 We can extend graphing to three dimensions by using a third coordinate. Since we usually call the first two coordinates x and y, we'll call the third coordinate, um, how about z? And since the x and y axes are perpendicular to each other, the z-axis will be perpendicular to both. And we usually draw from the perspective of the first quadrant. Wait, no, that's because the plane is split into four regions. We call them quadrant. Now space is split into eight regions, so we'll call this the first octant. Now, even though this is the first octant, nobody ever talks about the others in part because we can't agree on how to number them. All our ideas from Cartesian rectangular coordinates carry over. So the coordinates x, y, z specify distances along or parallel to the principal axes, and a relationship defines a set of points. So for example, if we want to plot the point 1, 3, 2, and negative or negative 1, 1, so for the first, we'll go out 1 along the x-axis. And if we're looking for the viewpoint of the first octant, positive x means towards us. Next, we'll go three units parallel to the y-axis, and again from the viewpoint of the first octant, positive y means to the right. And then finally, two units parallel to the z-axis, and again from the viewpoint of the first octant, that means two units upward. And again, if it's not written down, it didn't happen. So let's go ahead and label that point. And similarly, negative 4, negative 1, 1, well that's four units backwards away from us, one unit to the left, and one unit up. And again, if it's not written down, it didn't happen. Label. And it doesn't take a lot of graphing to realize there's going to be a problem trying to represent a three-dimensional space on a two-dimensional surface. What our graph will look like depends very much on what our viewpoint is. So these two points may appear close, but if we're looking at these two points from a different octant, their apparent distance will vary quite a bit. So it's less important to graph these points accurately than to get some sort of sense of what the surface looks like. So for example, let's find four points on the graph of z equals 5, and let's describe the figure that's produced by this graph. So remember, a point has coordinates x, y, and z, and it's on the graph if z is equal to 5. So as long as our z-coordinate is 5, any one of these points will be on the graph. So let's pick random values for the x and y coordinates. How about... And you can pick whatever values you want as long as your z-coordinate is 5. And if we graph these points... So for 0, 0, 5, don't go anywhere along the x or y axes, but do go 5 units straight up. 0, 1, 5, that don't go anywhere along the x axis, go 1 unit along y and 5 units straight up. 3, 1, 5, that's 3 along x, 1 parallel to y, and 5 units vertically. Negative 1, negative 1, 5, that's negative 1x, negative 1y, and 5 vertical. The thing to notice here is that no matter where we go along the x or y axis, we always end up by going upward by 5 units. And so notice that these points are all 5 units high, and that suggests that the graph of z equals 5 is a plane parallel to the x-y plane. Let's try a more complicated graph. How about that produced by the graph of 3x plus 2y equals 12? So again, a point has coordinates x, y, z, and it's on the graph if 3x plus 2y is equal to 12. So let's find a few points that are not on the graph. How about the point 5, 7, 2? Well, this is not on the graph because 3 times x plus 2 times y is not equal to 12. Likewise, the point 1, 5, 9 is not on the graph because 3 times x plus 2 times y is not equal to 12. On the other hand, the point 0, 6, 4 is on the graph because 3 times x plus 2 times y is in fact equal to 12. Now, you might notice that the z-coordinate of this point wasn't used in our verification. And what that means is that we can pick any z-value that we want to. So the point 0, 6, 8. Well, that's also on the graph since 3 times x plus 2 times y is equal to 12. And likewise, 0, 6 minus 2 is on the graph. And let's think about this a little bit more. Since the z-coordinate is the height of the point, all of these points we've found are above or below what we might call 0, 6 in the xy plane. So if we want to graph these, let's go to 0, 6, then go up or down by differing amounts. So we can go to 0, 6 and then up 4, or again to 0, 6 and then up 8, or to 0, 6 and then down 2. And really, we could go to 0, 6 and any distance above or below. And so what we actually get is a straight line through this point 0, 6 in the xy plane. How about a few more points? Well, if x is equal to 4 and y is equal to 0, then 3x plus 2y does in fact equal 12, and z could be anything at all. So a few more points on the graph are 4, 0, something. And we can pick our some things. And so we get all points above or below 4, 0 on the xy plane. So we'll go 4 along x, 0 along y, and then up or down some amount. And so we get all points above or below 4, 0 on the xy plane. And let's think about this a little bit more. Since 3x plus 2y equals 12 is a straight line in the xy plane, and z can have any value, the graph consists of all points above or below a straight line in the xy plane. So it might look something like this. How about going backwards? Let's start with the geometric object and try and figure out what its equation is. So let's find the equation of the xy plane. So again, we need to specify x, y, and z for any point on the xy plane. Well, if you're on the xy plane, x and y could be anything. That's kind of what it means to be on the plane. But in order to be on the plane, we do require that z is equal to 0, and there's our equation. And by a similar argument, we note that the yz plane has equation x equals 0. The xz plane has equation y equals 0, and the xy plane again has equation z equals 0. So what about more complicated graphs? We'll take a look at that next.