Graph Transformations about the X-axis and Y-axis

Okay, in this video, I just wanna finish
off kind of the general vertical and

horizontal stretching and reflecting.

In this one,
I'm gonna talk about reflecting, and

so what we're gonna look at
are conditions 5 and 6 here.

The idea is, if you basically plug
a big O negative sign out in front of

your formula, it's gonna reflect
your graph about the x-axis.

If you replace all of your x's with -x's,
so, you kinda change, well,

not necessarily change the sign,
you're gonna reflect about the y-axis.

So kind of generally, again.

So this will be probably the easiest
of all of them, kind of a general,

just again, a general idea.

Again, if the negatives on the outside,
it flips it about the x-axis.

And if you think about it,
I mean, that really makes sense.

I mean, what's gonna happen?

You plug x in, you get some y value out.

That's what f(x) is, okay?

So suppose it was positive originally.

Well, if you plug a negative out front,
what's it gonna do?

It's going to change the sign and
reflect it down to the other side.

So, really that's what it's doing.

The negative out front changes
the sign on all the y values,

and that has the effect of
reflecting it about the x-axis.

Likewise, if it's on the inside.

[COUGH] You reflect
the graph about the y-axis.

And now, the idea is,
whatever you plugged in for 2 originally,

you're gonna get that same
output at now -2, and

that's gonna have the effect of
shifting it about the y-axis.

So just a quick little graph.

Let's graph, y equals negative f(x),
here real quick.

So again, all it does is, it kinda
preserves its sort of general shape.

All that happens is, again,
you're just flipping it.

So instead of this first
little salt used part.

It's now instead of going up,
it's gonna go down To -2..

So there's the original part
that was up is now down.

This original part that was down,
will now go up, okay?

So again, a very loose graph on these two.

And instead of being down here at (-1,2),
again,

now if that point flips,
it's gonna be up here +1, and

it will extend over a distance of 2 units.

And then if this portion
down here that was at (2,-2)

we will now flip up to the top,
and be up here at 2 and

it'll extend over a distance of 4 units.

Okay, so that's the graph of
y equals negative f of x.

Again, all it does is,
stuff that was above the x-axis

gets flipped below on the x-axis and
vice versa, okay?

So, Let's do the other
one here real quick.

So again, very general,
I'm gonna do some more kinda

concrete ones with functions
that you probably encountered,

and produce some other graphs as well.

Just the general idea here.

Okay, last but not least, again we said,
if the negatives on the inside all

that does is reflects
a the graph about the x-axis.

So now this portion that was
originally on the left side is gonna

get moved over to the right side.

Okay, there's my little arrow, and

this portion that was on the bottom
will get moved over to the left side.

So, if- Instead of going
over from 0 to -2,

if you reflect it,
it'll go over from 0 to 2, and

it'll still go up and then back down,
up here to height of 2.

And then if you think about the other
part, it was going down from -2 to -4.

So now, it'll go down and
come back up between +2 and +4, okay?

And lastly,
the part that was on the right side, well,

instead of going to the right like
it normally did, if it reflects,

it'll now point over to the left.

And then we'll extend over to
an x coordinate of negative 2,

and then we'll jump down here, and we'll
extend over until we get to negative 4.

Okay, so again,
this is supposed to be a flat little line,

very sloppy little graph.

But again, the idea is
the shape is exactly preserved.

The only thing that happens
is you're just reflecting.

I mean, basically,
if you have this on a piece of paper,

just turn your paper over, okay?

And that new graph that you see,
is gonna be f of negative x, okay?

So, that's what the new graph
will look like, all right?

So I hope these help.

Again, I'm gonna put all of this together,
the all the stretching and

reflecting and shifting and
transforming with some different graphs,

maybe trig functions, exponential
functions, x squared absolute value of x.

Do all that in another video.

So, feel free to dig around for
that as well.