Graphs of practical situations, linear vs non-linear and exponential growth: HSC Maths Standard 1 Year 12
“A3.2 Graphs of practical situations: distinguish between linear and non-linear relationships, recognise exponential (and other non-linear) relationships from the shape of their graphs, and construct and interpret graphs of practical situations, including distance-time graphs and the limitations of models”
Linear relationships graph as straight lines with constant change; non-linear ones curve. Recognise exponential growth (steepening), exponential decay (levelling off) and inverse relationships from their shapes, read distance-time graphs using gradient as speed, and remember models only apply over a limited domain.
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What this dot point is asking
You need to recognise the type of relationship from a graph or table, construct and read graphs of practical situations (especially distance-time graphs), and comment on how far a model can be trusted. NESA's topic guide says Standard 1 students are not expected to find the algebraic equation of an exponential function from its graph.
The answer
Linear or non-linear?
- Linear: a straight line; y changes by a constant amount for each equal step in x.
- Non-linear: a curve; the rate of change varies.
Recognising non-linear shapes
| Relationship | Graph shape | Example |
|---|---|---|
| Exponential growth | Rises slowly then increasingly steeply | Population growth, compound interest |
| Exponential decay | Falls quickly then levels off | Car depreciation, cooling, medicine leaving the body |
| Inverse (reciprocal) | Falls steeply then flattens, never reaching zero | Travel time against speed for a fixed distance |
Graphs of practical situations
- Distance-time graphs: the gradient is the speed; a horizontal section means stopped; a steeper line means faster.
- Filling containers: the rate the water level rises depends on the container's width (narrow sections fill faster).
- Other contexts: temperature over a day, mobile data used over a month, tank levels.
Limitations of models
Models may apply only over a particular domain (range of x-values). A person's height may be roughly linear with age for a few years but not for life; exponential growth cannot continue forever. Always ask whether the prediction is sensible.
The value of a car is modelled by exponential decay: , where is years.
- After 1 year: .
- After 3 years: .
- The graph falls steeply at first, then flattens: the car loses more dollars in the early years.
- The model never reaches zero, but in reality an old car may be scrapped, so the model is limited.
Calling any increasing graph exponential. Check whether the increases grow (exponential) or stay equal (linear).
Misreading the gradient on distance-time graphs: steepness is speed, not distance.
Extrapolating too far. Comment on the model's limits.
Practice questions
Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.
foundation3 marksFor each table, state whether the relationship is linear or non-linear: (a) x = 0, 1, 2, 3 and y = 5, 8, 11, 14; (b) x = 0, 1, 2, 3 and y = 2, 4, 8, 16.Show worked solution →
(a) Linear: y increases by 3 each time x increases by 1.
(b) Non-linear (exponential): y doubles each time, so the increases (2, 4, 8) get larger.
Marking guide: 1 mark each, 1 mark for a reason.
core4 marksMaya walks 2 km to a shop in 30 minutes, stays 15 minutes, then catches a bus 6 km further in 15 minutes. Describe the distance-time graph and find her speed in km/h for each moving section.Show worked solution →
- Graph
- a straight line rising from (0, 0) to (30 min, 2 km); a horizontal line from 30 to 45 minutes at 2 km; a steeper line rising from (45 min, 2 km) to (60 min, 8 km).
- Walking speed
- 2 km in 0.5 h = 4 km/h.
- Bus speed
- 6 km in 0.25 h = 24 km/h.
The bus section is steeper because the speed is greater.
Marking guide: 2 marks for describing the graph, 1 mark for each speed.
exam5 marksA town's population was 8000 in 2020 and grows by 3% per year. (a) Complete the population for 2021 and 2022. (b) Describe the shape of the graph of population against time. (c) Explain why this model may not be reliable for predicting the population in 2060.Show worked solution →
(a) 2021: . 2022: (to the nearest person).
(b) An exponential growth curve: increasing, and getting steeper over time because each year's increase is 3% of a larger population.
(c) The model assumes the 3% growth rate continues for 40 years. In reality, growth depends on housing, jobs, services and migration, which can change. Exponential models also grow without limit, which is unrealistic for a town. Predictions far outside the data are unreliable.
Marking guide: 2 marks for (a), 1 mark for (b), 2 marks for (c).