Understanding Periodic Phenomena and Graphs

Hey, starting unit 3 right now, this lesson is going to be pretty easy. It's called Periodic Phenomena. And listen, phenomena, that is a word... Listen, if your teacher is as old as the algebras... I'm pretty old. I'm not that old, but I'm mostly old. When they hear phenomena and then they... they think of the Muppets, this song comes up. And so if your teacher knows this song, they're officially old. And if they don't, then you're probably young still, right? I don't know if you guys know that song. But Periodic Phenomena is so... easy. We're going to be talking about that. Here's what the definition says. As input values increase, if the output values, the y values, demonstrated repeating pattern over successive equal length intervals, we have what's called a periodic relationship. What this means is if you go a certain distance and you notice a pattern, like this is a sine curve, and we go up and then down, and then we're back at the middle again, and then it repeats over the same distance for the x, right? It takes the exact same distance. to repeat, then that is called a periodic relationship. And another word to describe this would be cyclical. It's cyclic. To go through this pattern, there's a cycle. Okay, so real life examples of this, phases of the moon, high tide and low tide to the ocean. And I actually want to show you an amazing animation that was created about the moon. So in this animation, the sun is all the way to the left and it's shining on the earth and the moon. And on the right here, you can see the view of the moon from the earth. Earth. Okay, I'm going to have this go around two times. This is by Phil Hart, by the way. He's on YouTube. Give him a shout out. But we start right there where there's no light. And then as the moon travels around the Earth, you can see the entire moon, the illumination increases, and then it decreases again. And then this happens repeatedly over and over and over again. So in my internet Google searching, I did find this cool graph, which basically graphs exactly what we were looking at in that animation. We start out here with the... percent of the illumination of the moon and if we start with a new moon which means you can't really see it at all right there's zero percent illuminated and then eventually it goes up to 100 but then back down to zero percent and then it would repeat over and over again so now knowing what we know about periodic relationships and how they repeat we can take a single period or cycle and we can construct the rest of the graph based off of what that looks like so here are two examples numbers 1 and 2, they want you to sketch the rest of the graph based on one cycle that's given here. Okay, so the way that I like to do this is I just find the key points. I know that this cycle starts at 0, also ends at 0, and there's a 0 in the middle. Okay, I also know that it takes 8 units to complete. That's one full cycle here. So I would say the period equals 8. Okay, the period of a function is the smallest change in x values it takes for the function to repeat itself. All right, so we would say the period of this function is x. function is 8. And they want us to sketch the rest of the graph. And what I like to do is I like to play halves with our trig functions. I know that in between here, if I go halfway, it will be at a minimum. And if I know I'm halfway here, it'll be at a maximum. So that helps me sketch it a little bit. If I go over two units, it'll be at a max. So if I go left two units, it should be at a minimum. I know that the output values are the same. So this minimum here will be the same as this minimum. And if I go back two more units, it should be at a minimum. should be back up at zero and then I can come over here and kind of construct this graph and it would look something like this and then I could keep going into the future here right just like Elsa didn't Elsa do that into the into the what are we doing into the unknown this is known though so this is not like Elsa we know exactly what's going to happen it's very cyclical okay so then number two they give us a weird graph here but you know here's what we have to keep in mind we know it repeats It's the same value over and over again. So you know what I might want to do? If you can copy and paste in your brain, that would help you out. So look at the cycle that they give you. Let's talk about what the period is. The period of this cycle would be what? This relationship has a period of four, right? Because that's what one cycle takes to complete. And you can take this, and if you can just kind of copy and paste it, boom. How easy was that? And we can do the same thing working backwards. It's a copy and paste. So you have to get the same pattern. over and over and over again. Nothing changes. And if you can do that in your brain, then sketching these graphs is pretty easy. In fancy math terms, we'll say that the period of a function is the smallest positive value k such that if you add k to x, you get the same thing that you had before, which is f of x. All right, that's how I think about what this says. f of x