Gradient of linear graphs

What gradient measures

The gradient of a straight line is a single number that tells you how steep the line is and which way it slopes. It is the change in $y$ for every 1 unit you move to the right.

$$\text{gradient}=\frac{\text{change in}y}{\text{change in}x}=\frac{\text{rise}}{\text{run}}$$

Reading a graph from left to right:

• a line that goes up has a positive gradient;
• a line that goes down has a negative gradient;
• a horizontal line has gradient $0$, because $y$ does not change at all.

The size of the number tells you the steepness. A line with gradient $3$ is steeper than a line with gradient $\frac{1}{2}$, and a line with gradient $-4$ is steeper than one with gradient $-1$: it falls 4 units for each step right instead of 1. A vertical line has no gradient you can calculate, because the run is $0$ and you cannot divide by $0$.

Why any two points give the same answer. Pick two points on a straight line and draw the right-angled triangle between them. Pick two other points and draw another triangle. The two triangles have the same angles, so they are similar: one is an enlargement of the other. That means rise divided by run is the same for both. This is what makes a straight line straight, and it is why you are free to choose whichever two points are easiest to read.

Finding the gradient from a grid

When the line is drawn on a grid, find its gradient like this.

1. Choose two points where the line passes exactly through a grid crossing. Choose them far apart: the bigger the triangle, the smaller the effect of any reading error.
2. Draw a right-angled triangle with a horizontal side (the run) and a vertical side (the rise), with the line as its longest side.
3. Read the run and the rise from the numbers on the axes, not by counting squares. On many exam grids one square across is 1 unit but one square up is 2, 5 or 10 units.
4. Decide the sign. Moving left to right, if the line goes down, the rise is negative.
5. Divide the rise by the run, and simplify the fraction.

Step 3 is where most marks are lost. Always check the scale on each axis before you count anything. If the $y$-axis goes up in 10s and the $x$-axis in 1s, a triangle that looks 5 squares by 5 squares is really a rise of 50 over a run of 5, so the gradient is 10, not 1.

A gradient can be a fraction or a decimal. A line that rises 3 units for every 4 across has gradient $\frac{3}{4}$, or $0.75$. Leave it as a fraction on the non-calculator paper unless it converts neatly.

A gradient triangle

From $(1,1)$ to $(4,7)$ the run is $4-1=3$ and the rise is $7-1=6$, so the gradient is $\frac{6}{3}=2$. Try the triangle from $(0,-1)$ to $(2,3)$: the run is 2, the rise is 4, and the gradient is again $\frac{4}{2}=2$.