Algebra: graphing lines 1

Welcome to the presentation
on graphing lines.

Let's get started.

So let's say I had the
equation-- let me make sure

that this line doesn't
show up too thick.

Let's say I had the equation--
why isn't that showing up?

Let's see.

Oh, there you go.

y is equal to 2x plus 1.

So this is giving a
relationship between x and y.

So say x equals 1, then y would
be 2 times 1 plus 1 or 3.

So for every x that we can
think of we can think

of a corresponding y.

So let's do that.

If we said that-- put
a little table here.

x and y.

And let's just throw out
some random numbers for x.

If x was let's say, negative
1, then y would be 2 times

negative 1, which
is negative 2.

Plus 1, which would
be negative 1.

If x was 0 that's easy.

It'd be 2 times 0, which is 0.

Plus 1, which is 1.

If x was 1, y would be
2 times 1, which is 2.

Plus 1, which is 3.

If x was 2, then I think
you get the idea here.

y would be 5.

And we could keep on going.

Obviously, there are an
infinite number of x's we

could choose and we could
pick a corresponding y.

So now you see we have a
little table that gives the

relationships between x and y.

What we can do now is actually
graph those points on

a coordinate axis.

So let me see if I can draw
this somewhat neatly.

I'll use this line so
I get straight lines.

Okay, that's pretty good.

Okay, let me draw some
coordinate points.

So let's say that's 1,
that's 2, that's 3.

This is negative 1,
negative 2, negative 3.

So this is the x-axis.

We have 1, 2, 3.

Notice we could keep going.

1, 2, 3, and this
is the y-axis.

And this would be 1,
2, 3, and so on.

This would be negative 1.

I think you get the idea.

So we can graph each
of these points.

So if we have the point x is
negative 1, y is negative 1.

So x, we go along the x-axis
here, and we go to x is

equal to negative 1.

Then we go to y is equal to
negative 1, so the point

would be right here.

Hope that makes sense to you.

That's the point.

I'll label it: negative
1 comma negative 1.

It's a little messy.

That says negative 1
comma negative 1.

That point I just
x'ed right there.

Let's do another one.

That's this point.

I'll do it in a different
color this time.

Let's say we had the
point 0 comma 1.

Well, x is 0, which is here.

And y is 1, so that
point is right there.

Let's do one more.

If we have the point 1 comma 3.

Well, 1 comma 3, x is
1 and we have y is 3.

So we have the
point right there.

Hope that's making
sense for you.

And we could keep graphing
them, but I think you see here,

and especially if I had drawn
this a little bit neater, that

these points are
forming a line.

Let me draw that line in.

The line looks
something like this.

That's not a good line.

Let me do it better than that.

The line looks
something like this.

You see that?

Well, that's actually a pretty
bad line that I just drew.

So it would be a line that goes
through-- let me change tools.

It'd be a line that goes
through here, through

here, and through here.

I don't know if I'm making
this clear at all.

Let me make these
points a little bit.

You see the line will go
through all of these points,

but it will also go through the
point 2 comma 5, which will

be up here some place.

For any x that you can think
of, if you had x is equal to

10,380,000,000 the
corresponding y will

also be on this line.

So this pink line, and it
keeps going on forever, that

represents every possible
combination of x's and y's that

will satisfy this equation.

And of course, x doesn't
have to just be whole

numbers or integers.

x could be pi-- 3.14159.

In which case it would be
someplace here and in which

case y would be 2 pi plus 1.

So every number that x could
be there's a corresponding y.

Let's do another 1.

So if I had the equation y is
equal to-- that's an ugly y.

y is equal to negative 3x plus 5.

Well, I'm going to draw it
quick and dirty this time.

So that's the x-axis.

That's the y-axis.

Let's put some values here.

x and y.

Let's say if x is negative
1, then negative 1 times

negative 3 is 3 plus 5 is 8.

If x is 0, then y is 5.

That's pretty easy.

If x is 1, negative 3
times 1 is negative 3.

Then y is 2.

If x is 2, negative 3
times 2 is negative 6.

Then y is 1.

Is that right?

Negative 6-- no, no.

Negative 1.

I knew something
was wrong there.

So let's graph some
of these points.

So when x = -1, and I'm just kind of approximating, when x = -1,

y = -8, so that point will be someplace around here.

And there's a whole module I'm
graphing coordinates if you're

finding the graphing a
coordinate pair to be

a little confusing.

Oh, wait.

I just made a mistake.

When x is negative 1, y is 8.

Not negative 8, so
ignore this right here.

When x is negative
1, y is positive 8.

So y being up here someplace.

When x is 0, y is 5.

So it'd be here someplace.

When x is 1, y is 2.

So it's like here.

When x is 2, y is negative 1.

So as you can see-- and
I've approximated it.

If I had graphing paper or if I
had a better drawn chart you

could have seen it and it would
have been exactly right.

I think this line
will do the job.

That every point that satisfies
this equation actually

falls on this line.

And something interesting
here I'll point out.

You notice that this line
it slopes downwards.

It goes from the top left
to the bottom right.

While the line we had drawn
before had gone from the

bottom left to the top right.

Is there anything about this
equation that seems a little

bit different than the last?

I'll give you a little
bit of a hint.

This number-- the negative 3,
or you could say that the

coefficient on x-- that
determines whether the line

slopes upward, or the line
slows downward, and it tells

you also how steep the line is.

And that actually,
negative 3 is the slope.

And I'm going to do a whole
nother module on slope.

And this number here is
called the y-intercept.

And that actually tells
you where you're going

to intersect the y-axis.

And it turns out here,
that you intersect the

axis at 0 comma 5.

Let's do one more real fast.

y is equal to 2--
we already did 2x.

y is equal to 1/2 x
plus 2 So real fast.

x and y.

And you only need two
points for a line, really.

So you could just say
let's say, x equals 0.

That's easy. y equals 2.

And if x equals 2
then y equals 3.

So before when we were doing 3
and 4 points that was just to

kind of show you, but you
really just need two

points for a line.

So 0 comma 1 2.

So that's on there.

And then 1, 2 comma 3.

So it's there.

So the line is going to
look something like this.

So notice here, once again,
we're upward sloping and that's

because this 1/2 is positive.

But we're not sloping-- we're
not moving up as quickly as

when we had y equals
2x. y equals 2x looked

something like this.

It was sloping up much,
much, much faster.

I hope I'm not confusing you.

And then the y intercept of
course is at 0 comma 2,

which is right here.

So if you ever want to graph
a line it's really easy.

You have to just try out some
points and you can graph it.

And now in the next module I'm
going to show you a little bit

more about slope and
y-intercept and you won't

even have to do this.

But this gives you good
intuitive feel, I think,

what a graph of a line is.

I hope you have fun.