Introduction to slope

- [Voiceover] As we start to graph lines,

we might notice that they're
differences between lines.

For example, this pink or
this magenta line here,

it looks steeper than this blue line.

And what we'll see is
this notion of steepness,

how steep a line is, how
quickly does it increase

or how quickly does it decrease,

is a really useful idea in mathematics.

So ideally, we'd be
able to assign a number

to each of these lines
or to any lines that

describes how steep it is,

how quickly does it increase or decrease?

So what's a reasonable way to do that?

What's a reasonable way to assign a number

to these lines that
describe their steepness?

Well one way to think
about it, could say well,

how much does a line increase

in the vertical direction

for a given increase in
the horizontal direction?

So let's write this down.

So let's say if we an increase

increase, in vertical,

in vertical,

for a given increase in horizontal

for a given increase

a given increase

in horizontal.

So, how can this give us a value?

Well let's look at that
magenta line again.

Now let's just start at an arbituary point

in that magenta line.

But I'll start at a point

where it's going to be easy for me

to figure out what point we're at.

So if we were to start right here,

and if I were to increase
in the horizontal

direction by one.

So I move one to the right.

To get back on the line, how much

do I have to increase in
the vertical direction?

Well I have to increase in
the vertical direction by two.

By two.

So at least for this magenta line,

it looks like our increase in vertical is

two, whenever we have an increase in

one in the horizontal direction.

Let's see, does that
still work if I were to

start here, instead of
increasing the horizontal

direction by one, if I were increase in

the horizontal direction...

So let's increase by three.

So now, I've gone plus three

in the horizontal direction,

then to get back on the line,

how much do I have to increase
in the vertical direction?

I have to increase by one,
two, three, four, five, six

I have to increase by six.

So plus six.

So when I increase by three

in the horizontal direction,

I increase by six in the vertical.

We were just saying,
hey, let's just measure

how much to we increase in vertical

for a given increase in the horizontal?

Well two over one is just two

and that's the same
thing as six over three.

So no matter where I start on this line,

no matter where I start on this line,

if I take and if I increase
in the horizontal direction

by a given amount,

I'm going to increase twice as much

twice as much

in the vertical direction.

Twice as much in the vertical direction.

So this notion of this
increase in vertical

divided by increase in horizontal,

this is what mathematicians
use to describe

the steepness of lines.

And this is called the slope.

So this is called the slope of a line.

And you're probably
familiar with the notion

of the word slope being
used for a ski slope,

and that's because a ski slope
has a certain inclination.

It could have a steep
slope or a shallow slope.

So slope is a measure for
how steep something is.

And the convention is, is
we measure the increase

in vertical for a given
in increase in horizontal.

So six two over one is
equal to six over three

is equal to two, this
is equal to the slope

of this magenta line.

So let me write this down.

So this slope right over
here, the slope of that line,

is going to be equal to two.

And one way to interpret that,

for whatever amount you increase in the

horizontal direction,
you're going to increase

twice as much in the vertical direction.

Now what about this blue line here?

What would be the slope of the blue line?

Well, let me rewrite another
way that you'll typically

see the definition of slope.

And this is just the
convention that mathematicians

have defined for slope

but it's a valuable one.

What is are is our change in vertical

for a given change in horizontal?

And I'll introduce a new notation for you.

So, change in vertical,

and in this coordinate,

the vertical is our Y coordinate.

divided by our change in horizontal.

And X is our horizontal coordinate

in this coordinate plane right over here.

So wait, you said change in but then you

drew this triangle.

Well this is the Greek letter delta.

This is the Greek letter delta.

And it's a math symbol used
to represent change in.

So that's delta, delta.

And it literally means, change in Y,

change in Y,

divided by change in X,

change in X.

So if we want to find the
slope of the blue line,

we just have to say, well
how much does Y change

for a given change in X?

So, the slope of the blue line.

So let's see, let me do it this way.

Let's just start at some point here.

And let's say my X changes by two

so my delta X is equal to positive two.

What's my delta Y going to be?

What's going to be my change in Y?

Well, if I go by the right by two,

to get back on the line,

I'll have to increase my Y by two.

So my change in Y is also
going to be plus two.

So the slope of this blue line,

the slope of the blue line,

which is change in Y over change in X.

We just saw that when our
change in X is positive two,

our change in Y is also positive two.

So our slope is two divided by two,

which is equal to one.

Which tells us however much we increase in

X, we're going to increase
the same amount in Y.

We see that, we increase
one in X, we increase

one in Y.

Increase one in X, increase one in Y.

>From any point on the line,
that's going to be true.

You increase three in X,

you're going to increase three in Y.

It's actually true the other way.

If you decrease one in X,

you're going to decrease one in Y.

If you decrease two in X,

you're going to decrease two in Y.

And that makes sense from
the math of it as well

Because if you're change
in X is negative two,

that's what we did right over here,

our change is X is negative two,

we went two back,

then your change in Y is going
to be negative two as well.

Your change in Y is
going to be negative two,

and negative two divided by negative two,

is positive one, which
is your slope again.