plus k equals f of x. If you add some value to x, you're going to end up in the same place that you were at. So in our first example here, that would be 8, right? k would be 8 here we're adding 8 and you end up in the same place. We're at the beginning of that function. And right here that we would say would be 4 because it takes 4 units to complete that function. How about that? So the next part they might ask you to identify is the period of the function just by looking at it. And identifying the period is pretty easy. You just have to find a common place in each cycle of the graph and figure out how long it takes to get there. So how far is it from 1 quarter to 3 quarter? And to figure that out we could do some simple math. We could take you know, three quarters, and we could subtract one quarter. Okay, that's going to give us what? Two quarters, which is one half. So it takes, you know, it's half a unit here. Look at that. By the time you're at one half, this whole function starts over again. But some students might be asking, why didn't you just start right here and then go to here? And the answer is, well, you could. You could do that as well. In fact, you can pick any two points that match up with each other, and you can use those points. And in fact, like on number four here, it actually is to our advantage not to use. use this point right here it's kind of hard to see where it would be it's not there right like one whole cycle if we start here then a whole cycle would be over here somewhere it's off the graph so why don't we just use our minimum values here i mean because i can see two minimum values but i can't see two maxes the middle part's a little bit it's a little uh blurry here but here are two minimum values and i know that one occurs that's hard to see but one occurs at negative four the other one occurs at 12 so what is the period of this function. It would be 12 minus negative 4, and 12 minus negative 4 is 16. That would be the period of this function. Easy enough, right? Could you use some other points? Well, sure, you could. I mean, if you notice that this point right here matches up with this point, because they're both on the way down, right, then you could use 8 and negative 8. You get the same answer, but why would you do that? Just pick the easy points. Periodic functions take on various characteristics of a function, such as increasing, decreasing, concavities. Recognize that any character... is found in one period will be in every period of course because they repeat themselves below is one cycle of a periodic function use the graph to answer the questions now we're going to look for patterns here too notice that this function starts up at the top at a maximum comes down goes back to the top and then it's periodic which means it's just going to repeat itself over and over and over again so i'm going to to answer some of these questions like the first question says is the function increasing decreasing or both on the interval from 18 to 20 Okay, well let's look at a pattern here. We know that the period is 4, so it takes 4 units to repeat. That means that a second cycle would be completed by 8 units. And if I did another cycle, it would be completed by 12 units, because I'm just adding 4, right? So that means that also 16 units, that's where another cycle would be completed, and then 20. So I know that 20 would be right here, because it's a multiple of 4, and it repeats every 4. units. That means that this would be 19, this would be 18. I know there'd be a minimum at 18, so from 18 to 20, we would say that the function is increasing. Easy enough, right? Well, the next one asks about concavity. Is it concave up or down or both on the interval from 31 to 33? We can use the same logic, right? If this is, you know, multiples of 4 are going to be at a maximum, so that means that 32 will be the maximum. All right, let's clear that out. I know that 32 will be a maximum. That means that 31 would be here, right, at 0. And then 33, because this is repeating, right, would come back down to 0. So it would look like this. All right, is that concave up, down, or both? Both don't make no sense. So we would say concave down from 31 to 33. And lastly, is there a relative max or min or neither at the point x equals 82? And again, 82, well, you know what I'm going to do? I'm going to use the fact that I know my multiples of 4. 4 multiples of 4 are going to be at a max and so I know that x equals 80 should be a max right because that's multiple of 4 I know that right in my head I don't even have to do anything there so if 80 is a max that would be like right here then I go over 2 1 2 it's going to be at a minimum the other way you could think about it is 84 is a max and that would be right here we can go back to but that should be a relative minimum and that's basically it easy peasy right that's a pretty easy easy, straightforward lesson. Hey, listen, whenever you're talking about periodic phenomena, think about that Muppets clip, Phenomena. Phenomena. You know the one I'm talking about. This is Mr. Kelly. Remember, it's nice to be important, but it's more important to be nice. See ya!

