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- [Voiceover] Hello everyone.

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So what I'd like to do here is to describe

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how we think about
three-dimensional graphs.

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Three-dimensional graphs
are a way that we represent

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certain kind of multi-variable function

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that kind of has two inputs,

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or rather a two-dimensional input,

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and then one-dimensional
of output of some kind.

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So the one that I have pictured here

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is f of (x, y) equals x
squared plus y squared.

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And before talking
exactly about this graph,

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I think it would be helpful, by analogy,

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we take a look at the
two-dimensional graphs and

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kinda remind ourselves how those work,

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what it is that we do,
because, it's pretty much

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the same thing in three-dimensions,

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but it takes a little bit
more of visualization.

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So the two-dimensional graphs,

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they have some kind of function, you know,

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let's see you have f of
x is equal to x squared,

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and anytime you visualizing
a function, you trying to

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understand the relationship between

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the inputs and the outputs.

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And here those are both just numbers,

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so you know you input a number like two,

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and it's gonna output four,

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you know you input negative
one it's gonna output one.

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And you're trying to
understand all the possible

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input-output pairs.

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And the fact that we can do this,

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that we can get a pretty
good intuitive feel for

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every possible input-output
pair is pretty incredible,

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the way we go about this
with graphs is you think

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we just plotting these
actual pairs, right?

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So you're gonna plot the
point, let's say we are gonna

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plot the point (2,4), so we
may kind of mark our graph,

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two here, one, two, three, four,

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so you wanna mark somewhere here (2,4),

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and that represents an input-output pair.

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And if you do that with,
you know, negative one, one,

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you go negative one, one.

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And when you do this for every
possible input-output pair,

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what you end up getting, I
might not draw this super well,

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is some kind of smooth curve.

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The implication for doing this
is that we typically think of

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what is on the x-axis as
being where the inputs live,

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you know, this would be, we
think of as the input one,

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and this is the input two, and so on,

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and then you think of the
output as being the height

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of the graph above each point.

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But this is kind of a
consequence of the fact

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where we just listing
all of the pairs here.

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Now if we go to the world
of multi-variable functions,

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you know, not gonna show
the graph right now,

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let's just think we've got
three-dimensional space

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at out disposal to do with what we will.

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We still want to understand
the relationship between

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inputs and outputs of
this guy, but this case,

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inputs are something that we
think of as pair of points,

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we might have a pair of points like (1,2),

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and the output there is gonna be

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one squared plus two squared,
and that equals is five.

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So how do we visualize that?

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Well if we wanna pair these
things together, the natural way

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to do that is to think of
a triplet of some kind.

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So in this case, you wanna
plug the triplet (1, 2, 5),

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and to do that in three-dimensions,

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we'll take a look over
here, we think of going one

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in the x direction, this
axis here is the x-axis,

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so we want to move distance one there,

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and we want to go two in the y direction,

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so we kinda think of
going distance two there,

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and then five up, and then

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that's gonna give us some
kind of point, right?

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So we think this point in space

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and that's a given input-output pair.

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But we could do this for a lot, right,

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a couple different
points that you might get

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if you start plotting
various different ones,

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look something like this,
and of course there is

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infinitely many that you can
do and it'll take forever

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if you try to just draw each
one in three-dimensions,

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but what's really nice here
is that you know get rid of

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those lines, if you imagine doing this

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for all of the infinite
many pairs of inputs

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that you could possibly have,
you end up drawing a surface.

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So in this case the surface
kind of looks like a

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three-dimensional parabola,
that's no coincidence,

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it has to do with the
fact that we are using

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x squared and y squared here.

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And now the inputs like (1,
2), we think of as being

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on the xy-plane, right?

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So you think of the inputs living here,

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and then what corresponds
to the output is that

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height of a giving point
above the graph, right?

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So it's very similar to
two-dimensions, you think,

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you know, we think of the
inputs as being on one axis,

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and the height gives the output there.

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So just to give an example of

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what the consequence of this
is, I want you to think about

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what might happen if we change
our multi-variable function

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a little bit, and we multiply
everything by half, right?

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So I'll draw in red here, let's
see that we have a function,

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but I'm gonna change it so
that it outputs one half

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of x squared plus y squared.

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What's gonna be the shape of
the graph for that function?

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And what it means is the
height of every point

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above this xy-plane is gonna
have to get cut in half.

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So it's actually just the modification

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of what we already have,
but everything kind of

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sloops on down to be
about half of what it was.

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So in this case instead
of that height being five,

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it'll be two-point-five.

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You could imagine, let's
say we did this, you know,

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is even more extreme,
instead of saying one-half,

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you cut it down by like one-twelfth,

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maybe I'll use the same
color, by one-twelfth,

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that would mean that everything, you know,

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sloops very flat, very flat
and close to the xy-plane.

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So the graph being very
close to xy-plane like this

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corresponds to very small outputs.

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And one thing that I'd like
to caution you against,

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it's very tempting to try to think of

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every multi-variable function as a graph,

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cause we are so used to
graphs in two-dimensions

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and we are so used to
trying to find analogies

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between two-dimensions and
three-dimensions directly,

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but the only reason that
this works is because

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if you take the number of
dimension in the input,

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two-dimensions, and then
the number of dimensions

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in the output, one-dimension,
it was reasonable

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to fit all of that into
three, which we could do.

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But imagine if you have
a multi-variable function

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with, you know, a three-dimensional input,

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and a two-dimensional
output, that would require

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a five-dimensional graph,
but we are not very good

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at visualizing things like that.

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So there's lots of other
methods, and I think

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it's very important to
kinda of open you mind

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to what those might be.

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In particular, another one
that I'm gonna go through soon,

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lets us think about 3-D graphs but kind of

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in a two-dimensional setting,
and we are just gonna

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look at the input space,
that's called a contour map.

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Couple of other ones,
like parametric functions,

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you just look in the output space;

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things like vector space,

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you kind of look at the input
space but get all the outputs.

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There's lots of different
ways, I'll go over those

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in the next few videos.

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And that's three-dimensional graphs.