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Interpreting trigonometric graphs in context

When a trigonometric function models a real-world relationship, we can assign meaning to its midline, amplitude and period. Created by Sal Khan.

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Video transcript

- [Instructor] We're told Alexa is riding on a Ferris wheel. Her height above the ground in meters is modeled by H of t where t is the time in seconds. And we can see that right over here. Now what I want to focus on this video is some features of this graph. And the features we're gonna focus on, actually the first of them, is going to be the midline. So pause this video and see if you can figure out the midline of this graph, or the midline of this function. And then we're going to think about what it actually represents. Well, Alexia starts off at five meters above the ground and then she goes higher, and higher, and higher, and gets as high as 25 meters, and then goes back as low as five meters above the ground, and as high as 25 meters. And what we can view the midline as is the midpoint between these extremes, or the average of these extremes. Well, the extremes are she goes as low as five and as high as 25. So what's the average of five and 25? Well, that would be 15. So the midline would look something like this. And I'm actually gonna keep going off the graph. And the reason is is to help us think about what does that midline even represent. And one way to think about it is it represents the center of our rotation in this situation, or how high above the ground is the center of our Ferris wheel? And to help us visualize that, let me draw a Ferris wheel. So I'm going to draw a circle with this as the center. And so the Ferris wheel would look something like this. And it has some type of maybe support structure. So the Ferris wheel might look something like that. And this height above the ground, that is 15 meters, that is what the midline is representing. Now the next feature I want to explore is the amplitude. Pause this video and think about what is the amplitude of this oscillating function right over here and then we'll think about what does that represent in the real world, or where does it come from in the real world? Well the amplitude is the maximum difference or the maximum magnitude away from that midline. And you can see it right over here. Actually right when Alexa starts we have starting 10 meters below the midline, 10 meters below the center, and this is when Alexa is right over here. She is 10 meters below the midline. And then after, it looks like 10 seconds, she is right at the midline. So that means that she is right over here. Maybe the Ferris wheel is going this way, at least in my imagination it's going clockwise. And then after another 10 seconds she is at 25 meters. So she is right over there. And you could see that. She is right over there, I drew that circle intentionally of that size. And so we see the amplitude in full effect. 10 meters below to begin the midline and 10 meters above. And so it's the maximum displacement or the maximum change from that midline. And so over here it really represents the radius of our Ferris wheel, 10 meters. And then from this part she starts going back down again. And then over here she's back to where she started. Now the last feature I want to explore is the notion of a period. What is the period of this periodic function? Pause this video and think about that. Well the period is how much time does it take to complete one cycle. So here she's starting at the bottom. And let's see, after 10 seconds, not at the bottom yet. After 20 seconds, not at the bottom yet. After 30 seconds, not at the bottom yet. And then here she is, after 40 seconds, she's back at the bottom and about to head up again. And so this time right over here, that 40 seconds, that is the period. And if you think about what's going on over here, she starts over here five meters above the ground. After 10 seconds she is right over here, and that corresponds to this point right over here. After 10 more seconds she's right over there. That corresponds to that point. After 10 more seconds she's over here. That corresponds to that. And after 10 more seconds, or total of 40 seconds, she's back to where she started. So the period in this example shows how long does it take to complete one full rotation. Now we have to be careful sometimes when we're trying to visually inspect the period because sometimes it might be tempting to say, "Start right over here, "and say okay, we're 15 meters above the ground. "All right, let's see, we're going down. "Now we're going up again. "And look, we're 15 meters above the ground. "Maybe this 20 seconds is a period." But when you look at it over here it's clear that that is not the case. This point represents this point at being 15 meters above the ground, going down, that's getting us to this point, and then after another 10 seconds we get back over here. Notice, all this is measuring is half of a cycle going halfway around. In order to go all the way around not only do we have to get to the same exact height, but we have to be moving in the same direction. We're at 15 meters and going down. Here we're at 15 meters and going up. So we have to keep going anther 20 seconds in order to be 15 meters in the air, and going down.