Matching Graphs to Real Situations
Show real-world situations on graphs for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-01). The shape of a graph tells a story: a straight line shows steady change, a flat line shows no change, and a curve shows change that speeds up or slows down.
You will match graphs to situations and sketch them, read distance-time graphs for trips that move away, stop and come back, and see how the depth of water rises over time as containers of different shapes are filled — plus everyday curves like a hot drink cooling down.
Theory
The shape of a graph describes what is happening in a real situation. This NSW Year 12 Mathematics Standard 1 guide shows how straight lines, flat lines and curves match travel, filling containers, savings and cooling, and how to sketch the right graph.
The shape of a graph tells the story of a real situation. A straight line shows steady change, a horizontal line shows no change, and a curve shows change that speeds up or slows down.
On a distance-time graph, a rising line means moving away, a flat line means stopped, and a line sloping down means coming back. A steeper line means faster.
When containers are filled at a steady rate, the depth rises quickly where the container is narrow (steep graph) and slowly where it is wide (gentle graph).
On a distance-time graph the steepness is the speed:
| Shape of the graph | What it means |
|---|---|
| Straight line rising | Steady increase; steeper means faster |
| Horizontal line | No change (on a distance-time graph: stopped) |
| Line sloping down | Decreasing (on a distance-from-home graph: coming back) |
| Curve getting steeper | Change speeding up (e.g. doubling) |
| Curve levelling off | Change slowing down (e.g. a hot drink cooling) |
How to match a graph to a real situation
- Identify what each axis shows (usually time across, the quantity up).
- Describe each section by its shape: rising, flat, falling, getting steeper or levelling off.
- Compare steepness: the steeper section is the faster change.
- Translate each shape into words for the situation (walking away, stopped, coming home, filling quickly).
A: rising — walking 800 m to the shop in 10 min. B: flat — stopped at the shop for 5 min. C: falling — walking 800 m home in 5 min.
| \(\text{A: } 800\div 10\) | \(=\) | \(80 \text{ m/min}\) |
| \(\text{C: } 800\div 5\) | \(=\) | \(160 \text{ m/min}\) |
A: going out, B: stopped, C: returning home; C is fastest (the steepest).
Both get the same volume each minute. In a narrow container that volume stacks up higher, so the depth rises faster and the line is steeper.
Line P is steeper than line Q, so container P is narrower.
The wide bottom needs a lot of water for each centimetre, so the depth rises slowly at first. Higher up the flask is narrow, so the depth rises faster and faster. In the straight neck the width stops changing, so the graph ends with a steep straight line.
A curve that rises slowly at first, then more and more steeply, ending in a steep straight line (the neck).
(a) A phone battery runs down at a steady rate.
(b) A savings balance grows faster and faster.
(c) Hot soup cools down to room temperature.
Match each description of the change to a shape.
(a) Steady decrease \(\Rightarrow\) a straight line sloping down. (b) Growth that speeds up \(\Rightarrow\) a curve rising more and more steeply. (c) Cools quickly while very hot, then slower \(\Rightarrow\) a curve falling quickly, then levelling off.
Common pitfalls
Frequently asked questions
What does a flat line on a distance-time graph mean?
It means the person or object has stopped. The distance from the starting point stays the same while time keeps going.
How can you tell which part of a travel graph is fastest?
The steepest section is the fastest, because the most distance is covered in the least time. You can check by dividing the distance by the time for each section.
Why does a narrow container fill faster than a wide one?
Water goes in at the same rate, so each minute adds the same volume. In a narrow container that volume stacks up higher, so the depth rises faster and the graph is steeper.
What does a curved graph mean in a real situation?
A curve shows change that is not steady. If it gets steeper the change is speeding up, like savings growing faster and faster. If it flattens out the change is slowing down, like a hot drink cooling.
What graph shows something cooling down?
A curve that falls quickly at first and then levels off towards room temperature. It does not drop below room temperature.