Slope Formula
m = (y₂ − y₁) / (x₂ − x₁) = rise / run
Line equation: y = mx + b
y-intercept: b = y₁ − m·x₁
Angle: θ = arctan(m) in degrees
Line equation: y = mx + b
y-intercept: b = y₁ − m·x₁
Angle: θ = arctan(m) in degrees
Slope Interpretations
| Slope Value | Meaning | Line direction |
|---|---|---|
| m > 0 | Positive slope | Rising left to right ↗ |
| m < 0 | Negative slope | Falling left to right ↘ |
| m = 0 | Zero slope | Horizontal line → |
| undefined | Vertical line | Straight up ↑ (x₁=x₂) |
| m = 1 | 45° slope | Equal rise and run |
Parallel & Perpendicular Lines
Parallel lines: same slope (m₁ = m₂)
Perpendicular lines: slopes are negative reciprocals (m₂ = −1/m₁)
Perpendicular lines: slopes are negative reciprocals (m₂ = −1/m₁)
Frequently Asked Questions
What does a slope of 2 mean?
A slope of 2 means for every 1 unit you move right (run = 1), you move 2 units up (rise = 2). As a percentage grade (used for roads), slope × 100 = 200% — extremely steep. Road grades are typically 0–8%; mountain roads up to 12%.
What is the slope of a horizontal line?
A horizontal line has slope = 0 (no rise over any run). Its equation is y = c (a constant). A vertical line has an undefined slope because the run = 0 and division by zero is undefined. Its equation is x = c.
How is slope used in real life?
Road grade (steepness), roof pitch (rise/run ratio), ramp accessibility (ADA requires ≤1:12 = 8.33%), ski trail difficulty ratings, rate of change in data (e.g., $50/month spending growth = slope 50 on a spending graph), and engineering drawings all use slope.