- [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.