Dependent and independent variables | Year 11

01 / LEARN

Understand the idea.

Which quantity is the input?

The independent variable is the input in the stated model; the dependent variable is calculated from it. For $C = 3 n + 8$ , choosing the number of items $n$ determines the cost $C$ . Put input on the horizontal axis and output on the vertical axis.

Use the meanings of the quantities, rather than their letters or positions in an equation. Asking for the time at a given height reverses a calculation, but the original height model can still treat time as its independent variable.

Context restricts the model

Inputs must fit the situation. A ticket count takes whole-number values from zero to the available capacity; elapsed time may vary continuously. Include units, and check whether a fractional or negative value has a sensible interpretation.

A formula describes how its chosen quantities are related. It does not establish a cause merely by naming one quantity as an input. Likewise, joining discrete points with a line does not make the values between them permitted inputs.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

Worked solution

1
Identify what is chosen and what is calculated.
Hire time $h$ , measured in hours, is independent; the charge $C$ , measured in dollars, depends on that time.
2
Evaluate the charge at zero hours.
$C (0) = 12 + 4 (0) = 12$
The constant term is the modelled initial charge in dollars.
3
Compare charges one hour apart.
$\begin{aligned} C (h + 1) - C (h) & = [12 + 4 (h + 1)] - [12 + 4 h] \\ = 4 \end{aligned}$
Every additional hour adds four dollars, so the coefficient has units dollars per hour.

Answer

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

EXAMPLE 02

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?

Worked solution

1
Specify the kind of allowed input.
$n \in {0, 1, 2, 3, \dots}$
Seats are counted in whole numbers and the theatre capacity supplies an additional upper bound.
2
Reject the proposed fractional input.
$2.5 \notin {0, 1, 2, 3, \dots}$
Selling two and a half seats is outside the stated counting model.
3
Substitute the permitted count.
$R = 18 (25) = 450$
Twenty-five seats at eighteen dollars each produce revenue of 450 dollars.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Area A.

The radius $r$ is chosen first, and $A = π r^{2}$ calculates its corresponding area. Therefore $A$ is dependent and $r$ is independent.

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Which quantity is the input?

Context restricts the model

See the method in action.

A practical fee model

Identify what is chosen and what is calculated.

Evaluate the charge at zero hours.

Compare charges one hour apart.

Discrete or continuous?

Specify the kind of allowed input.

Reject the proposed fractional input.

Substitute the permitted count.

Try it yourself.

Check your understanding

Keep practising