What Is Slope?
Slope (usually denoted by the letter m) describes the steepness and direction of a straight line on a coordinate plane. It is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Slope is a fundamental concept in algebra, calculus, physics, engineering, economics, and everyday measurements like ramps and roofs.
The Slope Formula
Given two points (x₁, y₁) and (x₂, y₂), the slope is calculated as:m = (y₂ − y₁) / (x₂ − x₁)This is often remembered as "rise over run." The numerator is the change in y (rise) and the denominator is the change in x (run).
Types of Slope
- Positive slope (m > 0): the line rises from left to right
- Negative slope (m < 0): the line falls from left to right
- Zero slope (m = 0): the line is horizontal
- Undefined slope: the line is vertical (division by zero)
Slope-Intercept Form
The equation of a straight line is commonly written in slope-intercept form:y = mx + bWhere m is the slope and b is the y-intercept — the value of y where the line crosses the y-axis. Given two points, you can find the slope, then plug one point back in to find b.
Step-by-Step Example
Find the slope, y-intercept, and equation of the line passing through (2, 3) and (5, 9):
- Calculate slope: m = (9 − 3) / (5 − 2) = 6 / 3 = 2
- Plug one point into y = mx + b: 3 = 2(2) + b → 3 = 4 + b
- Solve for b: b = 3 − 4 = −1
- Equation: y = 2x − 1
- The y-intercept is −1; the x-intercept is found when y=0: 0 = 2x − 1 → x = 0.5
Real-World Applications
Slope appears constantly in the real world. In construction, the pitch of a roof, the gradient of a road, and the incline of a wheelchair ramp are all measured using slope. In economics, the slope of a demand curve shows how price affects quantity demanded. In physics, the slope of a position-vs-time graph is velocity, and the slope of a velocity-vs-time graph is acceleration. In statistics, the slope of a regression line represents the predicted change in the dependent variable per unit change in the independent variable.
Frequently Asked Questions
How do I find slope from a graph?
Pick two clear points on the line, count the vertical change (rise) between them, count the horizontal change (run), and divide rise by run. Make sure to check the sign — if the line goes up to the right, the slope is positive.
What is the slope of a horizontal line?
A horizontal line has zero slope because there is no vertical change (rise = 0), and 0 divided by any non-zero run equals 0.
What about parallel and perpendicular lines?
Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other — their product equals −1. For example, a line with slope 2/3 is perpendicular to a line with slope −3/2.