Graphs of Two-Variable Functions

A weather map provides a practical example of a function of two variables. Across a wide region such as the United States, temperatures vary from one location to another. Each location can be identified by two geographic coordinates: longitude and latitude. Since a single temperature value is assigned to each coordinate pair, the situation can be represented mathematically as a function with two inputs and one output.

In mathematical notation, longitude and latitude can be labeled as x and y, while temperature can be labeled as z. The relationship is written as z = f(x, y).

This means that for every point (x,y) in the region, the function assigns a corresponding temperature value z. For example, two cities with different longitude and latitude coordinates may have different temperatures because their positions on the map are different.

To visualize this relationship, the coordinate pair (x, y) is placed in the horizontal plane, and the temperature value z is plotted vertically. Higher temperatures appear as higher points on the graph, while lower temperatures appear closer to the horizontal plane. When all of these temperature values are connected, they form a three-dimensional surface.

Because temperature usually changes gradually over space, the resulting surface is often smooth and continuous. This surface provides a visual representation of how temperature changes across the region. Steeper parts of the surface indicate areas where temperature changes quickly over short distances, while flatter regions show areas where temperature remains relatively constant.

This type of graph connects real-world geographic data with a mathematical structure. It shows how a function of two variables can transform coordinate information into a surface, making regional temperature patterns easier to interpret and compare.