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

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:

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.

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.

Why the second piece carries a constant

In the electricity model the second rule is $0.45(u - 300) + 90$, not simply $0.45u$. The $(u - 300)$ measures usage beyond the threshold, and the $+90$ is the accumulated cost of the first 300 kWh. This structure - a base amount plus a new rate applied only to the excess - is exactly how tax brackets and tiered tariffs work in practice. Writing the excess term explicitly keeps the model continuous and stops you from wrongly charging the higher rate on the whole amount. It is the single most important idea for piecewise charge models.

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.