Representations of Functions in Three Dimensions

In this section, we start with a situation and explore how we can represent the same situation with a formula, a table, a contour, and a surface. We explore the strong and the weak points of each representation. We continually emphasize that they all are trying to represent the same set of points that satisfy the given situation. And we attempt to incorporate geometric visualization whenever it is appropriate.

The Situation

Assume that two parents work. The father earns $5.00 an hour and the mother earns $10.00 an hour. Let

x = Number of hours that mom works in a week

y = Number of hours that dad works in a week, and

z = f (x, y) = Weekly salary for the family

If we know how many hours mom and dad worked during a given week, we would like to find a means to obtain the family salary for the week given the number of hours that mom has worked and the number of hours that dad has worked. To do this we are going to use a formula, a table, a contour diagram, and a surface.

Representations of Functions with a Formula

To find a formula for the family salary, we can divide the salary into two parts.

Mom's Salary = (Mom's hours)(10) = 10x

Dad's Salary = (Dad's hours)(5) = 5y

In this manner we can conclude that the family salary, z, is related to the weekly salaries of mom and dad by the formula z = f (x, y) = 10x + 5y

Representing Functions with a Table

To represent the family salary associated with x and y using a table, we can start by selecting various values of x and y.

	y = 0	y = 1	y = 2	y = 3
x = 0
x = 1
x = 2
x = 3

For each x and y value, we can obtain the family salary associated. For example, if mom works 1 hour and dad works 2 hours the family salary is twenty dollars. Upon finding the value for z associated with all the combinations of x and y. we can complete the table.

	y = 0	y = 1	y = 2	y = 3
x = 0	0	5	10	15
x = 1	10	15	20	25
x = 2	20	25	30	35
x = 3	30	35	40	45

Geometrically, we can see the information contained in the table by first placing a point for each (x, y) in the table on the xy plane of our 3-D space

Then we can raise each point to its appropriate z value (height) in 3 dimensions.

Representing Functions with a Contour Diagram

To obtain a contour diagram for the situation, we start by looking for ways in which we can obtain a family of $10. A few ways of obtaining this are (0, 2), (.5, 1), and (1, 0). With a little work, we can conclude that all (x, y), which can give us a family salary of $10, are contained in the following graph:

We then look for all values of (x, y) which can give us a family salary of $20, $30, $40, and $50 in a similar manner. When all these graphs are obtained, we put them together to form the following graph which is called a contour diagram.

Geometrically, we can see the information contained in the table by first placing each curve on the (x, y) plane of our 3-D space, clearly labeling each curve with its associated height.

We can then raise each curve to its appropiate height in 3 dimensions to visualize the contour in three dimensions.

Representing Functions with a Surface

To obtain a surface, which contains all the points associated with the situation, we wish to obtain enough points to guess the rest. The best way to do this is to start with the contours we have already obtained, The following points in 3 - space we know satisfy our situation.

If we wish more points before guessing the surface, we can consider all the points in 3 space where mom does not work x = 0.

We can also consider all the points in 3 - space where dad does not work y = 0.

Placing all of these points that we have determined will satisfy the situation together in our 3 dimensional systems, we obtain the following:

Considering this skeleton of a surface, it is not difficult to guess the remaining points and place a surface that is consistent with these data.