of this function is. How much do you have
to have a change in x to get to the same
point in the cycle of this periodic function? So I encourage you to pause
the video now and think about those questions. So let's tackle
the midline first. So one way to think
about is, well, how high does this function go? Well, the highest y-value for
this function we see is 4. It keeps hitting 4 on
a fairly regular basis. And we'll talk about
how regular that is when we talk
about the period. And what's the lowest value
that this function gets to? Well, it gets to y
equals negative 2. So what's halfway
between 4 and negative 2? Well, you could eyeball
it, or you could count, or you could,
literally, just take the average between
4 and negative 2. So 4-- so the midline is going
to be the horizontal line-- y is equal to 4 plus
negative 2 over 2. Just literally the
mean, the arithmetic mean, between 4 and negative 2. The average of 4 and
negative 2, which is just going to
be equal to one. So the line y equals
1 is the midline. So that's the midline
right over here. And you see that it's kind
of cutting the function where you have half of the
function is above it, and half of the
function is below it. So that's the midline. Now, let's think
about the amplitude. Well, the amplitude is
how much this function varies from the midline--
either above the midline or below the midline. And the midline
is in the middle, so it's going to be the
same amount whether you go above or below. One way to say it is, well,
at this maximum point, right over here, how far above
the midline is this? Well, to get from 1
to 4 you have to go-- you're 3 above the midline. Another way of thinking
about this maximum point is y equals 4 minus y equals 1. Well, your y can go as much
as 3 above the midline. Or you could say your
y-value could be as much as 3 below the midline. That's this point right over
here, 1 minus 3 is negative 1. So your amplitude right
over here is equal to 3. You could vary as much as
3, either above the midline or below the midline. Finally, the period. And when I think
about the period I try to look for a relatively
convenient spot on the curve. And I'm calling this
a convenient spot because it's a nice-- when
x is at negative 2, y is it one-- it's at a
nice integer value. And so what I want to do is
keep traveling along this curve until I get to the same y-value
but not just the same y-value but I get the same
y-value that I'm also traveling in the same direction. So for example, let's
travel along this curve. So essentially our
x is increasing. Our x keeps increasing. Now you might say, hey, have
I completed a cycle here because, once again,
y is equal to 1? You haven't completed a cycle
here because notice over here where our y is increasing
as x increases. Well here our y is
decreasing as x increases. Our slope is positive here. Our slope is negative here. So this isn't the same
point on the cycle. We need to get to the point
where y once again equals 1. Or we could say, especially in
this case, we're at the midline again, but our
slope is increasing. So let's just keep going. So that gets us to
right over there. So notice, now we have
completed one cycle. So the change in x needed
to complete one cycle. That is your period. So to go from negative 2
to 0, your period is 2. So your period here is 2. And you could do it again. So we're at that point. Let's see, we want to
get back to a point where we're at the
midline-- and I just happen to start right
over here at the midline. I could have started
really at any point. You want to get
to the same point but also where the
slope is the same. We're at the same point
in the cycle once again. So I could go-- so if I travel
1 I'm at the midline again but I'm now going down. So I have to go further. Now I am back at that
same point in the cycle. I'm at y equals 1 and
the slope is positive. And notice, I traveled. My change in x was the
length of the period. It was 2.