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# Graphing proportional relationships: unit rate

Sal graphs the equation of a line that represents a proportional relationship given a unit rate. Created by Sal Khan.

Video transcript

Graph the line that represents
a proportional relationship between y and x with
a unit rate 0.4. That is, a change
of one unit in x corresponds to a change
of 0.4 units in y. And they also ask
us to figure out what the equation of
this line actually is. So let me get my scratch pad
out and we could think about it. So let's just think about some
potential x and y values here. So let's think about some
potential x and y values. So when we're thinking about
proportional relationships, that means that y is going to be
equal to some constant times x. So if we have a
proportional relationship, if you have zero x's, it doesn't
matter what your constant here is, you're going
to have zero y's. So the point 0, 0
should be on your line. So if this is the point 0, 0,
this should be on my line right over there. Now, let's think about what
happens as we increase x. So if x goes from 0
to 1, we already know that a change of 1
unit in x corresponds to a change of 0.4 units in y. So if x increases by 1, then
y is going to increase by 0.4. It's not so easy to
graph this 1 comma 0.4. The 0.4 is hard to graph on this
little tool right over here. So let's try to get this
to be a whole number. So then when x
increases another 1, y is going to
increase by 0.4 again. It's going to get to 0.8. When x increases
again by 1, then y is going to increase
by 0.4 again. It's going to get to 1.2. If x increases again, y is
going to increase by 0.4 again. So just to 1.6. Notice, every time x
is increasing by 1, y is increasing by 0.4. That's exactly what
they told us here. Now, if x increases
by 1 again to 5, then y is going to
increase 0.4 to 2. And I like this point because
this is nice and easy to graph. So we see that the point 0,
0 and the point 5 comma 2 should be on this graph. And I could draw it. And I'm going to do it on
the tool in a second as well. So it'll look
something like this. And notice the slope of
this actual graph over here. Notice the slope of
this actual graph. If our change in x is 5. So notice, here our
change in x is 5. Our change in x is 5. You see that as well. When you go from 0 to 5,
this change in x is 5. Change in x is equal to 5. What was our
corresponding change in y? Well, our corresponding change
in y when our change in x was 5, our change
in y was equal to 2. And you see that here,
when x went from 0 to 5, y went from 0 to 2. So our change in y in this
circumstance is equal to 2. So our slope, which
is change in y over change in x, is
the rate of change of your vertical
axis with respect to your horizontal axis, is
going to be equal to 2 over 5, or 2/5. Which if you wrote it as
a decimal is equal to 0.4. So this right over
here is your slope. So I'm going to do
this with the tool. But first, let's
also think about what the equation of this
line is going to be. Well, we know that y is equal
to some constant times x. And we know that
the point 5, 2 is on this line right over here. So we could say, well, when x
is equal to 5, y is equal to 2. Or, when y equals 2, we
have k times 5, or k is equal to-- dividing both sides
by 5, you can't see that. If I divide both sides by 5,
I'm left with k is equal to 2/5. Which makes sense. We're used to seeing this. When we have y is equal
to something times x, this something right over
here is going to be our slope. So the equation of the
line is y is equal to 0.4x. So let's fill this in. Let's actually do
the exercise now. So we had two points,
one was the point 0, 0. When x is 0, y is 0. And when x is 5, y
is 0.4 times that. So it's y is equal to 2. And we said the equation is
y is equal to 0.4 times x. So let's check our answer.