100, you end up with about five thousand features.

If f(n) is in little o(g(n)), that's kind of like saying f(n) is strictly less than g(n).

But I think you get it at this point.

Our equation from before, 2n 2 n just becomes Θ(n).

This is equal to plus or minus b over a x.

So a hyperbola, if that's the x, that's the y axis, it has two asymptotes.

That's not the exact word, but it would look something like that where it's below the asymptote here.

It's going to come down and then go off, and never quite touch the asymptote, but approach it.

When you're taking the positive square root, we're always going to be slightly below the asymptote.

And you notice that the sigmoid function, well, it asymptotes at one, and asymptotes at zero as

And here, it's going to keep getting closer and closer to the asymptote on that side and then on that side.

Asymptotes of a Hyperbola

So x is equal to 1 y is equal to minus 1.

So that's one asymptote.

So we're always going to be a little bit above the asymptote.

Okay, lets move on the, the third and final guiding principle.

It just gets closer and closer and closer, arbitrarily close to the asymptote.

The two asymptotes of a hyperbola.

Oh woops, not using my line tool.

Let's draw our asymptotes.

Go out there.

I know you can't read that.

Which essentially b over a x, plus or minus b over a x.

So that tells us that were always going to be slightly below the asymptote on the positive square root, and we're always going to be slightly above the asymptote on the negative square root.

Because it will never, a hyperbola will never cross the asymptotes.

Hyperbola by Asymptotes Point

So that line would look like that.

Good enough.

Got to have a steady hand.

And then go like that.

So it'll go 3.

The asymptote is this thing, but we're always going to be slightly below it.

So it's going to get really close to the asymptote, and then go off, and go off in that direction.

So it's approximately linear around F of X equals zero, but it levels off to zero and one as we go to the extremes.

Let's do that.

So if these are the axes, then if I were to draw let me draw some asymptotes.

So the two asymptotes this is centered at the origin, it hasn't been shifted are going to be 16 over 9, so it's going to be fairly steep asymptotes, it's going to look something

Construct a conic with this asymptote

Construct the asymptotes of this conic

Square root of 4 over square root of 9, times x.

And this would've been the two lines of the asymptotes if we were centered at 0.

Well, as the universe expands, time goes to infinity, the scale factor will go to infinity, and one over a scale factor will go to zero.

The positive square root is the top line.

It's going to look something like that.

Let me see if I can draw it.