How Do You Find the Gradient of a Line?

The gradient of a straight line measures its steepness and direction. Using two points, calculate the change in the vertical coordinate divided by the change in the horizontal coordinate:

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

The Reasoning

Gradient is often described as rise over run:

$$m=\frac{\text{vertical change}}{\text{horizontal change}}=\frac{\Delta y}{\Delta x}.$$

A positive gradient means the line rises from left to right, while a negative gradient means it falls. A larger value of $\lvert m\rvert$ indicates a steeper line.

When reading a graph, use labelled grid intersections rather than unclear points, and check that the horizontal and vertical scales match.

For example, suppose a line passes through $A(2,3)$ and $B(6,11)$. Then

$$m=\frac{11-3}{6-2}=\frac{8}{4}=2.$$

Therefore, every increase of $1$ unit in $x$ corresponds to an increase of $2$ units in $y$.

A horizontal line has gradient $0$, because its vertical change is zero. A vertical line has an undefined gradient, because $x_2-x_1=0$, and division by zero is undefined.

Gradient also identifies relationships between lines. Parallel lines have equal gradients, while non-vertical perpendicular lines satisfy $m_1m_2=-1$. For example, a line perpendicular to a line with gradient $2$ has gradient $-\frac{1}{2}$. These results are especially useful when finding an unknown line equation.

A common misconception is reversing the subtraction in only one part of the fraction. You may calculate either point minus the other, but the order must be consistent in both the numerator and denominator.

IB Exam Technique

This is Mathematics: Analysis and Approaches SL 2.1: Equations of straight lines, parallel and perpendicular, and it can be tested on Paper 1 or Paper 2. For a command term such as find, show the substitution into the gradient formula rather than writing only the final value; this secures the method mark even if an arithmetic error follows.