Cause and effect
In many situations, one quantity depends on another.
- The independent variable is the one you choose or change (the input).
- The dependent variable responds to it (the output).
Example: A plumber charges by the hour. The cost depends on the hours worked.
- Independent: hours
- Dependent: cost
A quick test
Fill in the sentence: "[dependent] depends on [independent]."
| Situation | Sentence | Independent | Dependent |
|---|---|---|---|
| Plant growth and sunlight | Height depends on sunlight | sunlight | height |
| Distance driven and fuel used | Fuel used depends on distance | distance | fuel |
| Time studied and test score | Score depends on time studied | time | score |
| Temperature and ice cream sales | Sales depend on temperature | temperature | sales |
If the sentence sounds wrong the other way around ("Sunlight depends on plant height"), you’ve got it right.
On a graph
- Independent variable → horizontal axis (x).
- Dependent variable → vertical axis (y).
In equations and function notation
In y = 12x + 30 or C(h) = 12h + 30, the independent variable is the input (x or h) and the dependent variable is the output (y or C).
In experiments
Scientists change the independent variable on purpose and measure the dependent variable. Everything else is kept the same: these are controlled variables.
Example: Does the height of a ramp affect how far a car rolls?
- Independent: ramp height
- Dependent: distance rolled
- Controlled: same car, same surface, same release method
Time is usually independent
When time is involved, it’s almost always the independent variable: distance after t seconds, money after t years, temperature at time t.
Describing the relationship
Describe the relationship in words: "As the number of hours increases, the cost increases by $12 per hour."
Worked examples
Example 1: A phone plan’s cost and data used. Independent? Data used.
Example 2: Which variable goes on the y-axis in a graph of age vs height for children? Height (it depends on age).
Example 3: In d = 80t, which is dependent? d (distance).
Common mistakes
- Putting the variables on the wrong axes.
- Choosing the variable mentioned first rather than thinking about cause and effect.
- Forgetting controlled variables when designing experiments.
Try it yourself
- Number of people at a party and amount of pizza needed: which is dependent?
- In P = 15h, which is independent?
- A study tests how much water plants get and how tall they grow. Name one controlled variable.
- Which axis does the independent variable go on?
Answers: 1) pizza 2) h 3) e.g. type of plant, sunlight, soil 4) horizontal (x)
Quick recap
- The independent variable is the input; the dependent variable is the output.
- Graph independent on the horizontal axis, dependent on the vertical axis.
- Experiments change one variable and keep controlled variables the same.