Skill: Identify independent and dependent variables

When two quantities change together, identify their roles by asking:

• Independent variable: the input or quantity chosen first.
• Dependent variable: the output, whose value depends on the input.

In a graph, the independent variable goes on the horizontal axis and the dependent variable goes on the vertical axis. The variables do not have to be called $x$ and $y$.

Example: A taxi charges a starting fee of $3 plus $2 for each kilometre travelled. The relationship is

$$F=3+2d,$$

where $d$ is the distance travelled in kilometres and $F$ is the total fare in dollars.

Step 1: Find the input.
The number of kilometres, $d$, is the quantity that can be selected or measured first. This is the independent variable.

Step 2: Find the output.
The fare, $F$, changes according to the distance. This is the dependent variable.

Step 3: Label the graph, if needed.

• Horizontal axis: distance, $d$ (kilometres)
• Vertical axis: fare, $F$ (dollars)

For example, if $d=4$,

$$F=3+2(4)=11.$$

So travelling $4$ kilometres results in a fare of $11. The fare depends on the distance, so $d$ is independent and $F$ is dependent. This describes how the quantities are related; it does not automatically prove that one causes the other.