Horizontal And Vertical Graph Stretches and Compressions Part 1 of 3

All right, in this video,

I'm gonna talk about some more
graph transformation stuff.

I'm gonna talk about vertical and
horizontal stretching and reflecting.

So a couple things here, I've kind of
written them all down generically.

It's definitely gonna take me more than
one video to get through all this stuff.

So the first one is, if you multiply the
function, basically by a number out front,

it has the effect of stretching
it vertically, up and down, okay?

If the number is bigger than 1,
it's gonna stretch it.

If the number you're multiplying it
by is a fraction between 0 and 1,

it's going to squish it together.

So those are the first two conditions.

The next two conditions maybe
we can even label them.

So conditions 3 and conditions 4,
it basically say if you multiply

the x by a number, if you multiply
it by a number bigger than 1.

What it basically does is it
compresses it horizontally, okay, so

kind of in and out to the left and right.

The next condition says, basically,
if you multiply it by a number

a fraction between 0 and 1,
it has the effect of stretching it, okay?

So it stretches it out.

And the last two conditions are just
conditions that says if you multiply out

front by a negative number,
it flips the graph about the x-axis.

And if you plug a negative inside,
you flip it about the y-axis, okay?

So I'm gonna do one real generically here

I've got a real generic one
here that I'm gonna do.

And so this blue function
is my function f of x, and

these are all supposed
to be straight lines

Okay, eventually, I'm going to do some
other videos where I put all of this stuff

together and some more complicated ones,
but for now, just kind of the bare bones.

Okay, so it's kind of this little
sawtooth function on the left hand side.

It's got an x intercept at -4,
when it's at -3, it's down here at -2.

When it's at -2, it goes through 0, when
it's at -1, it's up here at positive 2.

When it's at 0, it's at 0,
it should be an open circle here at -1.

And then it's just supposed to be
a flat line extending over to 2,

and then it jumps down, it starts at 2,
-2, and again is a flat line from 4 to 2.

So what I'm gonna do is I'm gonna graph
all six of these functions that I have

down here on the bottom left.

So 2 times the function a half,
the 2 inside,

the half inside, and
a negative out front, and then f of -x.

So let me see if I can't do them all here.

All right, so
let's do the first one first.

Okay, I'm gonna try to graph what 2
times this function would look like

based on the original function.

Again, there's no way I'm gonna be able
to get through all of these in one video.

But basically what happens is,
if you take 2 times the function,

what you're really doing
is you're multiplying.

Think about f of x as being y,

you're multiplying all
the original y coordinates by 2.

So instead of, at -1,
instead of being at the y value of 2,

you're now multiplying it by 2, so that
you'll be up here at a height of 4, okay?

If you think about the x coordinate of -2,
its original y coordinate is 0.

If you multiply that by 2,
well, you're still at 0.

At -3 originally your y coordinate was -2,
but now if you multiply it again by 2,

you're gonna be down here
at a y coordinate of -4.

Lemme see if I can squeeze it in here,
so here's -3.

Again, a real rough graph,
sorry, I'm not an artist.

And then at -4,
my original y-coordinate is 0,

if I multiply it by 2, well,
I am still at 0, okay?

And then it's still gonna have that
sawtooth shape associated with it.

Okay, again It doesn't really look like
it, but it's supposed to be stretched out,

look where the y coordinate is now.

This is -1,4 and
down here at the bottom left,

this is -3,-4,
it's still a y intercept of -4.

On the right hand side,
it's gonna have the same effect,

it's gonna multiply the original y
coordinates all by this value of 2.

So instead of being at -1,
it'll now be at -2.

It's It'll still extend over the same
distance and then it'll jump and

again instead of being at -2
it'll now be down here at -4 and

will extend over a distance
of 4 units as well.

Okay, so that's the basic idea here
when you multiply the function by a 2.

It just stretches everything
out by a factor of 2, okay?

Sorry, I'm trying to bring it back
into focus here for a second, okay?

So hopefully that's a little better.

All right, let me see if I can at
least do one more in this video.

Instead of multiplying it by 2,

suppose we multiply it all
by one half out front, okay?

It's gonna have the exact same, well,

I guess obviously not the exact same
effect, but the idea is the same.

It's still gonna go out the same distance,
it's still gonna go out to -4.

Okay, and it's still going to extend
out to 4 on the right hand side, but

the difference now being is I'm now
multiplying the y coordinates by one half,

so all the original y coordinates are now
going to get multiplied by a half.

So instead of being at -1,2,

I'm only going to go up a distance
of 1 now, so I'll be at -1,1.

The original y coordinate associated
with -2 was 0, so I'll still be there.

The original y coordinate
associated with -3 was -2.

Again, if I multiply that by half,
I'm at -1.

The original y coordinate
associated with -4 is 0,

if you multiply that by a half,
again, you're at 0.

So I think this one does look a little
bit better maybe than my last one.

It does look, I think,
a little more squished.

So I multiply the function
by a number smaller,

basically a fraction between between 0 and
1.

And it's gonna squish the graph,
it's gonna compress it vertically.

On the right hand side, the same thing's
gonna happen, instead of going down to

a height of -1, now we'll simply be
down at a height of -1/2 open circle.

It'll still be a flat line over to 2,
okay, it's getting a little covered up.

And then at 2, instead of being
at the y coordinate of -2,

we'll now jump down here to -1.

So this is 2,-1, and then that line
will extend all the way over to 4.

Okay, and that would be the the graph
of one half times the function,

all right, so I'll try to do
the other two in one other video.

So dig around for that,
I'll do f of where the 2 is inside,

where the one half is inside, and

then we'll do a reflecting about the x and
y axis as well.