A nonlinear relationship is a type of relationship between two entities in which change in one entity does not correspond with constant change in the other entity. This might mean the relationship between the two entities seems unpredictable or virtually absent. However, nonlinear entities can be related to each other in ways that are fairly predictable, but simply more complex than in a linear relationship.

## Understanding Linear Relationships

A linear relationship exists when two quantities are proportional to each other. If you increase one of the quantities, the other quantity either increases or decreases at a constant rate. For example, if you get paid $10 an hour, there is a linear relationship between your hours worked and your pay. Working another hour always results in a $10 pay increase, regardless of how many hours you already worked.

## Differentiating Linear and Nonlinear Relationships

Any relationship between two quantities that doesn't fit the definition of a linear relationship is called a nonlinear relationship. The easiest way to differentiate a linear relationship from a nonlinear relationship is by mapping them on a graph. Use the x-axis of the graph to represent one of the quantities and the y-axis to represent the other. Using the previous example, plot hours worked on the x-axis and money earned on the y-axis. Then plot some known data points on the graph, such as one hour worked = $10, two hours worked = $20, and three hours worked = $30. Since you can connect the points to form a straight line, you know you have a linear relationship.

## Types of Nonlinear Relationships

Some nonlinear relationships are monotonic, meaning they always increase or decrease, but not both. Monotonic relationships differ from linear relationships because they do not increase or decrease at a constant rate. When graphed, they appear as curves. If a monotonic relationship occurs where increases in one entity cause a decrease in the other entity, this is called an inverse relationship. However, nonlinear relationships can also be too irregular to fit any of these categories.

## Examples of Nonlinear Relationships

Nonlinear relationships, and often monotonic relationships, arise regularly when comparing geometrical measurements of a single shape. For example, there is a monotonic nonlinear relationship between the radius of a sphere and the volume of that same sphere. Nonlinear relationships also appear in real world situations, such as in the relationship between the value of a motorcycle and the amount of time you owned the motorcycle, or in the amount of time it takes to do a job in relation to the number of people there to help. If your boss raises your hourly rate to $15 per hour when you work overtime, the relationship of your hours worked to your pay acquired might become nonlinear.