3.3 Linear Relations

Look at the examples above. You will notice that in each case, as the independent variable changes, the dependent variable also changes by equal amounts. Each one therefore qualifies as a linear relation. You can also see, however, that the relations have significant differences.

There is a significant difference between graphs 1 and 2. Obviously, graph 2 goes through the origin of the graph, while graphs 1 and 3 do not. Mathematically, the point at which a graph crosses the vertical axis is called a y-intercept. In the physical world, the y-intercept has some physical meaning. Specifically, it is the value of the dependent variable when the independent variable is zero. While all three graphs are linear relationships, only one of them illustrates a proportional relationship. A direct proportion occurs when, as one variable increases by a certain factor, the other variable increases by the same factor. Graphically, therefore, a direct proportion must not only be linear but must also go through the origin of the axes. When one variable is zero, the other variable must also be zero. When the independent variable doubles, the dependent variable doubles. When one variable triples, the other variable triples, and so on. Graph 2 is therefore the only example of a direct proportion while graphs 1 and 3 are simply linear relationships.

In graph 2 on the Direct Proportions page, we can say that Stretch is directly proportional to Mass. In mathematical symbols this would be stated:

stretch ∝ mass

Let us now look at a specific example that illustrates how to determine the specific mathematical relationship that is suggested by the linear relationship with a y-intercept.