F&A - 22 Dependent and independent variables: The terms dependent and independent variables describe how the variables in a relationship are associated to each other. The value of the dependent variable, as its name implies, is determined by the value of the independent variable.

Example 22.1:

A soccer team is holding a car wash to raise money. They are charging a flat rate of $10.00 for each car that gets washed.

In this situation, the amount of money raised (revenue) is dependent on the number of cars washed. Therefore, the amount of money raised (revenue) is the dependent variable, and the number of cars washed is the independent variable.

Graph of revenue from car wash

Note: In a situation involving one independent variable and one dependent variable, typically the independent variable is associated with the values on the horizontal axis and the dependent variable is associated with the variables on the vertical axis.

Sometimes distinguishing the dependent variable from the independent variable is not as clear cut as in this example; Examples 22.2 and 22.3 help illustrate this dilemma.

Example 22.2 – Unlimited resources ($):

Apples are sold for $2.00 a pound.

If people had unlimited resources when purchasing apples, then the total cost of apples purchased would be dependent upon the number of pounds of apples purchased.

Graph of cost of apples

Example 22.3 – Limited Resources ($):

Apples are sold for $2.00 a pound.

However, most people do not go shopping with an unlimited amount of money. In this case, the amount (number of pounds) of apples purchased depends upon how much money a shopper has to spend.

Graph of price of apples

Example 22.4 – In equations:

Note: Examples 22.1 – 22.3 are examples that can be expressed as equations.

As shown in Example 22.1, a soccer team is holding a car wash to raise money. They are charging a flat rate of $10.00 for each car that gets washed.

An equation that represents this situation is…

Discussion of dependent and independent variables

Example 22.5:

Example 22.2 and 22.3 are interesting cases. Both situations appear the same – “Apples are sold for $2.00 a pound.” However, in Example 22.2 the resources ($) are unlimited, and in Example 22.3 the resources ($) to buy the apples are limited. These differences result in different equations to represent each of the situations.

Unlimited vs limited resources

Note: In a situation with one independent and one dependent variable, it is standard to label the independent variable as “x” and the dependent variable as “y.” However, students should develop flexibility with different labeling schemes, as variables are often given different names. It is the relationship between the variables, not the labels of the variables that is important.

In Examples 22.2 and 22.3 some assumptions are made about the availability of resources (both money and pounds of apples available). In order to model real life situations, it is important to consider any restrictions on the variables involved. This requires an understanding of domain and range. (See F&A – 23, F&A – 24.)

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