Hey, starting unit 3 right now, this lesson is going to be pretty easy. It's called Periodic Phenomena. And listen, phenomena, that is a word...

Listen, if your teacher is as old as the algebras... I'm pretty old. I'm not that old, but I'm mostly old.

When they hear phenomena and then they... they think of the Muppets, this song comes up. And so if your teacher knows this song, they're officially old.

And if they don't, then you're probably young still, right? I don't know if you guys know that song. But Periodic Phenomena is so... easy. We're going to be talking about that.

Here's what the definition says. As input values increase, if the output values, the y values, demonstrated repeating pattern over successive equal length intervals, we have what's called a periodic relationship. What this means is if you go a certain distance and you notice a pattern, like this is a sine curve, and we go up and then down, and then we're back at the middle again, and then it repeats over the same distance for the x, right?

It takes the exact same distance. to repeat, then that is called a periodic relationship. And another word to describe this would be cyclical.

It's cyclic. To go through this pattern, there's a cycle. Okay, so real life examples of this, phases of the moon, high tide and low tide to the ocean.

And I actually want to show you an amazing animation that was created about the moon. So in this animation, the sun is all the way to the left and it's shining on the earth and the moon. And on the right here, you can see the view of the moon from the earth. Earth.

Okay, I'm going to have this go around two times. This is by Phil Hart, by the way. He's on YouTube. Give him a shout out.

But we start right there where there's no light. And then as the moon travels around the Earth, you can see the entire moon, the illumination increases, and then it decreases again. And then this happens repeatedly over and over and over again.

So in my internet Google searching, I did find this cool graph, which basically graphs exactly what we were looking at in that animation. We start out here with the... percent of the illumination of the moon and if we start with a new moon which means you can't really see it at all right there's zero percent illuminated and then eventually it goes up to 100 but then back down to zero percent and then it would repeat over and over again so now knowing what we know about periodic relationships and how they repeat we can take a single period or cycle and we can construct the rest of the graph based off of what that looks like so here are two examples numbers 1 and 2, they want you to sketch the rest of the graph based on one cycle that's given here. Okay, so the way that I like to do this is I just find the key points.

I know that this cycle starts at 0, also ends at 0, and there's a 0 in the middle. Okay, I also know that it takes 8 units to complete. That's one full cycle here. So I would say the period equals 8. Okay, the period of a function is the smallest change in x values it takes for the function to repeat itself.

All right, so we would say the period of this function is x. function is 8. And they want us to sketch the rest of the graph. And what I like to do is I like to play halves with our trig functions.

I know that in between here, if I go halfway, it will be at a minimum. And if I know I'm halfway here, it'll be at a maximum. So that helps me sketch it a little bit. If I go over two units, it'll be at a max. So if I go left two units, it should be at a minimum.

I know that the output values are the same. So this minimum here will be the same as this minimum. And if I go back two more units, it should be at a minimum.

should be back up at zero and then I can come over here and kind of construct this graph and it would look something like this and then I could keep going into the future here right just like Elsa didn't Elsa do that into the into the what are we doing into the unknown this is known though so this is not like Elsa we know exactly what's going to happen it's very cyclical okay so then number two they give us a weird graph here but you know here's what we have to keep in mind we know it repeats It's the same value over and over again. So you know what I might want to do? If you can copy and paste in your brain, that would help you out.

So look at the cycle that they give you. Let's talk about what the period is. The period of this cycle would be what? This relationship has a period of four, right?

Because that's what one cycle takes to complete. And you can take this, and if you can just kind of copy and paste it, boom. How easy was that? And we can do the same thing working backwards. It's a copy and paste.

So you have to get the same pattern. over and over and over again. Nothing changes.

And if you can do that in your brain, then sketching these graphs is pretty easy. In fancy math terms, we'll say that the period of a function is the smallest positive value k such that if you add k to x, you get the same thing that you had before, which is f of x. All right, that's how I think about what this says. f of x plus k equals f of x. If you add some value to x, you're going to end up in the same place that you were at.

