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Graph interpretation word problem: basketball

When a function models a real-world context, we can learn a lot about the context from the function's graph. In this video, we interpret the y-intercept of a graph that models a basketball free throw. Created by Sal Khan.

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  • male robot hal style avatar for user Scott Fitzgerald
    If y(x) is a representation of the dependency of the y value at a certain (x), which of x or y(x) is the dependent variable and which is independent?
    (12 votes)
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  • leaf green style avatar for user Karan Bhullar
    What if the function is not defined for x=0? What will be the Y-intercept then?
    (5 votes)
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    • leaf blue style avatar for user Stefen
      That's a great question!
      Suppose y IS defined for all x values near x=0, but not AT x=0, then what we do is start somewhere near x=0, say at x=0.1, and see what values y has as we get closer to x=0. Then we do the same thing for x=-0.1. If, as we get closer and closer to x=0 from the positive and negative sides, y seems to be getting closer to some value, let's call it 'a', then we say the limit of y, as x gets closer to 0 is 'a', which is what y would be if the function was defined at x=0. This is a bit of an oversimplification of the process and is just meant to give you an intuition of the concept. You will learn more about this in precalculus and calculus, where quite a bit of time is spent on the definition of the limit.
      (10 votes)
  • starky tree style avatar for user Cian Knight
    If Mr. Theisen is farther away, does it affect the outcome of the problem?
    (4 votes)
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  • duskpin ultimate style avatar for user Izzy
    I do not quite comprehend the meaning "of the significance of the y value". I understand everything so far, but this description seems vague and the true meaning quite elusive. Why is the relative maximum not a significant value of y? Why does the hoop height not carry any significance?
    (4 votes)
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    • starky ultimate style avatar for user Ron Joniak
      I think you might have read the question wrong. The question is not "of the significance of the y-value" it is instead: "of the significance of the y-intercept". There is an important distinction here.

      The y-intercept is directly talking about where x=0, aka the starting point in this example. Thus, the significance deals with the starting value of the ball, or the height at which the ball is released. That is the significance of the y-intercept.

      If we look at the significance of the y-value, then the answer would be something like: It describes the height of the ball at any given x.
      (6 votes)
  • blobby green style avatar for user manoj.raghavan
    At , he said goal. Shouldn't it be basket?
    (3 votes)
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  • starky tree style avatar for user stu215834
    Is there an easier way to do these type of problems?
    (2 votes)
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  • leafers seed style avatar for user dylan depew
    how do you find the domain'
    (2 votes)
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  • male robot donald style avatar for user Akshu
    Please explain me as I don't know that what's a y-intercept or a x-intercept......
    (1 vote)
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    • mr pink green style avatar for user David Severin
      Y intercepts are where x = 0 (this is all along the y axis) so where a function crosses the y axis is the y intercept or on a table or ordered pairs, it would be where x value is 0. X intercept is where y = 0 (along the x axis), so where the function crosses the x axis or on table and ordered pairs where 0 is in the y value.
      (3 votes)
  • leaf green style avatar for user Omor Almamun
    I didn't really understand the "significance of the y-intercept". To me it seemed like all the answers were correct.
    (1 vote)
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    • orange juice squid orange style avatar for user graciousartist
      It seems you may be confusing 2 different concepts. It is true to say that all of the answer are correct. They all relate some how to the graph and at least 3 of them relate to the y-axis, however, only one of them relates to the y-intercept.

      The y-intercept is the point at which the function (y(x)) moves through the y-axis. In this situation it is the height at which the ball is at in Mr Theisen hands. This is why the answer is only the first one.
      (2 votes)
  • blobby green style avatar for user ab03107
    Does the same rules apply for y(x) as f(x) or is that not a thing.
    (1 vote)
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    • mr pink green style avatar for user David Severin
      the letter in front does not matter, they are both some function in terms of x. f(x) and g(x) are the most common as labels for functional notation. y is usually used apart from functional notation as just the equation, so you will almost never see y(x) =, you will just see y =, but most letters are possible to be used (if you tried x(x) it would not make sense). .
      (1 vote)

Video transcript

Mr. Theisen is honing his deadly three-point precision on the basketball court. For one of his shots, the height of the ball in feet as a function of horizontal distance, in feet, y of x-- so here y is a function of x. So the height must be y because that's the thing that is a function of something else. So this right over here is height. So our y-axis is going to represent height. And it is a function of x. So x must represent horizontal distance because height is a function of horizontal distance. So this right over here is horizontal distance. Now, it's plotted below. Mr Theisen is standing at x equals 0. So he's standing right over here. This is Mr. Theisen, as I draw my best attempt to draw a little stick figure version of Mr. Theisen. That's not even an acceptable stick figure right over there. So this is Mr. Theisen, and he's standing at x equals 0. And at x equals 0, he is shooting a basketball. And you see from the function right over here that where the graph intersects the y-axis, that tells us that's essentially the height of the ball when x is equal 0, where it's where Mr. Theisen is standing. And if we look at this, this looks like it's 2, 4, 6 feet high. So that's really the initial position of the ball when Mr. Theisen is about to let go of it. Then he lets go of it, and the ball goes in this parabolic trajectory. It increases, increases, increasing, increases. It looks like it hits a maximum point right around there, roughly. That looks like it's at about 16 feet. And then it starts to go down. And right over here-- and this looks like it's about, let's see, 22, 24, 26 feet out-- it looks like it hit something. And considering that something is 10 feet high, it's reasonable to assume that the thing that it hits is the goal. And especially because the question states that he has deadly three-point precision, we can assume it's not crazy that he actually makes the goal. And so that's where it goes into the net. And then the net forces the ball to go down at a much steeper trajectory. And this is exactly, of course, 10 feet high, the height of the goal. Now let's see which of these interpretations are consistent with the interpretation that we just did. The ball is released from Mr. Theisen's hand at a height of 6 feet. Well, that looks exactly right. When x is equal to 0, the ball is 6 feet. And not only is that right, but that is the significance of the y-intercept of this function. The y-intercept is the value of y, the height when x is equal to 0. So that is indeed the significance of the y-intercept. Let's look at these other things. Mr. Theisen is shooting the basketball from 26 feet away. Well, that's right. He's at x equals 0. The goal is at 26 feet away. But that's not the significance of the y-intercept. That would be the significance of where we saw this little point here where the ball dropped down at a steeper angle. The rim of the basketball hoop is 10 feet high. Once again, that's true. You can look at it. You can see it right over there. But that's not the significance of the y-intercept. The maximum height that the ball reaches is 16 feet. Well, once again, that is true, but that's the significance of this maximum point on the curve. That's not the significance of the y-intercept. So we'll go with this first choice.