Okay, in this video, I wanna talk more about vertical and
horizontal stretching and reflecting.
And I'm gonna definitely talk about case 3 and case 4,
which deals with horizontal stretches or compressions.
And then 5 or 6 deals with reflections, maybe we'll get to that one, we'll see.
Again, the basic idea and, 3 with these horizontal compressions or stretches.
If you multiply by a number bigger than 1 on the inside,
it's actually gonna squish your graph together.
If you multiply by a number between 0 and 1,
it's gonna actually pull it apart, okay?
So let's look at at least two examples.
So again, here's my original graph, here at the top,
my blue graph, this is the one I'm gonna tweak to come up with a new one.
So it's this little sawtooth function, -4, 0, -3, -2, -2,
0, -1, 2, 0, 0, and then it's this little step function, okay?
So, what I'm gonna graph now is the function f of 2x, and
what this again does, since I'm multiplying the xs by 2,
it's actually gonna compress the graph.
It's gonna basically compress it by a factor of 2.
So instead of going out from 0 to -4 and 0 to 4,
it's now gonna go from 0 to -2 and 0 to 2.
I think this is the one that always confuses people as well, cuz, hey,
you're multiplying by 2, that should make things bigger.
But you can't really think about it like that, or I guess if you do so,
it's not correct.
Think about it this way, if I plug -1 in,
if I let x equal -1, what would I get on the inside?
I would get a value of -2, but according to the original graph,
it says if you plug -2 in, you should get 0 out.
So if I plug -1 in, I'm gonna get -2 on the inside,
which should give me an output of 0, okay, and that's the basic idea.
You're gonna kind of cut the x coordinates in half.
So, originally, at -2 there was an x coordinate of 0.
If I cut that x coordinate in half It now becomes -1, keep the same y coordinate.
At -4, the original x coordinate is -4,
if you cut that in half, you'll get -2.
Okay, originally, at- 1 it was up here at a y value of 2.
If you cut -1 in half you're at negative one-half, and then I'm up here at 2,
likewise, at negative three-halves, I'm gonna be down here at -2, okay?
So it still has the same height, but everything has
gotten squished together by a factor of 2, okay.
And then that's gonna be the same thing that happens on the right hand side.
Instead of extending out a distance of 0 to 2,
it's only gonna go out from 0 to 1.
And then instead of going out from 2 to 4, it's only gonna go out from now 1 to 2.
Okay, again, imagine chopping the interval 0 to two-half, you get 0 to 1.
Imagine chopping the interval 2 to 4 in half,
you would get the interval 1 to 2, okay?
So, again, notice the heights are the same, but
it should look definitely a little more squished together.
Okay, so let's do another one of these.
Now, what I'm gonna do is, again, very much a similar thing.
I'm gonna multiply now the inside by one-half, and now,
if you multiply by a one half, instead of compressing it by a factor of 2,
you actually stretch it by a factor of 2.
So I can't kind of keep the scale correct on this graph,
cuz I've got -4 to 4.
So now the idea is, instead of getting squished together,
you're gonna pull out by a factor of 2.
So that means, I'm gonna go, instead of from out to -4,
I'm gonna go all the way out here to -8,
and instead of +4, I'm gonna go all the way out here to +8.
Okay, and now basically, you do the same thing, originally, at -2,
that's where I got a 0.
If you multiply that x coordinate by 2, cuz we're stretching
it by a factor of 2, you're gonna go out to -4, okay?
And at-1, you originally had a y coordinate of 2, well,
now if you multiply that x coordinate by 2, you're gonna be at -2.
And then we'll be up here at a height of 2, likewise,
at -6, we're gonna be down here at a height of -2.
Again, if you multiply that original x coordinate by 2,
you're gonna keep that same y value, but
the x value turns into -6, and the x value stays the same.
So again, play connect the dots.
Obviously, again, I'm not a great artist, so forgive my artistry.
Again, it doesn't really look stretched out because
just the proportions of my graph.
But if you had a bigger piece of paper, I think you would definitely see
this thing looking more elongated, so try that for yourself, and
then we'll go from, whoops, we'll go all the way out to 4 this time.
So instead of being this little horizontal line at -1 from 0 to 2,
again, now I double that, so it's gonna be looking like that all the way out to 4.
And then I jump down to -2 at 4, and that extends all the way out to the value of 8.
Okay, so this would be the graph, again, of f one half of x.
And I'm definitely gonna put all of this stuff together in some more
concrete examples.
Again, just trying to give you a general idea of what's going on, let's see.
I don't know if I can do the other two real quick.
We'll save the other two for one other video.
So this, again, deals with horizontal stretches or compressions.
Next, I'll basically deal with flips about the x-axis and
flips about the y-axis, so look for another video.
And then again,
stay tuned I'll do some more general ones where I do all the compressions,
and stretching ,and rotating, and shifting, and all of that stuff combined.