Limitations of extrapolation | Year 11

01 / LEARN

The idea, step by step

Where does the evidence apply?

Interpolation predicts inside the observed input range; extrapolation goes outside it. Interpolation is often less risky, but neither is guaranteed correct. A changed policy, machine or physical condition can alter the relationship.

A calculator can evaluate a formula at an input where the model has no practical meaning. Separate the numerical evaluation from the claim that it is a reliable forecast, and state what conditions would need to remain unchanged.

Challenge predictions with bounds and evidence

Check the sign, size and units. A negative volume or an amount above a stated capacity suggests a model limitation, not just a rounding problem. Find when the rule reaches the bound and explain its valid range.

Compare later observations with predictions. If errors have a pattern, consider a changed mechanism or another model. Explain which assumption is doubtful and what extra information would help.

A common trap

Reporting a formula value as reliable without checking its assumptions.

Instead: Check the domain and mechanism before accepting a forecast.

02 / WORKED EXAMPLES

See the thinking

Example 1

The mechanism changes

Temperature follows $T = 18 - 2 t$ degrees for observations from 0 to 4 h. A heater is switched on at 4 h. Can the rule reliably forecast time 6?

The old rule evaluates to $T (6) = 6$ degrees.
Time 6 is outside the observed range, and the heater changes the process.
The evaluated value should not be reported as a reliable post-heater prediction.

The old rule gives 6 degrees, but the forecast is not justified.

Example 2

A capacity bound

A habitat has capacity 1000, while a model predicts $P = 100 {(1.3)}^{n}$ . Find the first whole year exceeding capacity.

At 8 the prediction is about 815.73.
At 9 it is about 1060.45.
The unrestricted growth assumption exceeds capacity by year 9.

Year 9.

Check your understanding

Data were collected for ages 12 to 16. Is a prediction at 14 interpolation?

Show answer

Yes.

14 lies inside the observed input interval.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Data cover times 2 to 10. Classify predictions at 6 and 15.
2 marks
A model is $V = 50 - 5 t$ . Explain its physical domain for an initially full draining tank.
2 marks
A sales forecast assumes 10% growth every month. State two assumptions needed for a year-ahead prediction.
2 marks
A model predicts 5000 units, but machine capacity is 3000. Interpret this.
2 marks
A tariff is $C = 5 + 3 d$ only for trips up to 20 km. Evaluate at 30 km and explain the limit.
2 marks
Residuals from three new observations are all positive and steadily increasing. What should be checked?
3 marks

Read the worked solutions

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

Data cover times 2 to 10. Classify predictions at 6 and 15.
2 marks
6 lies inside the interval, but 15 lies outside.
6: interpolation; 15: extrapolation.
A model is $V = 50 - 5 t$ . Explain its physical domain for an initially full draining tank.
2 marks
At 10 min it reaches zero.
Later formula values would be negative volumes.
$0 \leq t \leq 10$ min.
A sales forecast assumes 10% growth every month. State two assumptions needed for a year-ahead prediction.
2 marks
Give conditions about the process, not just calculator precision.
For example, the rate stays 10% and supply/customer conditions permit that growth.
A model predicts 5000 units, but machine capacity is 3000. Interpret this.
2 marks
Compare with the stated 3000-unit bound.
A constrained model or revised operating plan is needed.
The unrestricted prediction exceeds capacity and is not feasible.
A tariff is $C = 5 + 3 d$ only for trips up to 20 km. Evaluate at 30 km and explain the limit.
2 marks
$5 + 3 (30) = 95$ .
30 km is outside the stated tariff range.
Formula value 95 dollars; actual fare is not established.
Residuals from three new observations are all positive and steadily increasing. What should be checked?
3 marks
The pattern suggests the rule increasingly underpredicts.
Collect relevant observations and compare a different mechanism.
Whether the rate or model type is changing.

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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 ↗

Back to Linear and exponential models