Why graphs are worth learning
A graph turns numbers into a picture. Patterns that are hard to spot in a table, such as steady growth, a sudden jump or a plateau, often become obvious on a graph. Reading one correctly takes a few skills: understanding the axes, locating points, and interpreting the shape of a line.
Axes and coordinates
Most graphs in basic math have two number lines that cross at right angles.
- The x-axis runs horizontally (left to right).
- The y-axis runs vertically (bottom to top).
- They meet at the origin, where both values are zero. Its coordinates are (0, 0).
A point is described by a pair of numbers in brackets, written (x, y). The first number tells you how far to move along the x-axis; the second tells you how far to move up or down. To plot (3, 2), start at the origin, move 3 to the right, then 2 up. A helpful memory aid is that you go along the corridor before you go up the stairs, so x comes before y.
Negative values and the four quadrants
The axes extend below zero as well. Left of the origin the x-values are negative, and below the origin the y-values are negative. The axes divide the plane into four regions called quadrants.
| Quadrant | x | y | Example |
|---|---|---|---|
| First (top right) | positive | positive | (3, 2) |
| Second (top left) | negative | positive | (−3, 2) |
| Third (bottom left) | negative | negative | (−3, −2) |
| Fourth (bottom right) | positive | negative | (3, −2) |
Take the point (−4, 1): move 4 to the left, then 1 up. A common error is to swap the two numbers, so check which is which before you plot.
Reading the scale
Before reading anything off a graph, look at the labels and the scale. Each gridline might represent 1, 5, 10 or 100. If the x-axis goes up in 2s, the point halfway between 4 and 6 is 5. Both axes may use different scales, which changes how steep a line looks without changing the underlying numbers.
A few questions to ask:
- What does each axis measure, and in what units?
- Does the scale start at zero, and is it evenly spaced?
- Where is the data from, and how many points are plotted?
A scale that starts partway up the axis can make small changes look dramatic, a problem covered in more depth in the article on misleading charts. The article on basic charts also discusses how to make graphs that read clearly.
What a line tells you
When two quantities are linked by a steady relationship, the graph is a straight line. Consider a hypothetical journey: a cyclist rides at a steady pace and the graph plots distance (y) against time in hours (x). If the line passes through (0, 0) and (2, 60), the cyclist covered 60 km in 2 hours.
From a straight-line graph you can read:
- Values: at x = 1, the line is at y = 30, so after 1 hour the distance is 30 km.
- Direction: the line rises, so distance increases with time.
- Steepness: a steeper line means a faster change.
Slope (gradient)
The slope measures how steep a line is. It is the change in y divided by the change in x, often remembered as “rise over run”.
Take two points on a line, (1, 2) and (3, 6).
- Rise: 6 − 2 = 4
- Run: 3 − 1 = 2
- Slope: 4 ÷ 2 = 2
A slope of 2 means that for every 1 step to the right, the line goes up 2. For the cyclist, a slope of 30 (60 km ÷ 2 hours) means 30 km per hour: the slope is the speed. In many real graphs, the slope carries a meaning like this, a rate of change.
The sign tells you the direction:
- Positive slope: the line rises left to right.
- Negative slope: the line falls left to right.
- Zero slope: the line is horizontal, so y does not change.
- A vertical line has no defined slope, since the run is zero and you cannot divide by zero.
The link between slope and speed is a specific case of the idea of a ratio, which is the topic of ratios and proportions.
The y-intercept and the equation of a line
The point where a line crosses the y-axis is the y-intercept. For the line through (1, 2) and (3, 6), the slope is 2 and the line passes through the origin, so the rule is y = 2x. Check: when x = 1, y = 2; when x = 3, y = 6. Both match.
A line with slope 2 that crosses the y-axis at 1 has the rule y = 2x + 1. In general, y = mx + c, where m is the slope and c is the y-intercept. Reading this equation off a graph and solving for a value uses the same ideas as the article on linear equations. The letters m, x, y and c are variables in the sense described in algebra basics.
Check y = 2x + 1: at x = 0, y = 1, so the line crosses the y-axis at 1. At x = 3, y = 7. Plot (0, 1) and (3, 7) and the slope is (7 − 1) ÷ (3 − 0) = 2, as expected.
Practicing
Draw your own axes on squared paper, plot a few points and join them. Then pick two points and compute the slope. If you use a spreadsheet to produce a scatter plot, you can check your hand-drawn version; the basics are in spreadsheet basics. Working from a graph to a story (“what happened here?”) and from a story to a graph builds fluency in both directions.