Interpreting trigonometric graphs in context | Trigonometry | Algebra || | Khan Academy

We're told Alexa is riding on a Ferris wheel. Her height above the ground in meters is modeled by ( h(t) ), where ( t ) is the time in seconds, and we can see that right over here. Now, what I want to focus on in this video is some features of this graph. The features we're going to 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. Then, we're going to think about what it actually represents. Well, Alexa starts off at five meters above the ground, and then she goes higher and higher, gets as high as 25 meters, and then goes back as low as five meters above the ground and as high as 25 meters.

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 going to keep going off the graph, and the reason is to help us think about what does that midline even represent.

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. To help us visualize that, let me draw a Ferris wheel. So, I'm going to draw a circle with this as the center. 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. 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. You can see it right over here.

Actually, right when Alexa starts, we have her starting 10 meters below the midline, 10 meters below the center. This is when Alexa is right over here; she is 10 meters below the midline. Then after, it looks like 10 seconds, she is right at the midline, so that means that she's right over here.

Maybe the Ferris wheel is going this way—at least in my imagination, it's going clockwise. After another 10 seconds, she is at 25 meters, so she is right over there. You can see that 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.

So, it's the maximum displacement or the maximum change from that midline. Over here, it really represents the radius of our Ferris wheel: 10 meters. 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. Then here she is after 40 seconds—she's back at the bottom and about to head up again. This time right over here, that 40 seconds, that is the period.

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.

After 10 more seconds, or a total of 40 seconds, she is back to where she started. So, the period in this example shows how long it takes 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 start right over here and say, "Okay, we're 15 meters above the ground." Alright, 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 going to get 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 15 meters and going up. So, we have to keep going another 20 seconds in order to be 15 meters in the air and going down.