Slope (mathematics, geometry, and applied contexts)

Basic algebraic definition and forms

For a non-vertical straight line in a Cartesian plane, the slope m between points (x1,y1) and (x2,y2) is m = (y2 - y1)/(x2 - x1). Common algebraic forms that feature slope include the slope–intercept form y = mx + b and the point–slope form y - y1 = m(x - x1). Vertical lines have undefined slope because their horizontal change is zero.

Signs, special cases and geometric relation

Sign and value of the slope convey geometric orientation: a positive slope rises left to right, negative falls, zero is perfectly horizontal, and undefined indicates vertical. The slope of a line is the tangent of its angle of inclination θ to the positive x-axis: m = tan(θ), so θ = arctan(m) when the angle is measured from a horizontal baseline. Percent grade, often used in roads and railways, expresses slope as 100·(rise/run) and differs from the angular measure.

Calculus and multivariable generalizations

In differential calculus the instantaneous slope of a curve y = f(x) at a point is the derivative f'(x), interpreted as the slope of the tangent line. In several variables the gradient vector collects partial derivatives and points in the direction of steepest ascent; directional derivatives give slope in a specified direction. Slope fields (direction fields) visualize first‑order differential equations by showing the slope at many points.

Applications and examples

• Engineering: road and rail grades, roof pitch, drainage designs use slope or percent grade to meet safety and performance standards.
• Statistics: in linear regression, the slope coefficient describes how much the dependent variable changes, on average, per unit change in the predictor.
• Geography and geology: terrain steepness affects erosion, stability and land use planning; slope angles guide construction and risk assessment.

Notable properties and distinctions

Perpendicular nonvertical lines have slopes whose product equals −1. Slope is dimensionless when both rise and run use the same units, but practical expressions (percent, degrees, permille) provide alternative, context‑sensitive measures. While everyday speech may use "grade," "pitch," "incline" or "slope" interchangeably, each term can emphasize different measurement conventions or applications.