A graph can have correct points, a neat curve and still be marked down. The usual reason is that the scale on an axis is uneven or badly chosen, so the picture no longer represents the numbers.
The fix is a short routine you run before plotting anything. It takes under a minute.
What does “wrong scale” actually mean?
There are four separate problems, and most students have only one of them.
- Uneven intervals. The numbers on the axis are not equally spaced, for example labels placed at the plotted values only.
- A scale that wastes the grid. The graph is squeezed into one corner because the scale was too small.
- A scale that runs off the page. The points do not fit because the scale was too large.
- An awkward step. Steps of 3 or 7 make reading and plotting slow and error-prone.
The scale routine, step by step
- Find the range. Note the smallest and largest x values and y values you need.
- Count the large grid squares available along each axis.
- Divide range by squares to see roughly how many units one square is worth.
- Round up to an easy step: 1, 2, 5, 10, 20, 50 and so on.
- Label at regular intervals, every 2 or 5 large squares, not at your data values.
- Check: does the graph use more than half the grid in each direction, and are the labels equally spaced?
The quadratic structure explorer lets you see how a curve looks under different choices, and the graph-model explorer shows how points and a fitted line relate.
Worked example
Plot y = x² for x from 0 to 6. The grid has 12 large squares across and 18 large squares up.
Step 1, table of values:
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| y | 0 | 1 | 4 | 9 | 16 | 25 | 36 |
Step 2, ranges: x runs from 0 to 6 and y from 0 to 36.
Step 3, units per square: across, 6 units over 12 squares means 1 unit per 2 large squares. Up, 36 units over 18 squares means 2 units per large square.
Step 4, label: x axis: 0, 1, 2, 3, 4, 5, 6 with two large squares between each. y axis: 0, 2, 4, 6 and so on up to 36.
Step 5, plot and draw: the point (5, 25) sits above x = 5, halfway between the 24 and 26 lines. Draw one smooth curve through all seven points.
Check: the curve fills the grid, both axes are evenly spaced, and (6, 36) sits in the upper right corner.
A mistake to avoid
What the student did: labelled the y axis at the plotted values only: 0, 1, 4, 9, 16, 25, 36, placed at equal gaps along the axis.
Why it looks right: the dots form a neat curve, because each dot sits at the right “label position”.
Why it is wrong: the gap between 1 and 4 is the same length as the gap between 25 and 36, although the numbers differ by 3 and 11. The axis no longer measures anything.
The correction is to label at regular numerical steps (0, 2, 4, 6 and so on) and to place each point by its value, not by the nearest label.
Self-check
1. A student plots y = 3x + 1 for x from 0 to 10 on a grid with 16 large squares upward. What y range is needed and what scale suits?
Show answer
When x = 10, y = 31, so y runs from 1 to 31. The range 0 to 32 over 16 squares gives 2 units per large square, a clean step that fits exactly.
2. Why is labelling an axis at 0, 1, 4, 9, 16 with equal gaps wrong?
Show answer
Equal gaps on the page must mean equal numerical differences. Here the differences are 1, 3, 5 and 7, so the axis distorts the graph.
3. True or false: both axes must always use the same scale.
Show answer
False. They only need equal scales when true shape matters, for example with circles or perpendicular lines. Each axis needs equal intervals along itself.
Where this leads next
Practise the core skill in plot coordinates using an appropriate scale, then read meaning from the graph with gradient and intercept in context. The parent topic is graphs and transformations.
A teacher who watches you choose a scale can spot the exact step where the axis goes wrong. Online one-to-one Mathematics tuition gives you that, using your own graphs.