To plot coordinates well, first look at the smallest and largest x-values and y-values, then choose a scale that fits all of them on the grid. Every axis needs an even scale, a label and numbers written at regular steps. This skill is used whenever a question asks you to draw a graph, from straight lines to curves.
It sits at the start of graphs and transformations. Every other lesson in the module assumes you can put points in the right place.
How do you choose a scale?
Start with the range. Find the lowest and highest value for each variable, including negatives, and work out the total length the axis must cover.
Then divide that length by the space you have. If the space is 12 cm for 60 units, each centimetre can stand for 5 units. Round to a friendly step: 1, 2 or 5 (or ten times those), never 3, 7 or 15.
Finally, draw the axes through zero when negatives are involved, number them at even intervals and label each axis with its quantity and unit.
Worked example
Plot A(−15, 40), B(0, 25), C(20, −10) and D(35, −30). You have a grid 12 cm wide and 16 cm tall.
Step 1, ranges: x runs from −15 to 35. y runs from −30 to 40.
Step 2, axis lengths: make the x-axis run from −20 to 40, which is 60 units. Make the y-axis run from −40 to 40, which is 80 units.
Step 3, scale: 60 units on 12 cm gives 5 units per cm. 80 units on 16 cm also gives 5 units per cm. So one scale, 1 cm = 5 units, works for both.
Step 4, positions from the origin:
| Point | x steps | y steps |
|---|---|---|
| A(−15, 40) | 3 cm left | 8 cm up |
| B(0, 25) | on the y-axis | 5 cm up |
| C(20, −10) | 4 cm right | 2 cm down |
| D(35, −30) | 7 cm right | 6 cm down |
Check each: 15 ÷ 5 = 3, 40 ÷ 5 = 8, 25 ÷ 5 = 5, 20 ÷ 5 = 4, 10 ÷ 5 = 2, 35 ÷ 5 = 7, 30 ÷ 5 = 6.
Step 5, mark and label each point with a small cross, and write the letter beside it.
The mistake to watch for
A common slip is to swap the coordinates, reading the pair the wrong way round.
Mistaken plot: A is placed at 40 on the x-axis and −15 on the y-axis.
The x-axis only runs to 40, so the point lands far to the right, nowhere near the other points.
The correction is to say it aloud: “x first, along the corridor; y second, up the stairs”. For A(−15, 40), go 15 units left, then 40 up. A quick look at the pattern of all points also catches this, because one point far from the rest is usually a swap.
Check yourself
Try these without a calculator, then open each answer.
1. A quantity runs from 0 to 180 and you have a 12 cm axis. Suggest a scale.
Show answer
1 cm = 10 units needs 18 cm, which is too long. 1 cm = 20 units needs 9 cm, which fits and is easy to read. So use 1 cm = 20 units.
2. Describe how to plot (−3, 7) and state which quadrant it is in.
Show answer
Start at the origin, move 3 units left, then 7 units up. A negative x and positive y is the second quadrant.
3. Three corners of a rectangle are (1, 1), (1, 4) and (5, 4). Find the fourth corner.
Show answer
The side from (1, 1) to (1, 4) is vertical and the side from (1, 4) to (5, 4) is horizontal. The fourth corner shares x = 5 with the third and y = 1 with the first, so it is (5, 1).
Where this leads next
Once plotting is automatic, move on to interpreting gradient and intercept in a context, then test the whole module with the graphs and transformations practice set. The map scale and gradient practice tool gives more work on reading scales, and the non-calculator working trainer helps when the arithmetic behind a scale gets fiddly.
Some students understand each step here but still produce cramped graphs under time pressure. That is exactly the kind of pattern our teachers look for in online one-to-one Mathematics tuition.