Maths⏱ 4 min read
How to Calculate Slope, Gradient and Rate of Change
Slope is the foundation of linear algebra, graphs, and real-world engineering. Here's how to calculate it from two points, what positive and negative slopes mean, and practical applications.
Slope — also called gradient in the UK — is one of the most fundamental concepts in mathematics. It measures steepness and direction of a line, and it underlies everything from roof pitches to economic models.
The Core Formula
Slope (m) = Rise ÷ Run
= (y₂ − y₁) ÷ (x₂ − x₁)
= Change in y ÷ Change in x
Where (x₁, y₁) and (x₂, y₂) are two points on the line.
Worked Example
A line passes through points (2, 3) and (6, 11). What is the slope?
m = (11 − 3) ÷ (6 − 2)
= 8 ÷ 4
= 2
For every 1 unit you move right, the line rises 2 units.
What Different Slope Values Mean
Slope ValueLine DirectionExample
Positive (e.g. +2)Rises left to rightRevenue growing over time
Negative (e.g. −3)Falls left to rightWeight decreasing on a diet
Zero (0)Perfectly horizontalConstant temperature
UndefinedPerfectly verticalInstantaneous change
Fraction (e.g. 0.5)Gentle riseShallow ramp
Large (e.g. 10)Steep riseNear-vertical cliff
The Equation of a Line
Once you have the slope and one point, you can write the full equation of the line in slope-intercept form:
y = mx + c
m = slope
c = y-intercept (where the line crosses the y-axis)
To find c: substitute a known point (x, y) and solve
Example: m = 2, point (2, 3)
3 = 2(2) + c → c = 3 − 4 = −1
Equation: y = 2x − 1
Gradient in Real-World Applications
Road and roof gradients: In construction, gradient is expressed as a ratio (1:20 means 1 unit rise per 20 units horizontal) or as a percentage.
Gradient % = (Rise ÷ Run) × 100
A 1:20 slope = (1 ÷ 20) × 100 = 5%
A steep road at 1:8 = 12.5%
Finance: The slope of a price chart between two dates is literally the rate of change — pounds per day, percentage per week. A steeper positive slope means faster growth.
Physics: On a velocity-time graph, slope = acceleration. On a distance-time graph, slope = speed.
Medicine: On a growth chart, the slope of a child's weight-for-age curve tells a paediatrician whether growth is accelerating, steady, or faltering.
Parallel and Perpendicular Lines
Parallel lines: same slope (m₁ = m₂)
Perpendicular lines: slopes multiply to −1
m₂ = −1 ÷ m₁
Example: slope 3 → perpendicular slope = −1/3