Piecewise practical models where relevant | Year 11

01 / LEARN

The idea, step by step

Intervals select the rule

A piecewise model assigns different expressions to different input intervals. Locate the input first, then substitute into that branch. Every allowed input needs exactly one output; an endpoint must not accidentally receive conflicting values.

A weight-band fee gives horizontal segments with jumps. A later per-unit rate can instead create a continuous graph with a changed slope. The words describing the charge determine which model to use.

Keep the earlier charge

An accumulated charge includes the amount already charged before the threshold. If a new rate applies after two hours, multiply it by $t - 2$ , not all $t$ hours. Check whether the branches agree at the boundary.

A fee per started hour is a step model. A fee charged proportionally for part of an hour is continuous within its interval. Do not swap these interpretations without permission.

A common trap

Using the cheaper branch after its interval has ended.

Instead: Read the inequality signs before substituting, especially at the boundary.

02 / WORKED EXAMPLES

See the thinking

Example 1

Extra-time charge

Parking costs 5 dollars for any positive stay up to 2 h, then 3 dollars per additional hour charged proportionally. Find the cost of 3.5 h.

Since $3.5 > 2$ , use $C = 5 + 3 (t - 2)$ .
$C = 5 + 3 (1.5) = 9.5$ .
At 2 h the later expression gives 5, matching the first charge.

9.50 dollars.

Example 2

At a weight boundary

Postage is 4 dollars for $0 < m \leq 500$ g, 7 dollars for $500 < m \leq 1000$ g, and 10 dollars for $1000 < m \leq 1500$ g. Find charges at 500 g and 501 g.

500 is included in the first band.
501 is in the second band.
The jump means the fee is not proportional to mass.

4 dollars and 7 dollars.

Check your understanding

A fee is 6 dollars for $0 < t \leq 3$ h, then $6 + 2 (t - 3)$ for $t > 3$ . Find its value at 4 h.

Show answer

8 dollars.

Use the second branch: $6 + 2 (4 - 3) = 8$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A fee is 8 dollars for $0 < t \leq 2$ h, then $8 + 4 (t - 2)$ for $t > 2$ . Find the fee at 5 h.
2 marks
A fee is 8 dollars for $0 < t \leq 2$ h, then $8 + 4 (t - 2)$ for $t > 2$ . Find the charge at exactly 2 h.
2 marks
Water costs 2 dollars/kL for the first 10 kL and 3 dollars/kL above 10. Write its rule for $u > 10$ .
2 marks
Water costs 2 dollars/kL for the first 10 kL, then 3 dollars/kL above 10. Find the bill for 14 kL.
2 marks
A parcel fee is 5 dollars for $0 < m \leq 1$ kg and 9 dollars for $1 < m \leq 2$ kg. Find fees at 1 kg and 1.1 kg.
2 marks
A proposed rule gives $C = 4 t$ for $0 \leq t \leq 2$ and $C = 3 t + 1$ for $t \geq 2$ . What is wrong?
3 marks

Read the worked solutions

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

A fee is 8 dollars for $0 < t \leq 2$ h, then $8 + 4 (t - 2)$ for $t > 2$ . Find the fee at 5 h.
2 marks
$8 + 4 (5 - 2) = 20$ .
20 dollars.
A fee is 8 dollars for $0 < t \leq 2$ h, then $8 + 4 (t - 2)$ for $t > 2$ . Find the charge at exactly 2 h.
2 marks
The first interval includes 2.
The second expression also approaches 8 there.
8 dollars.
Water costs 2 dollars/kL for the first 10 kL and 3 dollars/kL above 10. Write its rule for $u > 10$ .
2 marks
First-band charge is $2 (10) = 20$ .
Only excess usage is charged at 3 dollars/kL.
$C = 20 + 3 (u - 10)$ .
Water costs 2 dollars/kL for the first 10 kL, then 3 dollars/kL above 10. Find the bill for 14 kL.
2 marks
$20 + 3 (14 - 10) = 32$ .
32 dollars.
A parcel fee is 5 dollars for $0 < m \leq 1$ kg and 9 dollars for $1 < m \leq 2$ kg. Find fees at 1 kg and 1.1 kg.
2 marks
1 is in the first band; 1.1 is in the second.
5 dollars; 9 dollars.
A proposed rule gives $C = 4 t$ for $0 \leq t \leq 2$ and $C = 3 t + 1$ for $t \geq 2$ . What is wrong?
3 marks
Both intervals include 2.
The formulas disagree, so the rule is not a function.
It assigns both 8 and 7 dollars at $t = 2$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: algebraic and graphical modelling of linear/exponential change, with interpretation and evaluation.

Official curriculum source ↗

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