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Okay, in this video,
I wanna talk more about vertical and

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horizontal stretching and reflecting.

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And I'm gonna definitely talk
about case 3 and case 4,

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which deals with horizontal stretches or
compressions.

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And then 5 or 6 deals with reflections,
maybe we'll get to that one, we'll see.

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Again, the basic idea and, 3 with these
horizontal compressions or stretches.

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If you multiply by a number
bigger than 1 on the inside,

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it's actually gonna squish
your graph together.

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If you multiply by a number between 0 and
1,

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it's gonna actually pull it apart, okay?

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So let's look at at least two examples.

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So again, here's my original graph,
here at the top,

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my blue graph, this is the one I'm
gonna tweak to come up with a new one.

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So it's this little sawtooth function,
-4, 0, -3, -2, -2,

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0, -1, 2, 0, 0, and
then it's this little step function, okay?

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So, what I'm gonna graph now
is the function f of 2x, and

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what this again does,
since I'm multiplying the xs by 2,

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it's actually gonna compress the graph.

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It's gonna basically compress
it by a factor of 2.

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So instead of going out from 0 to -4 and
0 to 4,

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it's now gonna go from 0 to -2 and 0 to 2.

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I think this is the one that always
confuses people as well, cuz, hey,

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you're multiplying by 2,
that should make things bigger.

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But you can't really think about it
like that, or I guess if you do so,

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it's not correct.

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Think about it this way, if I plug -1 in,

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if I let x equal -1,
what would I get on the inside?

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I would get a value of -2, but
according to the original graph,

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it says if you plug -2 in,
you should get 0 out.

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So if I plug -1 in,
I'm gonna get -2 on the inside,

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which should give me an output of 0,
okay, and that's the basic idea.

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You're gonna kind of cut
the x coordinates in half.

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So, originally,
at -2 there was an x coordinate of 0.

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If I cut that x coordinate in half It now
becomes -1, keep the same y coordinate.

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At -4, the original x coordinate is -4,

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if you cut that in half, you'll get -2.

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Okay, originally,
at- 1 it was up here at a y value of 2.

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If you cut -1 in half you're at negative
one-half, and then I'm up here at 2,

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likewise, at negative three-halves,
I'm gonna be down here at -2, okay?

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So it still has the same height,
but everything has

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gotten squished together by a factor of 2,
okay.

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And then that's gonna be the same thing
that happens on the right hand side.

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Instead of extending out
a distance of 0 to 2,

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it's only gonna go out from 0 to 1.

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And then instead of going out from 2 to 4,
it's only gonna go out from now 1 to 2.

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Okay, again, imagine chopping the interval
0 to two-half, you get 0 to 1.

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Imagine chopping
the interval 2 to 4 in half,

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you would get the interval 1 to 2, okay?

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So, again,
notice the heights are the same, but

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it should look definitely
a little more squished together.

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Okay, so let's do another one of these.

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Now, what I'm gonna do is, again,
very much a similar thing.

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I'm gonna multiply now
the inside by one-half, and now,

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if you multiply by a one half, instead
of compressing it by a factor of 2,

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you actually stretch it by a factor of 2.

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So I can't kind of keep
the scale correct on this graph,

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cuz I've got -4 to 4.

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So now the idea is,
instead of getting squished together,

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you're gonna pull out by a factor of 2.

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So that means, I'm gonna go,
instead of from out to -4,

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I'm gonna go all the way out here to -8,

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and instead of +4,
I'm gonna go all the way out here to +8.

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Okay, and now basically,
you do the same thing, originally, at -2,

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that's where I got a 0.

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If you multiply that x coordinate by 2,
cuz we're stretching

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it by a factor of 2,
you're gonna go out to -4, okay?

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And at-1, you originally had
a y coordinate of 2, well,

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now if you multiply that x coordinate
by 2, you're gonna be at -2.

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And then we'll be up here
at a height of 2, likewise,

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at -6, we're gonna be down
here at a height of -2.

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Again, if you multiply that
original x coordinate by 2,

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you're gonna keep that same y value, but

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the x value turns into -6, and
the x value stays the same.

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So again, play connect the dots.

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Obviously, again, I'm not a great artist,
so forgive my artistry.

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Again, it doesn't really
look stretched out because

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just the proportions of my graph.

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But if you had a bigger piece of paper,
I think you would definitely see

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this thing looking more elongated,
so try that for yourself, and

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then we'll go from, whoops,
we'll go all the way out to 4 this time.

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So instead of being this little
horizontal line at -1 from 0 to 2,

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again, now I double that, so it's gonna be
looking like that all the way out to 4.

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And then I jump down to -2 at 4, and that
extends all the way out to the value of 8.

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Okay, so this would be the graph,
again, of f one half of x.

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And I'm definitely gonna put all of
this stuff together in some more

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concrete examples.

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Again, just trying to give you a general
idea of what's going on, let's see.

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I don't know if I can do
the other two real quick.

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We'll save the other two for
one other video.

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So this, again, deals with horizontal
stretches or compressions.

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Next, I'll basically deal with
flips about the x-axis and

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flips about the y-axis, so
look for another video.

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And then again,

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stay tuned I'll do some more general
ones where I do all the compressions,

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and stretching ,and rotating, and
shifting, and all of that stuff combined.