Slope as Rate of Change

Slope measures steepness: how much the line rises for each step to the right.

Count the rise (vertical change) and the run (horizontal change) between two points on the line, then divide. Up 3 and right 4 gives a slope of 3/4.

Direction shows up in the sign. Uphill left-to-right is positive, downhill is negative. A flat line has slope 0. A vertical line has no slope at all, because the run is zero and you cannot divide by zero.

In real terms, slope is a rate: dollars per hour, miles per gallon, degrees per minute.

For two distinct points on a line, slope is the ratio of differences:

m = (y₂ − y₁)/(x₂ − x₁) = Δy/Δx

That this ratio is the same for every pair of points on a given line is the defining property of a line, provable from similar triangles: any two "rise over run" triangles drawn on the same line are similar, so their leg ratios agree.

Slope carries units, and those units are the units of y divided by the units of x. On a distance-time graph slope is speed; on a cost-quantity graph it is price per item. Reading the units is usually the fastest way to interpret a slope in an applied problem.

A vertical line has Δx = 0, making the ratio undefined — distinct from a horizontal line's slope of 0. "No slope" and "zero slope" describe opposite situations and should not be used interchangeably.