Slope and Rate of Change

In this video I'm going to do
a bunch of example slope

problems. Just as a bit of
review, slope is just a way of

measuring the inclination
of a line.

And the definition-- we're going
to hopefully get a good

working knowledge of it in this
video-- the definition of

it is a change in y divided
by change in x.

This may or may not make some
sense to you right now, but as

we do more and more examples,
I think it'll make a good

amount of sense.

Let's do this first
line right here.

Line a.

Let's figure out its slope.

They've actually drawn two
points here that we can use as

the reference points.

So first of all, let's
look at the

coordinates of those points.

So you have this point
right here.

What's its coordinates?

Its x-coordinate is 3.

Its y-coordinate is 6.

And then down here, this
point's x-coordinate is

negative 1 and its y-coordinate
is negative 6.

So there's a couple of ways
we can think about slope.

One is, we could look at it
straight up using the formula.

We could say change in y-- so
slope is change in y over

change in x.

We can figure it out
numerically.

I'll in a second draw
it graphically.

So what's our change in y?

Our change in y is literally
how much did our y values

change going from this
point to that point?

So how much did our
y values change?

Our y went from here, y is at
negative 6 and it went all the

way up to positive 6.

So what's this distance
right here?

It's going to be your
end point y value.

It's going to be 6 minus your
starting point y value.

Minus negative 6 or 6 plus
6, which is equal to 12.

You could just count this.

You say one, two, three, four,
five, six, seven, eight, nine,

ten, eleven, twelve.

So when we changed our y value
by 12, we had to change our x

value by-- what was our
change it x over the

same change in y?

Well we went from x is equal
to negative 1 to

x is equal to 3.

Right? x went from
negative 1 to 3.

So we do the end point, which is
3 minus the starting point,

which is negative 1, which
is equal to 4.

So our change in y over change
in x is equal to 12/4 or if we

want to write this in simplest
form, this is the

same thing as 3.

Now the interpretation of this
means that for every 1 we move

over-- we could view this,
let me write it this way.

Change in y over change in x
is equal to-- we could say

it's 3 or we could
say it's 3/1.

Which tells us that for every
1 we move in the positive

x-direction, we're going to move
up 3 because this is a

positive 3 in the y-direction.

You can see that.

When we moved 1 in the x,
we moved up 3 in the y.

When we moved 1 in the x,
we moved up 3 in the y.

If you move 2 in the
x-direction, you're going to

move 6 in the y.

6/2 is the same thing as 3.

So this 3 tells us how quickly
do we go up as we increase x.

Let's do the same thing for the
second line on this graph.

Graph b.

Same idea.

I'm going to use the points
that they gave us.

But really you could use any
points on that line.

So let's see, we have one
point here, which is

the point 0, 1.

You have 0, 1.

And then the starting point-- we
could call this the finish

point-- the starting point right
here, we could view it

as x is negative 6 and
y is negative 2.

So same idea.

What is the change in y given
some change in x?

So let's do the change in
x first. So what is

our change in x?

So in this situation, what is
our change in x? delta x.

We could even count it.

It's one, two, three,
four, five, six.

It's going to be 6.

But if you didn't have a graph
to count from, you could

literally take your finishing
x-position, so it's 0, and

subtract from that your
starting x-position.

0 minus negative 6.

So when your change in x is
equal to-- so this will be 6--

what is our change in y?

Remember we're taking this as
our finishing position.

This is our starting position.

So we took 0 minus negative 6.

So then on the y, we have to
do 1 minus negative 2.

What's 1 minus negative 2?

That's the same thing
as 1 plus 2.

That is equal to 3.

So it is 3/6 or 1/2.

So notice, when we moved in the
x-direction by 6, we moved

in the y-direction
by positive 3.

So our change in y was 3 when
our change in x was 6.

Now, one of the things that
confuses a lot of people is

how do I know what order to--
how did I know to do the 0

first and the negative 6 second
and then the 1 first

and then the negative
2 second.

And the answer is you could've
done it in either order as

long as you keep
them straight.

So you could have also
have done change in y

over change in x.

We could have said, it's equal
to negative 2 minus 1.

So we're using this coordinate
first. Negative 2 minus 1 for

the y over negative 6 minus 0.

Notice this is a negative
of that.

That is the negative of that.

But since we have a negative
over negative, they're going

to cancel out.

So this is going to be equal to
negative 3 over negative 6.

The negatives cancel out.

This is also equal to 1/2.

So the important thing is if
you use this y-coordinate

first, then you have
to use this

x-coordinate first as well.

If you use this y-coordinate
first, as we did here, then

you have to use this
x-coordinate

first, as you did there.

You just have to make sure
that your change in x and

change in y are-- you're
using the same final

and starting points.

Just to interpret this, this is
saying that for every minus

6 we go in x.

So if we go minus 6 in x, so
that's going backwards, we're

going to go minus 3 in y.

