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# Graph interpretation word problem: basketball

CCSS.Math:

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.

## Want to join the conversation?

- 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)
- The value of y(x) changes depending on the value of x. Therefore y(x) is the dependent variable and x is the independent variable(8 votes)

- What if the function is not defined for x=0? What will be the Y-intercept then?(5 votes)
- 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)

- If Mr. Theisen is farther away, does it affect the outcome of the problem?(4 votes)
- 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)
- 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)

- At3:20, he said goal. Shouldn't it be basket?(3 votes)
- Sal said: "The goal is at 26 feet away" at3:14. In this context, the word "goal" means a target.(4 votes)

- Is there an easier way to do these type of problems?(2 votes)
- how do you find the domain'(2 votes)
- When you look at the graph in this video you can't tell exactly how far the ball went. If you assume that the ball stopped at 26.6 feet then the domain would be: {x∈R | 0 ≤ x ≤ 26.6}.

If you want to find out how to find domains from graphs watch this video: https://www.khanacademy.org/math/algebra/algebra-functions/domain-and-range/v/domain-and-range-from-graphs(1 vote)

- Please explain me as I don't know that what's a y-intercept or a x-intercept......(1 vote)
- 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)

- I didn't really understand the "significance of the y-intercept". To me it seemed like all the answers were correct.(1 vote)
- 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)

- Does the same rules apply for y(x) as f(x) or is that not a thing.(1 vote)
- 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.