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Graph interpretation word problem: basketball

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.