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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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  • piceratops seedling style avatar for user Rain
    in the following questions does anyone know what the term end behavior means?
    (5 votes)
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    • piceratops ultimate style avatar for user Just Keith
      Yes, end behavior is how a function behaves as x gets very large (both in the negative and positive direction). The main thing we are interested in is whether f(x) is increasing or decreasing as x grows very large.
      For example, f(x) = x² - x + 2 keeps increasing as x gets very large, so its positive end behavior is "increasing" or "upward" (different teachers use different word".
      Likewise, when x becomes very negative, this function keeps increasing so its negative end behavior is "increasing" or "upward".

      But, f(x) = x³ + 2x² - 3 is different. When x is very negative, it is decreasing or "downward". So its negative (sometimes called left hand) end behavior is downward or decreasing. However, when x if very large on the positive side, f(x) goes up, so its positive or right hand end behavior is increasing or upward.

      If you have a good algebra teacher, you will get a slightly more rigorous way of saying the same thing:
      Negative side, going down would be written as: f(x)→−∞ as x→−∞
      This is read as "f(x) approaches negative infinity as x approaches negative infinity"
      Similarly,
      Negative side, going up would be written as: f(x)→ + ∞ as x→−∞
      Positive side, going down would be written as: f(x)→−∞ as x→ +∞
      Positive side, going up would be written as: f(x)→ + ∞ as x→+∞

      There is a video on this:
      https://www.khanacademy.org/math/algebra2/polynomial_and_rational/polynomial-end-behavior/v/polynomial-end-behavior
      (5 votes)
  • male robot hal style avatar for user Scott Fitzgerald
    At , Sal says that T(v) = T(-v) defines the function as even. An odd function is defined as T(v) = -T(-v), right?
    (3 votes)
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    • leaf grey style avatar for user Brendan
      This is not super necessary to say but I just noticed that... Scott, you're right T(v)= -T(-v) but Gene is also right -T(v)=T(-v). Those two statements say the same thing. To convert from one to the other just multiply both sides by -1.
      (5 votes)
  • leaf green style avatar for user John Sung
    So magnitude is like absolute value?
    (3 votes)
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  • piceratops ultimate style avatar for user Andrew Park
    Did khan academy change their videos to their own? I liked the youtubes
    (2 votes)
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  • piceratops ultimate style avatar for user Sorrow
    So any quadratic function who's symmetry is the y axis is even?
    EDIT: And any absolute value function who's symmetry is the y axis is even?
    (1 vote)
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  • starky ultimate style avatar for user Damion Vanderschaegen
    At , How did they figure out the equation to use so you can get the answer? What would you use to get the equation?
    (1 vote)
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  • starky tree style avatar for user Riptide
    Could somebody explain in more detail?
    (1 vote)
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  • male robot donald style avatar for user Cobra Coder
    Wouldn't the function also be dependent of time? If you just sand for 3 seconds, it won't do much, but if you sand for an hour, it is a lot of sanded wood.
    (1 vote)
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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.