Year 11 General Linear Equations And Their Graphs

Piecewise Linear And Step Graphs

4 practice questions 2 video lessons Theory + worked examples

Theory

A piecewise linear function is made of two or more line segments, each with its own equation valid over its own range (domain). A step graph holds a constant value over a range, then jumps. Closed circles mean the endpoint is included, open circles mean it is not. Tiered tariffs charge each band's rate only on the portion within that band.

A piecewise linear function is a function defined by two or more straight-line formulas, each with its own domain — the range of $x$ values for which that formula applies. The pieces typically join end-to-end. A common write-up uses brace notation:

$f (x) = {\begin{cases} 2 x & 0 \leq x \leq 3 \\ x + 3 & 3 < x \leq 6 \end{cases}$

A step graph is a special kind of piecewise function where each piece is horizontal: the function holds a constant $y$ -value over a range of $x$ , then jumps to a new constant at a fixed $x$ . Endpoints are drawn with circles:

Filled (closed) circle: the $x$ value is included in this step.
Open circle: the $x$ value is not included — the function has already moved to the next step.

The domain of each piece is the range of $x$ values for which that piece's formula is used.

The first diagram is a piecewise linear graph: a tank fills fast in phase 1, then slower in phase 2 after $t = 3$ min. The two segments meet at $(3, 4.5)$ . The second is a step graph for car park fees, with closed circles marking included endpoints and open circles marking excluded ones.

Piecewise linear: two segments joined at

(3, 4.5)

, each with its own slope.

Step graph: parking fees jump at each hour. Closed circle = included; open = excluded.

There are no new formulas — just the brace notation for writing a piecewise function, and the rule for which piece to use.

Piecewise function — brace notation

$f (x) = {\begin{cases} formula 1 & condition 1 \\ formula 2 & condition 2 \\ formula 3 & condition 3 \end{cases}$

f (x) = \{\begin{matrix} formula 1 & if condition 1 \\ formula 2 & if condition 2 \end{matrix}

Endpoint conventions on a step graph

Symbol on graph	What it means	In inequality form
Filled (closed) circle	This $x$ value is included in this step	$\leq$ or $\geq$
Open circle	This $x$ value is not in this step	$<$ or $>$

Tiered tariff — total cost

For a quantity $q$ charged in bands at rates $r_{1}, r_{2}, r_{3}, \dots$ , with band widths $w_{1}, w_{2}, \dots$ , charge each band's rate only on the amount within that band, then add:

$total = r_{1} \cdot (amount in band 1) + r_{2} \cdot (amount in band 2) + \dots$

total = r_{1} \cdot w_{1} + r_{2} \cdot w_{2} + \dots

Evaluating a piecewise function at a given x

Check the conditions — find which branch contains your $x$ value.
Use only that branch's formula. Substitute $x$ and compute.
Make sure you respect strict vs non-strict inequalities ( $<$ vs $\leq$ ) at boundary points.

Reading a step graph

Locate your $x$ on the horizontal axis.
Move up to the step. If your $x$ lands exactly under a circle, use the circle's filled/open status: filled means this step applies, open means the next step applies.
Read off the constant $y$ -value of that step.

Computing a tiered tariff bill

Identify how the quantity is split across bands. For a total of $q$ , work out how much falls in each band, capped at that band's width.
Multiply each amount by its band's rate.
Add all the band-bills. The result is the total cost.

Break-even with a piecewise plan

For each plan, write a cost formula. If a plan changes rate at some point, write a piecewise formula.
Identify which branch of the piecewise plan you expect the break-even to fall in.
Set the two cost expressions equal on that branch and solve. Check the answer falls inside the assumed branch.

EXAMPLE 1 — EVALUATE A PIECEWISE FUNCTION

Given

V (t) = {\begin{cases} 12 t & 0 \leq t \leq 20 \\ 240 + 4 t & 20 < t \leq 60 \end{cases}

, find

V (10)

and

V (50)

SOLUTION

For $V (10)$ : $t = 10$ is in the first range $0 \leq t \leq 20$ , so use $V (t) = 12 t$ .

$V (10)$	$=$	$12 \times 10$
	$=$	$120 L$

For $V (50)$ : $t = 50$ is in the second range $20 < t \leq 60$ , so use $V (t) = 240 + 4 t$ .

$V (50)$	$=$	$240 + 4 (50)$
	$=$	$440 L$

Answer: $V (10) = 120$ L and $V (50) = 440$ L.

V (10) = 120, V (50) = 440

EXAMPLE 2 — READING A STEP GRAPH

A car park charges

$ 4

for

0 < t \leq 1

$ 7

for

1 < t \leq 2

$ 10

for

2 < t \leq 3

, and a flat

$ 15

(maximum) for

t > 3

. Find the cost for

2.5

hours and for

5

hours.

