Visualizing surfaces in three dimensions. Review: Trying to draw a function of two variables As a first step, we can plot the graph of f, which is a function of two variables. To determine the possible values of x and y, we can construct a graph of this equation. As with equations involving a single variable, the possible values of x and y depend on the shape of this graph. We plot a point for each of them in the xy- plane, whose height is the value of a function of these parameters. So we'll plot— let's say z equals f(x, y)—and now that will actually become a surface in space. So for each value of x and y, here we have (x, y) in the xyplane, then we'll plot the point in space at position (x y, z) equals f(x, y). If we combine all these points together, they will give us a surface that sits in space. Given a function of two variables, how do we represent it geometrically? Let's take our first example. We are given the function f of (x, y) equal to negative y. However, it does not depend on x, which is not a problem. It is still a valid function of x and y. The function z=−y is constant with respect to x. Because the curve is a plane, we can draw its graph by examining the plane in space defined by z being equal to −y. If we want to draw the graph of z = −y, we can use the axes shown on the blackboard. If we look at what happens in the yz-plane, which is the plane of the blackboard, we will see that it will look like a line that goes downward with slope 1. If we change x, nothing happens because x does not appear in this equation. If instead of setting x equal to 0, we set x equal to 1 or minus 1, it still looks exactly the same. Now we have a plane that contains the x-axis and slopes downward with a slope of 1. It's hard to draw, but you can see immediately what the big problem with graphs will be: They are hard to read.