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Interpret a quadratic graph

We can interpret what the features of a graph of a quadratic model mean in terms of a given context. Created by Sal Khan.

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  • aqualine ultimate style avatar for user Adam Kirsten
    Why is a quadratic equation called "quadratic"? The prefix "quad-" means four, but a quadratic equation only has a degree of two. Is there something else behind it?
    (12 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user Philip
      I have wondered why “4” was used for a “2”. It didn't seem to make any sense at first. However, in Latin:
      • Linear means a line, and having 1 dimension contains 1 solution
      • Quad refers to a square, which has 2 dimensions and thus has 2 solutions
      • Cubic refers to a cube, which has 3 dimensions and thus has 3 solutions
      So it seems that the "quad" is focusing on the number of sides the shape has (while not yet about the number of solutions); it means a "square-like" equation. This is why the “2nd-power” is called “squared”, because for whatever value is “squared” the result produces the shape of a square. A square has 4 sides, and this is likely what “quad” or “four” is coming from.
      (17 votes)
  • duskpin tree style avatar for user Witenko, Kinley
    Do you solve these any different if they have endpoints instead of continued lines?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user Philip
      A parabola does not really have endpoints because there are no restrictions to its function's domain or range of the general x^2. Even if there is a coefficient ≠ 1 at the x^2, a coefficient for the first degree, and/or a constant, there won't be undefined aspects. The curve may be stretched or squeezed by the coefficients, and shifted up or down by the constant, but any input to the x-variable will be valid. Unlike some other functions which have restrictions either because of the operation, such as square roots, or if there is some division by 0 due to a fraction's denominator, the "x^2" does not have any values of the x variable that will-produce something undefined. So any real value: positive, negative, zero, decimal or fraction, or even an irrational value, will not have restrictions (though due to the context, it is possible for some restrictions to occur, like how Sal Khan states time cannot really be "negative".

      Below are some examples for why a function may be undefined:
      •If we have say 3/(x-4), we while all values, positive and negative, are allowed, we cannot have 4 because 4-4 equals 0, and division by 0 is undefined.
      •Without imaginary values, we can't have the square root of negative values because we will end up with a positive value by squaring either a positive or negative but whenever a value is squared,for we have either a [positive]x[positive] or [negative]x[negative], and to really produce a negative we will be needing a negative times a positive.

      Hope this helps.
      (3 votes)

Video transcript

- [Instructor] Katie throws a ball in the air for her dog to chase. The function f models the height of the ball, in meters, as a function of time, in seconds, after Katie threw it. And we could see that right over here. This is our function f. So at time t equals zero, the height looks like it's a couple of meters. And then as we go forward in time, to a little under 2 1/2 seconds, the ball's going up. And then after a little under 2 1/2 seconds, the ball starts going down. And after, by the time you get to five seconds or close to five seconds, it looks like the ball is on the ground. Its height is zero meters. So then they ask us which of these statements are true? Choose all that apply. So pause this video, and see if you can work it out. All right, now let's look through the choices. So the first one says Katie threw the ball from a height of five meters. So let's see if that bears fruit or (chuckles) see if that's true. So if she threw it from a height of five meters, that means that the y-intercept would've been at five meters. At time t equals zero, we would've been at a height of five meters. Clearly, that is not the y-intercept. It looks like she threw it from a height of maybe 1 1/2 or two meters. So I'm assuming that Katie is not five meters tall. So she wasn't on a ladder or anything. She just threw it from her regular height. And so we can rule out this first choice. The second one says at its highest point, the ball was about 31 meters above the ground. Let's see if that is true. So the highest point is right over here. And yeah, that looks about 31 meters, so I like that choice. So I will select that one. The ball was in the air for about 2 1/2 seconds. So we can clearly see that that is not the case. The ball was going up for about 2 1/2 seconds, but then it was going down for roughly another 2 1/2 seconds. And so it was actually in the air for almost five seconds. So I would rule this one out. The ball reached its highest point in the air about 2 1/2 seconds after Katie threw it. So let's see, after Katie, 2 1/2 seconds after Katie threw the ball, it's right over there. It looks like it reached its highest point a little bit before that. But they said about, so they're speaking in rough terms. So I think that statement can be true. It's about 2 1/2 seconds. If they said exactly 2 1/2 seconds, I wouldn't have selected it 'cause it seems like it happened at like 2.45 seconds or something like that. But there you go, those are the ones that seem true.