Comparing linear with exponential change | Year 11

01 / LEARN

The idea, step by step

Different mechanisms

A linear model adds a constant amount in each equal step. An exponential model multiplies by a constant positive factor. Inspect consecutive differences and ratios rather than relying on the visual impression of a short graph.

An increase of 20 units per year is not an increase of 20% per year. The fixed amount is independent of the current size, while the percentage multiplier changes the absolute increase as the amount changes.

Compare fairly

Models with the same starting value can agree early, then diverge. Calculate both at the same times and use their practical domains. Tables can locate a change of order, followed by closer calculations if needed.

A faster-growing prediction is not automatically the better model. Check the mechanism, fit to observations and assumptions. Either simple model can fail beyond the period for which its conditions are reasonable.

A common trap

Treating 20% growth as adding 20 at every step.

Instead: A percentage acts on the changing amount; a fixed increase does not.

02 / WORKED EXAMPLES

See the thinking

Example 1

Same start, different change

Compare $L = 100 + 20 n$ and $E = 100 {(1.2)}^{n}$ at years 0 to 4.

n	0	1	2	3	4
L	100	120	140	160	180
E	100	120	144	172.8	207.36

L gives 100, 120, 140, 160, 180.
E gives 100, 120, 144, 172.8, 207.36.
They agree at 0 and 1; E is larger at the other listed years.

At year 4: L = 180 and E = 207.36.

Example 2

A later change of order

Compare $L = 200 + 30 t$ and $E = 200 {(1.1)}^{t}$ at times 8 and 9.

At 8, L = 440 and E is about 428.72.
At 9, L = 470 and E is about 471.59.
The continuous crossing lies between these times. The order has reversed by whole reporting time 9.

L is larger at 8; E is larger at 9.

Check your understanding

At consecutive inputs, outputs are 10, 15, 20, 25. Which simple model fits exactly?

Show answer

A linear model.

Differences are all 5; ratios vary.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Classify 40, 50, 60, 70 and give a rule starting at $n = 0$ .
2 marks
Classify 40, 60, 90, 135 and give a rule.
2 marks
Compare $A = 300 + 30 t$ and $B = 300 {(1.08)}^{t}$ at $t = 2$ .
2 marks
Find the differences for 100, 110, 121, 133.1 and identify its change.
2 marks
One tank loses 4 L/min; another loses 4% of its current volume per minute. Identify the model types.
2 marks
Why might neither an unrestricted linear nor an exponential model describe a population indefinitely?
3 marks

Read the worked solutions

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

Classify 40, 50, 60, 70 and give a rule starting at $n = 0$ .
2 marks
Constant difference is 10.
Linear; $L = 40 + 10 n$ .
Classify 40, 60, 90, 135 and give a rule.
2 marks
Each ratio is 1.5.
Exponential; $E = 40 {(1.5)}^{n}$ .
Compare $A = 300 + 30 t$ and $B = 300 {(1.08)}^{t}$ at $t = 2$ .
2 marks
Substitute the same time in both models.
A = 360; B = 349.92; A is larger.
Find the differences for 100, 110, 121, 133.1 and identify its change.
2 marks
Differences vary but consecutive ratios are all 1.1.
10, 11, 12.1; exponential factor 1.1.
One tank loses 4 L/min; another loses 4% of its current volume per minute. Identify the model types.
2 marks
One removes a fixed amount; the other applies factor 0.96.
The first is linear; the second is exponential.
Why might neither an unrestricted linear nor an exponential model describe a population indefinitely?
3 marks
A simple rule may ignore crowding, migration or limited food.
State a useful domain and check new observations.
Resources, capacity and conditions may change.

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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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