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YEAR 11 · LINEAR AND EXPONENTIAL MODELS · 18 MIN

Piecewise practical models where relevant

Choose the correct branch of a practical piecewise rule and handle threshold values consistently.

In this lesson

Before you start
  • Linear equations and formulas
01 / LEARN

Understand the idea.

Intervals select the rule

A piecewise model uses different rules on different input intervals. First locate the input in its interval, paying attention to < and ≤, then substitute into that branch. Each allowed input must have one unambiguous output.

A weight-band fee can jump when a threshold is crossed. A changed hourly rate can instead change the slope while keeping the charge continuous. Decide from the words of the pricing rule, rather than assuming all piecewise graphs behave alike.

Keep the earlier charge

When a new rate applies only after a threshold, charge it on the excess input. After two hours, the additional time is t−2. Keep the amount already charged for the first two hours.

“Charged proportionally” permits fractional hours at the stated rate. “Per started hour” would require a step calculation instead. Evaluate the expressions at a boundary to check continuity when the context calls for it.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

Worked solution

  1. 1

    Select the later branch.

    3.5>2,C=5+3⁢(t−2)

    The stay exceeds the two hours covered by the initial five-dollar fee.

  2. 2

    Calculate the extra time and its charge.

    t−2=3.5−2=1.5 h3⁢(1.5)=4.50

    The extra hourly charge is proportional, so one and a half hours is allowed.

  3. 3

    Add the original fee.

    C=5+4.50=9.50

    The total is the first-period fee plus the extra-time charge, in dollars.

  4. 4

    Check the threshold behaviour.

    5+3⁢(2−2)=5

    The later expression meets the initial charge at two hours.

Answer

9.50 dollars.

EXAMPLE 02

At a weight boundary

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

Worked solution

  1. 1

    Locate five hundred grams.

    0<500≤500

    The upper endpoint belongs to the first band.

  2. 2

    Read its charge.

    C⁢(500)=4

    Every positive mass in that band costs four dollars.

  3. 3

    Locate five hundred and one grams.

    500<501≤1000

    This mass is above the first threshold and inside the second band.

  4. 4

    Read the second charge.

    C⁢(501)=7

    Crossing the threshold raises the charge to seven dollars; it is not a per-gram rate.

Answer

4 dollars and 7 dollars.

03 / YOUR TURN

Try it yourself.

Check your understanding

A fee is 6 dollars for 0<t≤3 h, then 6+2⁢(t−3) for t>3. Find its value at 4 h.

Show answer & reasoning

8 dollars.

Four hours lies above the three-hour threshold. The extra time is one hour, giving 6+2⁢(4−3)=8 dollars.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions
Read the questions online
  1. A fee is 8 dollars for 0<t≤2 h, then 8+4⁢(t−2) for t>2. Find the fee at 5 h.
    2 marks
  2. A fee is 8 dollars for 0<t≤2 h, then 8+4⁢(t−2) for t>2. Find the charge at exactly 2 h.
    2 marks
  3. Water costs 2 dollars/kL for the first 10 kL and 3 dollars/kL above 10. Write its rule for u>10.
    2 marks
  4. Water costs 2 dollars/kL for the first 10 kL, then 3 dollars/kL above 10. Find the bill for 14 kL.
    2 marks
  5. A parcel fee is 5 dollars for 0<m≤1 kg and 9 dollars for 1<m≤2 kg. Find fees at 1 kg and 1.1 kg.
    2 marks
  6. A proposed rule gives C=4⁢t for 0≤t≤2 and C=3⁢t+1 for t≥2. What is wrong?
    3 marks
Read the worked solutions

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

  1. A fee is 8 dollars for 0<t≤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.
  2. A fee is 8 dollars for 0<t≤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.
  3. 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).
  4. 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.
  5. A parcel fee is 5 dollars for 0<m≤1 kg and 9 dollars for 1<m≤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.
  6. A proposed rule gives C=4⁢t for 0≤t≤2 and C=3⁢t+1 for t≥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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