Limitations of extrapolation | Year 11

01 / LEARN

Understand the idea.

Where does the evidence apply?

Interpolation estimates within the measured input range; extrapolation extends beyond it. First identify that range. A calculator can evaluate either input, but numerical evaluation alone does not establish that the forecast is justified.

Ask whether the original mechanism continues. Switching on a heater changes a cooling process; a rule fitted before the switch lacks evidence for later behaviour. Interpolation can also fail if conditions change inside the measured period.

Challenge predictions with bounds and evidence

Test predictions against hard bounds such as capacity, non-negative volume and available stock. If an unrestricted formula crosses a bound, identify when that happens and explain which modelling assumption must change.

For a first whole-year threshold, calculate the adjacent yearly values and justify why no earlier year qualifies. Constant positive growth makes values increase, so once a threshold is crossed all later unrestricted predictions remain above it.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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?

Worked solution

1
Evaluate the old formula at six.
$T (6) = 18 - 2 (6) = 6$
This is the numerical output of the pre-heater rule, in degrees.
2
Compare with the evidence interval.
$6 > 4$
Six hours is outside the zero-to-four-hour observation range.
3
Assess the changed conditions.
The heater is switched on at four hours, so the earlier cooling mechanism no longer supports a reliable forecast of six degrees.

Answer

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

EXAMPLE 02

A capacity bound

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

Worked solution

1
Evaluate year eight.
$P (8) = 100 {(1.3)}^{8} \approx 815.73 < 1000$
The prediction remains below the habitat capacity at this whole year.
2
Evaluate the next year.
$P (9) = 100 {(1.3)}^{9} \approx 1060.45 > 1000$
The unrestricted prediction now exceeds capacity.
3
Identify the first qualifying year.
$n = 9$
The factor is greater than one, so the sequence increases; all earlier non-negative years are no larger than year eight.
4
Interpret the model limit.
The habitat cannot sustain this unrestricted prediction, so continued thirty-percent growth needs revision by year nine.

Answer

Year 9.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Yes.

Yes. Age $14$ lies inside the measured interval $12 \leq a \leq 16$ , so the prediction is interpolation.

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6 worksheet questions · 13 marks

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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 ↗

In this lesson

Understand the idea.

Where does the evidence apply?

Challenge predictions with bounds and evidence

See the method in action.

The mechanism changes

Evaluate the old formula at six.

Compare with the evidence interval.

Assess the changed conditions.

A capacity bound

Evaluate year eight.

Evaluate the next year.

Identify the first qualifying year.

Interpret the model limit.

Try it yourself.

Check your understanding

Keep practising