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Symmetry of algebraic models

Sal interprets the significance of modeling function being even. Created by Sal Khan.

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Video transcript

Cid is experimenting with a piece of sandpaper and some wood. He tries scraping the piece of sandpaper over the wood in different ways to see how much is scraped off. The thickness of wood scraped off, in millimeters, as a function of the speed of the sandpaper, in meters per second. t of v. So this is the thickness scraped off. So that's the thickness, or how much is scraped off. And it is a function of speed. One, they're using v and also they're getting negative value. So we care about the direction. It's actually the velocity. So this is how much is scraped off as a function of velocity. It's shown below. And so if the velocity is greater than 0, that means that the sandpaper is moving to the right. That makes sense. That's the standard convention. And if the velocity is less than 0, it means the sandpaper is moving to the left. Fair enough. The function is even. What is the significance of the evenness of this function? Well, the fact that it's even means that t of v is equal to t of negative v. So that tells us that if our velocity is 8 meters per second to the left we're going to get as much scraped off as if we go 8 millimeters per second to the right. And we see that right over here. So that is equal to that. If we go at 6 meters per second to the left we're going to get just as much scraped off as we go 6 millimeters-- 6 meters per second, these are in meters per second-- to the right. So these two are going to be the same. So it's really telling us-- and we could say do it for 4 meters per second and negative 4-- is it doesn't matter if we go to the left or the right. What really matters is the magnitude of the velocity or the absolute value of it. But it doesn't matter if we're going to the left or the right. Whether we're going to the left or the right for a given magnitude of velocity we are going to get the same amount scraped off. Now let's see which of these choices are consistent with what I just said. Moving the sandpaper faster scrapes off more wood. Well, that's true. We see as the speed increases, or the magnitude of the speed increases, we scrape off more wood. As the magnitude of the speed, this negative 8, you might say, hey, that's lower than negative 2, but the magnitude is larger. We're going 8 meters per second to the left and we're scraping off more. So this is a true statement, but it's not the significance of the evenness of the function. This could have been true even if this was a seven, but then this function would no longer be even. The piece of wood is six millimeters thick. We actually don't get any of that from the function. Moving the sandpaper to the right has the same effect as moving it to the left. Well, that seems pretty close to what I had said earlier. That for a given speed to the right or to the left, we get the same amount that is taken off of the piece of sandpaper, or the piece of wood. So this looks like our answer. Keeping the sandpaper still doesn't scrape off any wood. Well, that is true but once again is not the significance of the evenness of this function.