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CCSS Math: 8.F.B.4, HSF.IF.C.7, HSF.IF.C.7a

Find the slope of
the line that goes through the ordered pairs
4 comma 2 and negative 3 comma 16. So just as a reminder, slope
is defined as rise over run. Or, you could view that rise
is just change in y and run is just change in x. The triangles here,
that's the delta symbol. It literally means "change in." Or another way, and you
might see this formula, and it tends to be
really complicated. But just remember it's just
these two things over here. Sometimes, slope will be
specified with the variable m. And they'll say that
m is the same thing-- and this is really the
same thing as change in y. They'll write y2 minus
y1 over x2 minus x1. And this notation tends
to be kind of complicated, but all this means
is, is you take the y-value of your endpoint
and subtract from it the y-value of your
starting point. That will essentially
give you your change in y. And it says take the
x-value of your endpoint and subtract from that the
x-value of your starting point. And that'll give
you change in x. So whatever of
these work for you, let's actually figure out
the slope of the line that goes through these two points. So we're starting
at-- and actually, we could do it both ways. We could start at this
point and go to that point and calculate the slope or
we could start at this point and go to that point
and calculate the slope. So let's do it both ways. So let's say that our starting
point is the point 4 comma 2. And let's say that our endpoint
is negative 3 comma 16. So what is the change
in x over here? What is the change in
x in this scenario? So we're going from
4 to negative 3. If something goes
from 4 to negative 3, what was it's change? You have to go
down 4 to get to 0, and then you have to go down
another 3 to get to negative 3. So our change in x
here is negative 7. Actually, let me
write it this way. Our change in x is equal
to negative 3 minus 4, which is equal to negative 7. If I'm going from 4 to
negative 3, I went down by 7. Our change in x is negative 7. Let's do the same thing
for the change in y. And notice, I implicitly
use this formula over here. Our change in x was this value,
our endpoint, our end x-value minus our starting x-value. Let's do the same thing
for our change in y. Our change in y. If we're starting at
2 and we go to 16, that means we moved up 14. Or another way you
could say it, you could take your ending
y-value and subtract from that your starting y-value
and you get 14. So what is the slope over here? Well, the slope is just
change in y over change in x. So the slope over
here is change in y over change in x, which
is-- our change in y is 14. And our change in
x is negative 7. And then if we want to simplify
this, 14 divided by negative 7 is negative 2. Now, what I want
to show you is, is that we could have done
it the other way around. We could have made
this the starting point and this the endpoint. And what we would have
gotten is the negative values of each of these, but then
they would've canceled out and we would still
get negative 2. Let's try it out. So let's say that our start
point was negative 3 comma 16. And let's say that our
endpoint is the 4 comma 2. 4 comma 2. So in this situation,
what is our change in x? Our change in x. If I start at negative
3 and I go to 4, that means I went up 7. Or if you want to
just calculate that, you would do 4 minus negative 3. 4 minus negative 3. But needless to say,
we just went up 7. And what is our change in y? Our change in y over here,
or we could say our rise. If we start at 16 and we end at
2, that means we went down 14. Or you could just say 2
minus 16 is negative 14. We went down by 14. This was our run. So if you say rise
over run, which is the same thing as change
in y over change in x, our rise is negative 14
and our run here is 7. So notice, these are
just the negatives of these values from
when we swapped them. So once again, this is
equal to negative 2. And let's just visualize this. Let me do a quick
graph here just to show you what a downward
slope would look like. So let me draw our two points. So this is my x-axis. That is my y-axis. So this point over
here, 4 comma 2. So let me graph it. So we're going to go
all the way up to 16. So let me save some space here. So we have 1, 2, 3, 4. It's 4 comma-- 1, 2. So 4 comma 2 is right over here. 4 comma 2. Then we have the point
negative 3 comma 16. So let me draw that over here. So we have negative 1, 2, 3. And we have to go up 16. So this is 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16. So it goes right over here. So this is negative 3 comma 16. Negative 3 comma 16. So the line that
goes between them is going to look
something like this. Try my best to draw a
relatively straight line. That line will keep going. So the line will keep going. So that's my best attempt. And now notice, it's
downward sloping. As you increase an x-value,
the line goes down. It's going from the top
left to the bottom right. As x gets bigger,
y gets smaller. That's what a downward-sloping
line looks like. And just to visualize our
change in x's and our change in y's that we dealt with
here, when we started at 4 and we ended at-- or when
we started at 4 comma 2 and ended at
negative 3 comma 16, that was analogous to starting
here and ending over there. And we said our change
in x was negative 7. We had to move back. Our run we had to move in
the left direction by 7. That's why it was a negative 7. And then we had to move
in the y-direction. We had to move in the
y-direction positive 14. So that's why our
rise was positive. So it's 14 over negative
7, or negative 2. When we did it the other way,
we started at this point. We started at this point,
and then ended at this point. Started at negative 3, 16
and ended at that point. So in that situation,
our run was positive 7. And now we have to go
down in the y-direction since we switched the
starting and the endpoint. And now we had to
go down negative 14. Our run is now positive 7 and
our rise is now negative 14. Either way, we got
the same slope.