But they're essentially
saying the same thing.

The slope of this line is 1/2.

Which tells us for every 2 we
travel in x, we go up 1 in y.

Or if we go back 2 in x,
we go down 1 in y.

That's what 1/2 slope
tells us.

Notice, the line with the 1/2
slope, it is less steep than

the line with a slope of 3.

Let's do a couple
more of these.

Let's do line c right here.

I'll do it in pink.

Let's say that the starting
point-- I'm just picking this

arbitrarily.

Well, I'm using these points
that they've drawn here.

The starting point is at the
coordinate negative 1, 6 and

that my finishing point is at
the point 5, negative 6.

Our slope is going to be-- let
me write this-- slope is going

to be equal to change in
x-- sorry, change in y.

I'll never forget that.

Change in y over change in x.

Sometimes it's said
rise over run.

Run is how much you're moving
in the horizontal direction.

Rise is how much you're moving
in the vertical direction.

Then we could say our change in
y is our finishing y-point

minus our starting y-point.

This is our finishing y-point.

That's our starting y-point,
over our finishing x-point

minus our starting x-point.

If that confuses you, all I'm
saying is, it's going to be

equal to our finishing y-point
is negative 6 minus our

starting y-point, which is 6,
over our finishing x-point,

which is 5, minus our starting
x-point, which is negative 1.

So this is equal to negative
6 minus 6 is negative 12.

5 minus negative 1.

That is 6.

So negative 12/6.

That's the same thing
as negative 2.

Notice we have a negative
slope here.

That's because every time we
increase x by 1, we go down in

the y-direction.

So this is a downward
sloping line.

It's going from the top left
to the bottom right.

As x increases, the
y decreases.

And that's why we got
a negative slope.

This line over here should
have a positive slope.

Let's verify it.

So I'll use the same
points that they

use right over there.

So this is line d.

Slope is equal to
rise over run.

How much do we rise when we go
from that point to that point?

Let's see.

We could do it this way.

We are rising-- I could
just count it out.

We are rising one, two, three,
four, five, six.

We are rising 6.

How much are we running?

We are running-- I'll do it
in a different color.

We're running one, two, three,
four, five, six.

We're running 6.

So our slope is 6/6,
which is 1.

Which tells us that every time
we move 1 in the x-direction--

positive 1 in the x-direction--
we go positive 1

in the y-direction.

For every x, if we go negative
2 in the x-direction, we're

going to go negative 2
in the y-direction.

So whatever we do in x, we're
going to do the same thing in

y in this slope.

Notice, that was pretty easy.

If we wanted to do it
mathematically, we could

figure out this coordinate
right there.

That we could view as our
starting position.

Our starting position is
negative 2, negative 4.

Our finishing position
is 4, 2.

So our slope, change in
y over change in x.

I'll take this point 2 minus
negative 4 over 4 minus

negative 2.

2 minus negative 4 is 6.

Remember that was just this
distance right there.

Then 4 minus negative
2, that's also 6.

That's that distance
right there.

We get a slope of 1.

Let's do another one.

Let's do another couple.

These are interesting.

Let's do the line
e right here.

Change in y over change in x.

So our change in y, when we
go from this point to this

point-- I'll just
count it out.

It's one, two, three, four,
five, six, seven, eight.

It's 8.

Or you could even take this
y-coordinate 2 minus negative

6 will give you that
distance, 8.

What's the change in y?

Well the y-value here is-- oh
sorry what's the change in x?

The x-value here is 4.

The x-value there is 4.

X does not change.

So it's 8/0.

Well, we don't know.

8/0 is undefined.

So in this situation the
slope is undefined.

When you have a vertical
line, you say

your slope is undefined.

Because you're dividing by 0.

But that tells you that you're
dealing probably with a

vertical line.

Now finally let's just
do this one.

This seems like a pretty
straight up vanilla slope

problem right there.

You have that point right
there, which is

the point 3, 1.

So this is line f.

You have the point 3, 1.

Then over here you have the
point negative 6, negative 2.

So our slope would be equal
to change in y.

I'll take this as our ending
point, just so you can go in

different directions.

So our change in y-- now
we're going to go

down in that direction.

So it's negative 2 minus 1.

That's what this distance
is right here.

Negative 2 minus 1, which
is equal to negative 3.

Notice we went down 3.

And then what is going to
be our change in x?

Well, we're going to go
back that amount.

What is that amount?

Well, that is going to be
negative 6, that's our end

point, minus 3.

That gives us that distance,
which is negative 9.

For every time we go back 9,
we're going to go down 3.

Which is the same thing as if
we go forward 9, we're going

to go up 3.

All equivalent.

And we see these cancel out and
you get a slope of 1/3.

Positive 1/3.

It's an upward sloping line.

Every time we run
3, we rise 1.

Anyway, hopefully that was a
good review of slope for you.