Sketching Informal Graphs
An informal graph tells a story with its shape rather than exact numbers: a section going up, staying flat or coming down shows how something changes over time. In NSW Year 8 maths these sketches build the reading skills behind distance–time graphs (NESA outcome MA4‑RAT‑C‑01).
You will match stories such as a bike ride, a bath or a Ferris wheel to the right sketch, explain what each section means, and read changes and rates from graphs of a burning candle, a cooling drink or a leaking tank. You will also decide how fast water rises in containers of different shapes, where the level rises faster in a narrow part, and see how a curved graph shows a change that speeds up or slows down. It is part of Year 8 Time and Travel.
Every question with a fully worked solution.
- Sketching Informal Graphs - Video - Distance Time Graphs Watch
Theory
An informal graph tells the story of how something changes. Up means increasing, flat means not changing, down means decreasing, and steeper means changing faster. A drone that drops \(40\) m in \(10\) s is coming down at \(40 \div 10 = 4\) m/s. This Year 8 maths page shows how to read and sketch these graphs.
Read the axes first: what is measured up the side, what goes across (usually time), and the units.
Up means the amount is getting bigger, flat means it is not changing, down means it is getting smaller. Steep means changing quickly, gentle slowly. Compare sections by how steep they are, not by how high up they sit.
A straight section is a steady rate: rate \(=\) change \(\div\) time, in units such as cm/min or \(^\circ\)C per hour.
Sketching from a story. Split the story into sections; for each decide up, flat or down, then steep or gentle. Draw them in order, joined up, with both axes labelled. A sketch needs the right shape, not exact numbers.
Curves. A curve getting steeper means changing faster and faster; one getting flatter means changing more and more slowly. A vertical jump means a sudden change; repeating humps mean something going up and down over and over.
Filling a container at a steady rate: a wide part rises slowly, a narrow part quickly. One width all the way up gives a straight line. Three times as wide across means nine times the area, so the depth rises nine times as slowly.
Reading a section:
| Section | Means |
|---|---|
| up, steep | increasing quickly |
| up, gentle | increasing slowly |
| flat | no change (stopped, waiting, hovering) |
| down | decreasing |
| curve flattening | change slowing down |
| vertical jump | sudden change |
Reading an informal graph
- Read both axes and their units.
- Split the graph into sections and describe each: up, flat or down; steep or gentle.
- Calculate a rate for a straight section: change \(\div\) time.
Sketching from a story
- List the parts of the story in order.
- Decide the shape of each part.
- Draw them joined up and label the axes.
(i) When did he start?
(ii) How far is the friend’s house?
(iii) What was the total distance?
(iv) Did he stop anywhere?
(v) How long was the whole trip?
| \(8 + 8\) | \(=\) | \(16\) |
| \(15 - 11\) | \(=\) | \(4\) |
(i) At 11:00
(ii) \(8\) km (reached at 12:00 noon)
(iii) \(16\) km there and back
(iv) Not at the friend’s house, but he stopped for an hour on the way back, \(5\) km from home, from 13:00 to 14:00.
(v) \(4\) hours (11:00 to 15:00)
(i) What is the drone doing in the flat section?
(ii) Which section is steepest?
(iii) How fast does it come down, in m/s?
| \(40 \div 10\) | \(=\) | \(4\) |
(i) Hovering at \(40\) m from \(20\) s to \(50\) s
(ii) The last: \(40\) m in \(10\) s (the climb took \(20\) s)
(iii) \(4\) m/s
On the bench: up, gentle. In the oven: up, steep. Cooling: down, fast at first then slower, so a curve that flattens.
A gentle rise, then a steep rise, then a fall that flattens out.
Three times as wide means \(3 \times 3 = 9\) times the area, so the top takes \(1 \times 9 = 9\) minutes and rises a ninth as fast.
A steep straight line for \(1\) minute, then a gentler straight line for \(9\) minutes.
(i) What happened from month \(4\) to month \(8\)?
(ii) Find her saving rate in the first \(4\) months.
(iii) She keeps up the month \(10\) to \(14\) rate. When does she first have \(\$600\)?
| \(400 \div 4\) | \(=\) | \(100\) |
| \((400 - 200) \div (14 - 10)\) | \(=\) | \(50\) |
| \(200 \div 50\) | \(=\) | \(4\) |
(i) Flat: no change
(ii) \(\$100\) per month
(iii) \(\$200\) more at \(\$50\) a month takes \(4\) months: month \(18\)
Common pitfalls
Frequently asked questions
What does a flat section on a graph mean?
The amount is not changing during that time, such as a car that is stopped.
What does a steeper line mean?
The amount is changing faster.
How do I find a rate from a straight section?
Divide the change in the amount by the time taken.
How do I sketch a graph from a story?
Break the story into parts, decide if each goes up, stays flat or goes down, and how steeply, then draw them in order.
Why does a narrow container fill faster?
The same volume of water spreads over a smaller area, so the depth rises more quickly.