Features of sinusoidal functions

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.