Dependent and independent variables | Year 11

01 / LEARN

The idea, step by step

Which quantity is the input?

The independent variable is the input chosen or observed first in the model; the dependent variable is the output calculated from it. In $C = 3 n + 8$ , the number of items n is the input and cost C is the output. This convention tells you which quantity to place on each axis: input horizontally and output vertically.

Identify the variables from their meanings, not simply from letters. Time is often independent, but not always. A question can instead ask which time gives a stated height. The original model may still treat time as its input.

Context restricts the model

A number of tickets is a non-negative integer, whereas the time since a journey began can vary continuously. State units for both variables and decide whether fractional inputs are meaningful. A straight-line drawing through whole-ticket values does not make half a ticket an allowed purchase.

A dependence in a formula describes a mathematical relationship. Observing two quantities change together does not, by itself, establish that one causes the other. Interpret only what the problem and model justify.

A common trap

Calling the variable on the left of every equation the independent variable.

Instead: Use the quantities and the stated model. Writing $C = 4 h + 12$ or $4 h + 12 = C$ does not swap their input-output roles.

02 / WORKED EXAMPLES

See the thinking

Example 1

A practical fee model

A bicycle-hire charge is $C = 12 + 4 h$ , where h is hire time in hours. Identify the variables and explain the coefficients.

The chosen hire time is the input, and the charge is calculated from it.
The constant 12 is the charge when h=0. Each extra hour increases C by 4 dollars.

h is independent; C is dependent; 12 dollars is the initial charge and 4 dollars per hour is the hourly rate.

Example 2

Discrete or continuous?

A theatre sells n seats at 18 dollars each, with $R = 18 n$ . Can n=2.5 be used? What is the revenue for 25 seats?

Seats are counted, so permitted n values are non-negative integers up to the theatre capacity.
$R = 18 (25) = 450$ dollars.

No; n is an integer. Revenue is 450 dollars.

Check your understanding

In $A = π r^{2}$ , with the radius chosen first, which variable is dependent?

Show answer

Area A.

The area is calculated from the chosen radius, so r is the input and A is the output.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A tank starts with 200 L and drains at 8 L/min: $V = 200 - 8 t$ . Name the independent and dependent variables and their units.
2 marks
For the tank model $V = 200 - 8 t$ , find the volume at 15 minutes.
2 marks
A postage model uses $P = 2 + 0.5 m$ for mass m in grams. Interpret 0.5 with units.
2 marks
For $R = 18 n$ ticket revenue, give the first four permitted inputs if no ticket has yet been sold.
2 marks
A phone battery model is $B = 100 - 5 t$ for $0 \leq t \leq 20$ , with t in hours. Explain why t=30 is not allowed.
2 marks
A table records shoe size and height for 30 people. Does calling height an output prove that shoe size causes height?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A tank starts with 200 L and drains at 8 L/min: $V = 200 - 8 t$ . Name the independent and dependent variables and their units.
2 marks
Elapsed time is the input; remaining volume is calculated from it.
Independent t (minutes); dependent V (litres).
For the tank model $V = 200 - 8 t$ , find the volume at 15 minutes.
2 marks
$V = 200 - 8 (15) = 80$ litres.
$80 L$
A postage model uses $P = 2 + 0.5 m$ for mass m in grams. Interpret 0.5 with units.
2 marks
The coefficient of the mass has units dollars/gram.
The price increases by 0.50 dollars per gram.
For $R = 18 n$ ticket revenue, give the first four permitted inputs if no ticket has yet been sold.
2 marks
Tickets are counted; inputs are whole numbers rather than arbitrary real values.
$n = 0, 1, 2, 3$
A phone battery model is $B = 100 - 5 t$ for $0 \leq t \leq 20$ , with t in hours. Explain why t=30 is not allowed.
2 marks
$B (30) = - 50$ , which is not a possible remaining percentage.
It lies outside the stated model interval and predicts a negative battery percentage.
A table records shoe size and height for 30 people. Does calling height an output prove that shoe size causes height?
2 marks
The input-output convention organises the data; it does not establish causation.
No.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Topic 1: Functions and graphs. Prerequisite algebra and coordinate geometry are embedded as supporting lessons.

Official curriculum source ↗

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