Construct, graph and interpret piecewise-linear models in which the rule changes over different intervals of the domain.

How to build a piecewise-linear model where different straight-line rules apply over different intervals, graph it, read its breakpoints, and use it to solve practical step-rate problems.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
What a piecewise-linear model is
Choosing the right piece
Graphing the model
Reading information back from the model
Solving equations with a piecewise model

What this dot point is asking

You must be able to write a model in pieces, state the interval each piece applies to, graph the segments, identify the breakpoints where the rule changes, and use the model to calculate and interpret values.

What a piecewise-linear model is

Many real charges and rates do not follow a single straight line. A taxable income, a phone plan, or a delivery fee may follow one rate up to a threshold, then a different rate beyond it. We write this as several linear rules, each tagged with the interval where it applies:

C(x) = \begin{cases} 2x + 10, & 0 \le x \le 20 \\ 3x - 10, & x > 20 \end{cases}

Each piece is just $y = mx + c$ . The values of $x$ where the rule changes are the breakpoints (here $x = 20$ ).

Choosing the right piece

The single most important skill is matching the input to its interval before you substitute.

Worked example

A two-rate electricity charge

An electricity retailer charges $0.30 per kWh for the first 300 kWh in a quarter, then$ 0.45 per kWh for any usage above 300 kWh. Find the cost of using 500 kWh.

The first 300 kWh cost a flat block. Build the model for total cost $C$ against usage $u$ :

C(u) = \begin{cases} 0.30u, & 0 \le u \le 300 \\ 0.45(u - 300) + 90, & u > 300 \end{cases}

The constant $90$ is the cost of the first 300 kWh, since $0.30 \times 300 = 90$ . For $u = 500$ we use the second piece because $500 > 300$ :

C(500) = 0.45(500 - 300) + 90 = 0.45(200) + 90 = 90 + 90 = 180

The cost is $180.

Graphing the model

Plot each segment only over its own interval. At a breakpoint, draw a filled dot for the endpoint that is included and an open dot for one that is excluded. A change in gradient appears as a corner ("kink") in the graph; a steeper gradient after the breakpoint means the quantity grows faster beyond that point.

Reading information back from the model

Examiners often ask you to interpret the graph in words:

Each segment's gradient is the rate over that interval (cost per kWh, fee per kilometre).
A breakpoint marks where conditions change (a usage threshold, a tax bracket boundary).
A horizontal segment ( $m = 0$ ) means the quantity stays constant over that interval, such as a fixed monthly fee regardless of usage.

Solving equations with a piecewise model

To find when $C(x)$ equals a target value, decide which interval the answer falls in, then solve that piece's equation. Always check the solution actually lies in that interval; if it does not, the answer belongs to a different piece.