Slope

Slope is a measure of the steepness or incline of a line or surface. In two-dimensional analytic geometry, the slope of a line is the ratio of the vertical rise to the horizontal run between two points on the line. If a line passes through points (x1, y1) and (x2, y2), its slope is m = (y2 − y1) / (x2 − x1). A positive slope means the line rises as it moves to the right; a negative slope means it falls. A zero slope indicates a horizontal line, while an undefined slope indicates a vertical line. The slope is also the derivative dy/dx for a function y = f(x), so it represents the instantaneous rate of change.

In algebra, the slope is the parameter m in the linear equation y = mx + b, where b is the y-intercept. In higher dimensions, slope generalizes to the gradient of a function, which points in the direction of steepest ascent with magnitude equal to the rate of increase.

Slope has broad applications, including determining road and rail grades, roof pitch, and drainage or erosion risk; in geology and geomorphology, for assessing slope stability; in GIS, slope maps characterize terrain; and in statistics, the slope of a regression line indicates the strength and direction of a relationship.

Common properties include that slope depends on units; horizontal lines have slope zero, vertical lines have undefined slope.