Gradient or Slope of a Line

The slope formula

For distinct points $A=(x_1,y_1)$ and $B=(x_2,y_2)$ on a non-vertical line,

The numerator is the signed vertical change, sometimes called the rise. The denominator is the signed horizontal change, or run. Either change can be negative. “Rise over run” is a reminder of the order, not a requirement that both movements be positive.

The ratio is constant along a straight line. Different right triangles built against the same line have proportional legs, so they give the same ratio. This is the geometric basis of the constant rate described in OpenStax’s linear-functions chapter.

Worked examples with positive and negative slopes

A positive slope

For $A=(2,1)$ and $B=(6,9)$,

The output rises 2 units for every 1-unit increase in the input. A larger run of 4 units produces a rise of 8 units; both descriptions express the same rate.

A negative slope

For $P=(-1,4)$ and $Q=(3,-4)$,

As $x$ increases by 1, $y$ decreases by 2. The sign communicates direction; the magnitude 2 communicates the rate of change in the chosen units.

Endpoint order must be consistent

Using $B$ first in the positive-slope example gives

The result is unchanged because both differences reverse sign. Reversing only the numerator or only the denominator incorrectly reverses the slope. Label the points before substituting when several negative coordinates are involved.

The ratio $y/x$ measures the slope from the origin to $(x,y)$ when $x\ne0$. It is not the general two-point slope formula. For a line not passing through the origin, ratios of individual coordinates generally differ from its slope.

Zero and undefined slope

A horizontal line through $(1,4)$ and $(6,4)$ has slope

Its equation is $y=4$. The output stays constant as the input changes.

A vertical line through $(3,-2)$ and $(3,5)$ would require $7/0$. Division by zero is undefined, so its slope is undefined and its equation is $x=3$. Calling its slope “infinity” does not make it a finite number that can be used in the slope formula.

Identical points are another separate issue: they do not determine a unique line. Both coordinate differences would be zero, so the formula would give $0/0$ rather than a meaningful slope.

Slope has units in a model

Suppose an idealized tank contains 120 L at time 2 min and 200 L at time 6 min, and its volume changes linearly over that interval. Then

More explicitly, the numerator is an 80 L volume change and the denominator is a 4 min time change. A positive result means the volume is increasing. It does not by itself identify whether that is caused by inflow, reduced outflow or some other modelled mechanism.

For a curved graph, the same two-point calculation is an average rate of change over an interval. It need not equal the instantaneous rate at either endpoint. Introduction to Differential Calculus develops that distinction.

Slope and an angle of inclination

On equally scaled perpendicular spatial axes, a non-vertical line making an inclination angle $\theta$ with the positive $x$-axis satisfies

Here $0\le\theta<\pi$, excluding $\theta=\pi/2$. An inclination between $90^\circ$ and $180^\circ$ produces a negative tangent and therefore a negative slope. A vertical line has inclination $90^\circ$, where tangent is undefined.

This angle interpretation assumes equal unit lengths on the two axes. Stretching the vertical scale changes a graph’s apparent angle without changing its labelled slope. Always calculate from coordinate values rather than estimating the angle of an arbitrarily scaled image.

Related topics

Intercepts of a Line identifies useful anchor points. Combine a point with a slope in Equation of a Straight Line, then compare directions in Parallel and Perpendicular Lines.