Constructing and interpreting straight-line models | Year 11

01 / LEARN

The idea, step by step

Turn observations into a rule

Name the input, output and units. If a constant rate is reasonable, calculate it using the change between the observations. Substitute one observation into $y = m x + b$ to find the constant. Check the second observation too.

Two points determine a line but do not prove that a real process is linear. Further measurements and knowledge of the mechanism are needed to support that assumption.

y = m x + b

A rule needs conditions

Time usually begins at a specified instant; item counts may be whole numbers; a container cannot hold a negative amount or exceed capacity. State these limits when giving a practical model.

Do not force a line through the origin unless the context supports zero output at zero input. A negative intercept can be mathematically valid even when zero lies outside the observed input range.

A common trap

Assuming every practical line passes through the origin.

Instead: A starting amount or fixed fee produces a nonzero intercept.

02 / WORKED EXAMPLES

See the thinking

Example 1

Filling a container

A container holds 12 L at $t = 0$ and 36 L after 3 min. It fills steadily to its 60 L capacity. Model its volume.

Rate $= (36 - 12) / 3 = 8$ L/min.
The initial amount is 12 L, so $V = 12 + 8 t$ .
Capacity occurs when $60 = 12 + 8 t$ , giving $t = 6$ .

V = 12 + 8 t

, for

0 \leq t \leq 6

min.

Example 2

A descending line

Find the line through $(2, 9)$ and $(6, 1)$ .

$m = (1 - 9) / (6 - 2) = - 2$ .
Use $9 = - 2 (2) + b$ to obtain $b = 13$ .
Check: $- 2 (6) + 13 = 1$ .

y = - 2 x + 13

Check your understanding

A plant starts at 15 cm and grows 2 cm per week in a model. Write its rule.

Show answer

$H = 15 + 2 t$ cm, where $t$ is weeks since the first measurement.

The starting amount is the constant; the weekly change is the coefficient.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the line through $(0, 6)$ and $(4, 18)$ .
2 marks
Find the line through $(3, 8)$ and $(7, 20)$ .
2 marks
A machine has a setup cost of 250 dollars and costs 7 dollars per item. Model total cost.
2 marks
A tank contains 80 L and loses 5 L/min. Write the rule and physical time domain.
2 marks
Explain why $d = 12 t$ passes through the origin when d is distance travelled and t is elapsed time.
2 marks
Are $(0, 2), (1, 5), (2, 10)$ exactly linear? Explain.
2 marks

Read the worked solutions

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

Find the line through $(0, 6)$ and $(4, 18)$ .
2 marks
Gradient $= (18 - 6) / 4 = 3$ .
The first point gives intercept 6.
$y = 3 x + 6$ .
Find the line through $(3, 8)$ and $(7, 20)$ .
2 marks
Gradient $= 12 / 4 = 3$ .
$8 = 3 (3) + b$ gives $b = - 1$ .
$y = 3 x - 1$ .
A machine has a setup cost of 250 dollars and costs 7 dollars per item. Model total cost.
2 marks
Separate fixed cost 250 from variable cost $7 n$ .
$C = 250 + 7 n$ , for non-negative integer $n$ .
A tank contains 80 L and loses 5 L/min. Write the rule and physical time domain.
2 marks
Set $V = 0$ to find emptying time 16 min.
Exclude negative volume.
$V = 80 - 5 t$ , $0 \leq t \leq 16$ .
Explain why $d = 12 t$ passes through the origin when d is distance travelled and t is elapsed time.
2 marks
Substitute $t = 0$ : $d = 0$ .
There is no starting-distance term.
At elapsed time zero, distance travelled is zero.
Are $(0, 2), (1, 5), (2, 10)$ exactly linear? Explain.
2 marks
Equal x-steps give increases 3 and 5.
A straight line needs a constant increase.
No.

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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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