SOLUTION

For $2.5$ hours: $2.5$ is in the range $2 < t \leq 3$ , so the cost is $$ 10$ .

For $5$ hours: $5$ is past $3$ , so the maximum applies: cost $=$ $$ 15$ .

Answer: $2.5$ hours costs $$ 10$ ; $5$ hours costs $$ 15$ .

C (2.5) = $ 10, C (5) = $ 15

EXAMPLE 3 — TIERED TARIFF

Water is billed as follows: the first

100

kL at

$ 1.50

/kL, the next

200

kL at

$ 2.50

/kL, and anything above

300

kL at

$ 4.00

/kL. Find the bill for

400

kL.

SOLUTION

Split $400$ kL across the three bands. Band 1 takes the first $100$ kL, band 2 the next $200$ kL, band 3 the remaining $100$ kL.

Band 1	$=$	$100 \times $ 1.50 = $ 150$
Band 2	$=$	$200 \times $ 2.50 = $ 500$
Band 3	$=$	$100 \times $ 4.00 = $ 400$
Total	$=$	$$ 150 + $ 500 + $ 400$
	$=$	$$ 1 050$

Answer: the bill for $400$ kL is $$ 1 050$ .

total = $ 1050

EXAMPLE 4 — BREAK-EVEN WITH A PIECEWISE PLAN

Plan A:

$ 0.20

per minute, flat. Plan B:

$ 0.50

per minute for the first

30

min, then

$ 0.10

per minute. Find when both plans cost the same.

SOLUTION

After the first $30$ min, Plan B has cost $30 \times $ 0.50 = $ 15$ . For $t > 30$ , the costs are:

Plan A cost	$=$	$0.20 t$
Plan B cost	$=$	$15 + 0.10 (t - 30)$

Set the costs equal:

$0.20 t$	$=$	$15 + 0.10 t - 3$
$0.20 t$	$=$	$12 + 0.10 t$
$0.10 t$	$=$	$12$
$t$	$=$	$120$

Check: $120 > 30$ , so we are in the correct branch of Plan B.

Answer: at $t = 120$ minutes, both plans cost $$ 24$ .

t = 120 min, cost = $ 24

Common pitfalls

Don't apply a formula outside its domain. Each branch of a piecewise function is only valid in its stated range. Using

V (t) = 12 t

when

t = 50

(which is outside

0 \leq t \leq 20

) gives a wrong answer.

Mind the open vs closed circles. On a step graph, an open circle at

t = 2

means the step does not include

t = 2

— at exactly

t = 2

you have already moved up to the next step.

Tiered tariffs are not "highest rate wins". If you use

400

kL of water, you do not pay

$ 4.00

per kL on all

400

kL. Only the portion above

300

kL is charged at the top rate; the rest pays its own band's rate.

Check that break-even falls in the assumed branch. When you set up a break-even equation for a piecewise plan on a particular branch, the solution must satisfy that branch's condition. If it doesn't, you used the wrong branch — try another.

Strict vs non-strict inequalities. At a boundary, the difference between

<

and

\leq

decides which branch applies. Read the brace notation carefully.

Frequently asked questions

What is a piecewise linear function?

A piecewise linear function is a function defined by two or more straight-line pieces, each valid over its own range of x values (its domain). The pieces are usually joined end-to-end so the graph is continuous, but they have different slopes.

How do I evaluate a piecewise function at a given x?

Find which branch the given x value falls in by checking the conditions, then substitute x into that branch's formula. Each branch only applies to its own domain — never use a formula outside its stated range.

What is a step graph?

A step graph is a function that holds a constant value over a range of x, then jumps suddenly to a new constant at a fixed point. The graph looks like a series of horizontal steps. Step graphs are commonly used for postage charges, parking fees and tax brackets.

What do open and closed circles on a step graph mean?

A filled (closed) circle at an endpoint means that x value is included — the step's price applies AT that point. An open circle means that x value is excluded — at that exact point you have already moved to the next step's price.

How do tiered tariffs work?

Tiered tariffs charge each band's rate only on the amount within that band, not on the full total. To compute a tiered bill, work out the cost for each band separately (band amount times band rate), then add them together. Do not charge the highest rate on the entire amount.

What is brace notation for a piecewise function?

Brace notation uses a single large left brace to group several formulas, with the condition for each formula written to its right. For example, f(x) equals 2x when x is less than 1, and f(x) equals x + 1 when x is greater than or equal to 1. It is a compact way to write a function with multiple rules.

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More in Linear Equations And Their Graphs

Piecewise Linear And Step Graphs

📖 Theory

Piecewise function — brace notation

Endpoint conventions on a step graph

Tiered tariff — total cost

Evaluating a piecewise function at a given x

Reading a step graph

Computing a tiered tariff bill

Break-even with a piecewise plan

Common pitfalls

Frequently asked questions

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Theory

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