Gradient as a constant rate of change

A straight line has constant gradient \(m=\dfrac{\text{vertical change}}{\text{horizontal change}}=\dfrac{y_2-y_1}{x_2-x_1}\), provided \(x_2\ne x_1\). Use the same subtraction order in both differences.

Example

Through \(A(1,2)\) and \(B(4,8)\), \(m=\dfrac{8-2}{4-1}=2\). Each increase of \(1\) in \(x\) increases \(y\) by \(2\).

A line rising from left to right has positive gradient; a falling line has negative gradient. A horizontal line has gradient \(0\). A vertical line has undefined gradient because its horizontal change is zero.