What Is Rise Over Run?
"Rise over run" is the most intuitive way to understand slope. The rise is the vertical change (how much the line goes up or down), and the run is the horizontal change (how much the line moves left or right). Dividing rise by run gives you the slope of the line connecting two points.
Mathematically, slope is defined as: m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two points on the line.
Types of Slope
Positive Slope
Line rises from left to right. Both rise and run have the same sign.
Negative Slope
Line falls from left to right. Rise and run have opposite signs.
Zero Slope
Horizontal line. No vertical change between points.
Undefined Slope
Vertical line. No horizontal change (division by zero).
Slope in Different Forms
Slope-Intercept Form
The equation y = mx + b uses slope (m) and y-intercept (b). This form makes it easy to graph a line and understand its behavior at a glance.
Point-Slope Form
The equation y - y₁ = m(x - x₁) is useful when you know the slope and a single point on the line.
Slope as a Percentage (Grade)
Multiplying slope by 100 gives the grade percentage. A 6% grade means the road rises 6 feet for every 100 feet of horizontal distance. Road signs, wheelchair ramps, and hiking trails often express steepness this way.
Real-World Applications
- Construction: Roof pitch, drainage slope, road grade.
- Geography: Hill steepness and terrain analysis.
- Economics: Rate of change in cost, revenue, or supply/demand curves.
- Physics: Velocity from position-time graphs, acceleration from velocity-time graphs.
- Accessibility: ADA ramp requirements specify maximum slopes.
Worked Example
Find the slope between (2, 3) and (6, 11):
- Rise = y₂ - y₁ = 11 - 3 = 8
- Run = x₂ - x₁ = 6 - 2 = 4
- Slope = Rise / Run = 8 / 4 = 2
- The line rises 2 units for every 1 unit to the right.