Slope

A road with a 6% grade rises 6 metres for every 100 metres across: a slope of 6/100 = 3/50.

Rise over run

Some lines climb steeply, some gently, some fall, and some lie flat. The number that measures this tilt is the slope, written m. OpenStax lists the places you meet it outside a classroom: the pitch of a roof, the grade of a highway and a ramp for a wheelchair. To find the slope of a line, pick two points on it, draw the right triangle between them, and measure two sides. The vertical change is the rise and the horizontal change is the run. The slope is m = rise/run.

With coordinates you do not need to count squares. For points (x₁, y₁) and (x₂, y₂), the rise is y₂ − y₁ and the run is x₂ − x₁, so m = (y₂ − y₁)/(x₂ − x₁). For (−4, −4) and (4, 2): m = (2 − (−4))/(4 − (−4)) = 6/8 = 3/4. Subtract in the same order on the top and bottom. If you start with the second point instead, both differences change sign and the slope is the same: (−4 − 2)/(−4 − 4) = −6/−8 = 3/4. A positive slope rises from left to right; a negative slope falls.

Flat lines and upright lines

On a horizontal line every point has the same y-coordinate, so the rise between any two points is 0 and the slope is 0/run = 0. On a vertical line every point has the same x-coordinate, so the run is 0. Division by zero is not defined, so OpenStax says the slope of a vertical line is undefined. Keep the two apart: a horizontal line such as y = 4 has slope zero, which is a number, while a vertical line such as x = 3 has no slope at all. In the graph lab, drag one point straight above the other and the panel reports exactly that.

Slope in buildings and roads

The grade of a road is its slope written as a percent. A 6% grade means a rise of 6 for a run of 100, which simplifies to 3/50. Builders describe a roof by how many inches it rises over a 12-inch run: a 4-12 pitch has slope 4/12 = 1/3. OpenStax also quotes a rule for wheelchair ramps of at most a 1-inch rise for every 12 inches of run. To climb 24 inches to a door, a ramp therefore needs at least 24 × 12 = 288 inches of run, which is 24 feet.

The size of the slope, ignoring its sign, tells you how steep the line is. A slope of 3 is steeper than a slope of 1/2, and a slope of −3 is just as steep as 3, only falling instead of rising. A slope of 1 climbs at the same rate it moves across, which on a grid with square cells is a 45 degree line. As the slope grows towards very large numbers, the line gets closer and closer to vertical, which is one way to see why a vertical line has no slope at all.