Strengths & Limitations of Models
Weigh up the strengths and weaknesses of models for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-01). A linear model adds the same amount each step and an exponential model multiplies by the same number each step — both give a simple rule for predicting, but only over a sensible range.
You will tell the two models apart from a table or graph, pick the one that suits a real situation, use it to predict, and explain when a prediction can be trusted and when it reaches too far beyond the data — such as a child's height or a growing bacteria colony.
Theory
A linear model changes by the same amount each step and an exponential model multiplies by the same number each step. This Year 12 Standard 1 (NSW) guide shows how to tell them apart from a table, use a model to make predictions, and judge the strengths and limitations of a model — especially when a prediction goes far beyond the data.
A model is a rule that describes a real situation so you can make predictions. It gives one simple rule for any input, but it only describes the real situation over a limited range — always check that a prediction makes sense.
A linear model changes by the same amount each step (a constant difference), so its graph is a straight line — rising if the same amount is added each step or falling if the same amount is taken away (e.g. a machine losing $500 a year), e.g. \(C=20+5n\). An exponential model is multiplied by the same number each step (a constant ratio), so its graph is a curve that gets steeper (growth) or flattens towards \(0\) (decay), e.g. \(N=20\times2^t\).
A model is most reliable inside the range of the data it was built from. Predicting far outside that range is called extrapolation and can give impossible answers, because real things have limits — a tank fills up, a paddock has a fixed size, and animals stop growing.
A linear model starts at \(a\) and adds \(b\) each step:
An exponential model starts at \(a\) and is multiplied by \(b\) each step:
How to use and judge a model
- Test the data: subtract neighbouring values for a constant difference (linear) or divide them for a constant ratio (exponential).
- Substitute the input into the model's rule, working out any power first, to make the prediction.
- Judge the answer: is the input inside the range of the data, and is the answer actually possible in real life?
| \(x\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Table A: \(y\) | 7 | 11 | 15 | 19 |
| Table B: \(y\) | 3 | 12 | 48 | 192 |
Subtract the values in Table A; divide the values in Table B.
| \(\text{A: } 11-7\) | \(=\) | \(4\) |
| \(15-11\) | \(=\) | \(4\) |
| \(19-15\) | \(=\) | \(4\) |
| \(\text{B: } \dfrac{12}{3}\) | \(=\) | \(4\) |
| \(\dfrac{48}{12}\) | \(=\) | \(4\) |
| \(\dfrac{192}{48}\) | \(=\) | \(4\) |
Table A is linear (adds \(4\) each time). Table B is exponential (multiplies by \(4\) each time).
Substitute, then check that the answer makes sense.
| \(\text{(i) } M\) | \(=\) | \(3+2.5(4)\) |
| \(\) | \(=\) | \(3+10 = 13\) |
| \(\text{(ii) } M\) | \(=\) | \(3+2.5(60)\) |
| \(\) | \(=\) | \(153\) |
A \(153\) kg dog is not realistic: dogs stop growing, and \(60\) months is far outside the data.
(i) \(13\) kg. (ii) Growth does not continue at a steady rate, so the model only works for young puppies.
Work out the power first, then multiply.
| \(\text{(i) } A\) | \(=\) | \(3\times2^{4}\) |
| \(\) | \(=\) | \(3\times16 = 48\) |
| \(\text{(ii) } A\) | \(=\) | \(3\times2^{7}\) |
| \(\) | \(=\) | \(3\times128 = 384\) |
\(384 > 200\), but the paddock is only \(200\text{ m}^2\).
(i) \(48\text{ m}^2\). (ii) \(384\text{ m}^2\) is impossible — the weeds cannot keep doubling once the paddock is full.
| \(\text{(i) } C\) | \(=\) | \(35+12(6)\) |
| \(\) | \(=\) | \(35+72 = 107\) |
(ii) One simple rule gives the cost for any number of weeks quickly.
(iii) \(w=6\) is inside the data; \(30\) and \(120\) are outside it (prices may change).
(i) \(\$107\). (ii) Quick, simple predictions. (iii) \(w=6\) weeks.
Common pitfalls
Frequently asked questions
What is the difference between a linear and an exponential model?
A linear model changes by the same amount each step, so its graph is a straight line. An exponential model is multiplied by the same number each step, so its graph is a curve that gets steeper for growth or flattens towards zero for decay.
How do you tell if a table is linear or exponential?
Subtract neighbouring values. If all the differences are equal, the table is linear. If not, divide neighbouring values. If all the ratios are equal, the table is exponential. For example 7, 11, 15, 19 adds 4 each time (linear) and 3, 12, 48, 192 multiplies by 4 each time (exponential).
What are the limitations of a mathematical model?
A model only describes the real situation over a limited range. Real things have limits, such as a tank filling up, a paddock having a fixed size or animals stopping growing, so a model can give impossible answers when it is used far outside the data.
What is extrapolation and why is it risky?
Extrapolation is using a model to predict far outside the range of the data it was built from. It is risky because the pattern may not continue. For example, a puppy growth model built from ages 1 to 6 months predicts an unrealistic 153 kg dog at 60 months.
What are the strengths of a linear model?
A linear model gives one simple rule, it makes quick predictions for any input, and it is reliable inside the range of the data. For example, a gym cost of C equals 35 plus 12w gives the cost for any number of weeks straight away.