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# How to interpret the y-intercept of a graph in context (example)

Sal discusses the meaning of the y-intercept of the graph of a function that models the height of a ball. Created by Sal Khan.

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.