Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 1 - Linear and nonlinear modelling
Linear and Nonlinear Modelling | Year 10 Mathematics
Choose a model from tables, graphs and contexts, then explain where the model is useful and where it breaks.
Updated 2026-06-15 - 4 min read
A mathematical model is a simplified way to represent a real situation. In Year 10, the important skill is choosing whether a relationship looks linear or nonlinear, then explaining what the model can and cannot tell you.
Linear models
A relationship is linear when the same change in the input produces the same change in the output.
Example: A taxi charges a $5$ booking fee plus $2$ per kilometre.
If $d$ is distance in kilometres and $C$ is cost in dollars, a model is:
$C = 5 + 2d$
The rate of change is $2$ dollars per kilometre. The graph is a straight line because each extra kilometre adds the same amount.
Nonlinear models
Nonlinear relationships do not have a constant rate of change. They might involve powers, percentages, repeated multiplication, curves or turning points.
Example: The area of a square with side length $s$ is:
$A = s^2$
When $s$ increases from $2$ to $3$, the area changes from $4$ to $9$. When $s$ increases from $3$ to $4$, the area changes from $9$ to $16$. The change is not constant, so the relationship is nonlinear.
Choosing a model
Look for evidence in three places:
- Table: are the output differences constant?
- Graph: is the pattern roughly straight or curved?
- Context: does the situation suggest constant addition, repeated multiplication, area, volume or another changing pattern?
If a table has equal input steps and output differences of $3, 3, 3, 3$, a linear model is reasonable. If the differences are $2, 4, 8, 16$, a nonlinear model is more likely.
Interpolation and extrapolation
Interpolation means estimating inside the range of data you have. Extrapolation means predicting outside the range of data.
If you measure plant height from day 1 to day 20, predicting day 15 is interpolation. Predicting day 100 is extrapolation.
Extrapolation is riskier because the pattern may change. A straight-line model might fit short-term data, but real contexts can have limits, turning points, saturation, shortages or changing conditions.
Senior maths and science often ask whether a model is reasonable. A good response does not only say "the equation works". It explains whether the input value is inside the observed domain and whether the context supports the pattern continuing.
Domain and limitations
The domain is the set of input values that make sense. A model may work only over a sensible range.
A model for plant height might fit the first 20 days, but it cannot be used forever. Plants do not keep growing at the same rate without limit.
Quick check
- A candle burns down by $1.5$ cm each hour. Is a linear model reasonable?
- The area of a circle is $A = \pi r^2$. Is that linear or nonlinear?
- A table has output differences $5, 5, 5$. What does that suggest?
Answers:
- Yes, if the burn rate stays constant over the time being modelled.
- Nonlinear, because the radius is squared.
- A linear model may be reasonable because the rate of change is constant.
Transfer task
Find a real situation that could be modelled mathematically: streaming costs, battery charge, distance travelled, plant growth, sports training or savings. Decide whether a linear or nonlinear model is more reasonable, and state one limitation.