So in our first example here, that would be 8, right? k would be 8 here we're adding 8 and you end up in the same place. We're at the beginning of that function.

And right here that we would say would be 4 because it takes 4 units to complete that function. How about that? So the next part they might ask you to identify is the period of the function just by looking at it. And identifying the period is pretty easy. You just have to find a common place in each cycle of the graph and figure out how long it takes to get there.

So how far is it from 1 quarter to 3 quarter? And to figure that out we could do some simple math. We could take you know, three quarters, and we could subtract one quarter.

Okay, that's going to give us what? Two quarters, which is one half. So it takes, you know, it's half a unit here. Look at that.

By the time you're at one half, this whole function starts over again. But some students might be asking, why didn't you just start right here and then go to here? And the answer is, well, you could.

You could do that as well. In fact, you can pick any two points that match up with each other, and you can use those points. And in fact, like on number four here, it actually is to our advantage not to use.

use this point right here it's kind of hard to see where it would be it's not there right like one whole cycle if we start here then a whole cycle would be over here somewhere it's off the graph so why don't we just use our minimum values here i mean because i can see two minimum values but i can't see two maxes the middle part's a little bit it's a little uh blurry here but here are two minimum values and i know that one occurs that's hard to see but one occurs at negative four the other one occurs at 12 so what is the period of this function. It would be 12 minus negative 4, and 12 minus negative 4 is 16. That would be the period of this function. Easy enough, right?

Could you use some other points? Well, sure, you could. I mean, if you notice that this point right here matches up with this point, because they're both on the way down, right, then you could use 8 and negative 8. You get the same answer, but why would you do that? Just pick the easy points. Periodic functions take on various characteristics of a function, such as increasing, decreasing, concavities.

Recognize that any character... is found in one period will be in every period of course because they repeat themselves below is one cycle of a periodic function use the graph to answer the questions now we're going to look for patterns here too notice that this function starts up at the top at a maximum comes down goes back to the top and then it's periodic which means it's just going to repeat itself over and over and over again so i'm going to to answer some of these questions like the first question says is the function increasing decreasing or both on the interval from 18 to 20 Okay, well let's look at a pattern here. We know that the period is 4, so it takes 4 units to repeat.

That means that a second cycle would be completed by 8 units. And if I did another cycle, it would be completed by 12 units, because I'm just adding 4, right? So that means that also 16 units, that's where another cycle would be completed, and then 20. So I know that 20 would be right here, because it's a multiple of 4, and it repeats every 4. units.

That means that this would be 19, this would be 18. I know there'd be a minimum at 18, so from 18 to 20, we would say that the function is increasing. Easy enough, right? Well, the next one asks about concavity. Is it concave up or down or both on the interval from 31 to 33?

We can use the same logic, right? If this is, you know, multiples of 4 are going to be at a maximum, so that means that 32 will be the maximum. All right, let's clear that out. I know that 32 will be a maximum.

That means that 31 would be here, right, at 0. And then 33, because this is repeating, right, would come back down to 0. So it would look like this. All right, is that concave up, down, or both? Both don't make no sense. So we would say concave down from 31 to 33. And lastly, is there a relative max or min or neither at the point x equals 82?

And again, 82, well, you know what I'm going to do? I'm going to use the fact that I know my multiples of 4. 4 multiples of 4 are going to be at a max and so I know that x equals 80 should be a max right because that's multiple of 4 I know that right in my head I don't even have to do anything there so if 80 is a max that would be like right here then I go over 2 1 2 it's going to be at a minimum the other way you could think about it is 84 is a max and that would be right here we can go back to but that should be a relative minimum and that's basically it easy peasy right that's a pretty easy easy, straightforward lesson. Hey, listen, whenever you're talking about periodic phenomena, think about that Muppets clip, Phenomena. Phenomena.

You know the one I'm talking about. This is Mr. Kelly. Remember, it's nice to be important, but it's more important to be nice. See ya!

Transcript for:Understanding Periodic Phenomena and Graphs

Transcript for:
Understanding Periodic Phenomena and Graphs