Graph of a function of two variables

 Goal: graph of a function of two variables

Graphing of a function of two variables

A function z = f(x,y) is a function with two independent variables x and y and one independent variable z. The graph of a function of one variable y = f(x) is made of of all the ordered pairs (x,y) in the cartesian plane. The pair is made of two elements  where the first element is an independent variable and the second element is a dependent variable. Similarly the graph of a function with two variables is made of triples (x,y,z) where x and y are independent variables and z is the dependent variable. The graph of a function z = f(x,y) is called a surface.

To understand the concept of graphing a function z = f(x,y) to obtain a surface in three dimensional space, let's imagine the coordinate system (x,y) laying flat. Every pair (x,y) in the domain of the function has a unique point z associated with it. If z is positive, then z is located above the xy plane. If z is negative, then z is located below the xy plane. The set of all the graphed points becomes the three dimensional surface that is the graph of the function.

Example :

 

Solution

In the example of the previous post, we determined that the domain and range of: g(x,y) = ⎷9-x²-y² are respectively:

and

When x² + y² = 9, g(x,y) = 0. Therefore, any point on the circle of radius 3 is mapped to the point z = 3 in R³. When x² + y² = 8. g(x,y) = 1. Therefore, any point on the circle centered at the origin and of radius 2⎷2 is mapped to z =1 in R³. As x² + y² gets closer to 0, z gets closer to 3. When x² + y² = 0, g(x,y) = 3. This is the origin in the xy plane. If x² + y² is equal to any other value between 0 and 9, g(x,y) takes any value between 0 and 3. The surface described by this function is an hemisphere of radius 3 as shown in the figure below.                                                                                             

This function contains also  x² + y². Setting this expression equal to various values starting at 0, we obtain circles of increasing radius. The minimum value of f(x,y) = x² + y² is 0 (obtained when x = 0 and y = 0). When x = 0, the function becomes z = y². When y = 0, the function becomes z = x². These are cross-sections of the graph and are parabolas. The graph of f is shown below: