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

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horizontal stretching and reflecting.

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

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

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

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

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

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

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So kind of generally, again.

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

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just again, a general idea.

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

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

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I mean, what's gonna happen?

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

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

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So suppose it was positive originally.

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

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

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

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

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

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Likewise, if it's on the inside.

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

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

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

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

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So just a quick little graph.

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

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

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

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So instead of this first
little salt used part.

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

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

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

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So again, a very loose graph on these two.

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

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

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it will extend over a distance of 2 units.

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

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

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it'll extend over a distance of 4 units.

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

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

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

46
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So, Let's do the other
one here real quick.

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

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concrete ones with functions
that you probably encountered,

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and produce some other graphs as well.

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Just the general idea here.

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

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

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

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get moved over to the right side.

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Okay, there's my little arrow, and

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

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

58
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if you reflect it,
it'll go over from 0 to 2, and

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

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

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

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

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

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it'll now point over to the left.

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

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

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

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very sloppy little graph.

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

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

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

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just turn your paper over, okay?

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

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

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So I hope these help.

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

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reflecting and shifting and
transforming with some different graphs,

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

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Do all that in another video.

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