Entrance to the paint ball court costs $5,
and paint balls are paid for separately.
A ticket for entrance with 5 balls, for example, costs $8.
So let's define some variables here.
Let's let p equal the price that I'm
paying for admission and the balls.
And let's let b equal the number of balls-- number of balls.
And we know that the price is going
to be a function of the number of balls.
We could make a little table here.
So let's make a table where, for a given number of balls,
let's figure out what the price is going to be.
So let's just start at 0 balls.
If we have 0 balls, we have to pay just
to get into the paint ball court, and that costs us $5.
Now they tell us a ticket for entrance with 5 balls--
so now we have entrance and 5 balls--
this is going to cost $8.
So let's think about what the incremental price per ball
here.
And we're going to assume that the price per ball
is constant after you pay for admission, after you
pay this first $5.
So What's.
The price per ball?
Well, when our number of balls increase by 5,
our price increases by $3-- by $3, so plus 3.
So we could say our change in price
over the change in the number of balls is equal to our change
in price is $3, when our change in the balls are 5.
So this right over here, you could
view this as your unit price per ball--
your unit rate of change of price per ball.
This is going to be $0.60 per ball-- $0.60 per ball.
So let's think about constructing an equation that
represents how much we will pay as a function of balls.
And we could write it in function notation, if we like.
We could write price as a function of balls
is going to be equal to.
Well, you're going to start off just paying
$5 just for getting into the place.
So you're going to start off getting $5,
and then you're going to pay more
depending on the number of balls you get.
And you're going to pay $0.60 per ball.
You see that right over here.
When you've got 5 balls, you paid $3.
$3 for 5 balls is the same thing as $0.60 per ball.
So you're then going to pay $0.60 per ball after that.
So this right over here is the equation that represents--
or that's a way of defining price as a function of balls.
And we can also try to graph it.
So let's do that.
So let's actually try to graph it, as well.
So that's our price axis, our vertical axis.
And let's say this is the number of balls axis.
So this is balls, and this is price as a function--
as a function of balls.
So when the balls are 0-- let me do this in a color
I haven't used yet.
When the balls are 0, the price, we've already seen, is $5.
Let's say this is $5, this is $10, this is $15.
So when we have 0 balls, we're going to pay $5.
And then when you see when we have 5 balls-- so let's
say that this is 5 right over here-- when you have 5 balls,
we paid $8, which gets us right around here.
And two points define a line, so this function
is a linear function, and it would look something like this.
And if you ask, what is the slope of this line?
Well, the slope is the rate of change
of the vertical axis with respect to the horizontal axis.
So what is that?
So in the horizontal axis-- and frankly,
we've already calculated this.
When the horizontal axis changed by 5, when your change in b
is equal to 5, your vertical axis-- so your vertical axis.
So your change in price was equal to $3.
So your change in price over your change
in the number of balls, which would
be the change in the vertical axis
over the change in the horizontal axis, which
is the definition of slope.
It's a way of defining the inclination of this line.
That is going to be equal to 3 over 5-- 3 over 5,
which is exactly what we calculated right over here,
just thinking about it in a slightly different way.
Now we've thought about this in multiple ways.
Let's actually try to answer their questions.
Let's figure out which of these actually apply.
So the first is that the relationship is proportional.
So if we had a proportional relationship between the two,
this function would look like this.
It would be p is equal to some constant times
the number of balls.
And by definition, in a proportional relationship--
in a proportional relationship-- when your number of balls is 0,
price would be 0.
And that's not the scenario we have here.
When your balls are 0, you're still paying $5.
So this, this is not-- this is not--
a proportional relationship.
Next statement-- entrance with 10 balls cost $16.
Well, we can test that out.
When b equals 10-- So the price for 10 balls is going to be
equal to 5 plus 0.60 times 10.
Well, this is 5 plus 6, which is equal to 11, not 10.
So this isn't right, either.
When the number of balls increases by 11,
the price increases by $6.60.
So remember, the change in price over the change
in the number of balls is going to be
equal to-- let we get it right in a good color here.
So let me write it here.
The change in price over the change in the number of balls
is always going to be equal to 3/5.
And so, they say number of balls increases by 11.
So 3/5 is equal to what over 11?
So let's think about that a little bit.
So 3/5 is equal to-- well, let's write it this way.
3/5 is the same thing as 0.6, so let
me write this-- 0.6 is equal to the price change over 11.
And I'll just write that as a question mark--
as a question mark over 11.
You multiply both sides by 11, you
get 6.6 is equal to question mark.
So this is absolutely true.
The price increases by $6.60 when the number of balls
increases by 11.
So this one right over here is absolutely true.
And then finally, when the x-axis represents
the number of paint balls, the slope
of the graph of the relationship is 5/3.
Well here, we did represent the x-axis
as the number of paint balls-- or the horizontal axis
as the number of paint balls-- but our slope wasn't 5/3.
It was 3/5.
So this last statement is absolutely not right.
So this is the only one that we can say applies.