13.2.1 Definitions and Formulas

Dimensions

For each variable (independent or dependent) in a model, you need one dimension in order to create a graph of the model. Thus, a model like y = f(x) needs two dimensions, one for y and one for x. A model like the general quadratic below needs three dimensions for its graph.

Surface Plot

A graphic representation of a function of one variable (two dimensions) is a scatterplot. Creating a similar type of graph for a function of two variables requires three dimensions. Each point has three coordinates, and the height of the point above the xy-plane is the value of the function. When the points are connected together, they form a surface in three dimensions.

General Quadratic Model

The general quadratic model we will use in this text is

In this, we assume that at least one of the B coefficients is non-zero. Other texts may refer to the model in slightly different terms, but the important things to note are that (1) this is a polynomial model (in two variables) and (2) the degree of each term (sum of the powers of each variable) is either 0, 1 or 2. For example, the terms with a B coefficient all have one variable raised to the second power and the other raised to the zeroth power, so they are degree 2. The cross term (the term with the C coefficient that involves both independent variables) has both variables raised to the first power, so its degree is 1 + 1 = 2 as well.

Discriminant

There are several mathematical objects that go by the name ”discriminant”. Each is used to discriminate between several alternatives. In this case, we are referring to a quantity that can be derived from the formula for the general quadratic that helps decide whether the graph will look like a bowl, a hill or a saddle. Using the symbols above, the discriminant is the quantity

The shape of the graph (as we will see in the examples), depends on this quantity in the following ways:

1. If D > 0 and B₁ > 0, then the graph will look like a bowl.
2. If D > 0 and B₁ < 0, then the graph will look like a hill.
3. If D < 0, then the graph will look like a saddle.
4. If D = 0, then the discriminant is not helpful.

There are two other possible shapes for the graph, which occur if the coefficients in front of all instances of one variable are zero. In that case, the graph looks like either a trough (if the remaining B coefficient is positive) or a speed bump (if the coefficient is negative).

Depending on your viewpoint and the exact values of your graph, you may not be able to see it has a particular shape, though (see example 5).