Slope Calculator
Slope Formula
Positive Slope
Line rises from left to right (↗)
Negative Slope
Line falls from left to right (↘)
Slope Types
Related Calculators
Understanding Slope
What is Slope?
Slope measures the steepness and direction of a line. It describes how much a line rises or falls as you move from left to right. In mathematics, slope is often represented by the letter "m" and is calculated as "rise over run."
Slope Formula
Forms of Linear Equations
Slope-Intercept Form
m = slope, b = y-intercept
Point-Slope Form
Uses one point and slope
Standard Form
A, B, C are integers
Interpreting Slope
| Slope Value | Meaning | Example |
|---|---|---|
| m > 0 | Line rises left to right | y = 2x + 1 |
| m < 0 | Line falls left to right | y = -3x + 5 |
| m = 0 | Horizontal line | y = 4 |
| m = 1 | 45° angle with x-axis | y = x |
| undefined | Vertical line | x = 3 |
Real-World Applications
Construction
- Roof pitch calculation
- Wheelchair ramp grades
- Road incline design
- Staircase rise/run
Science & Economics
- Rate of change in data
- Velocity calculations
- Cost per unit analysis
- Growth rate trends
Key Relationships
- Parallel lines have equal slopes (m₁ = m₂)
- Perpendicular lines have slopes that multiply to -1 (m₁ × m₂ = -1)
- The y-intercept is where the line crosses the y-axis (x = 0)
- The x-intercept is where the line crosses the x-axis (y = 0)
Frequently Asked Questions
What does a slope of 0 mean?
A slope of 0 means the line is perfectly horizontal - it doesn't rise or fall as you move left to right. The equation is simply y = b (a constant). Examples include the horizon line or a perfectly level floor. Every point on the line has the same y-value.
What is an undefined slope?
An undefined slope occurs with vertical lines where you would divide by zero (run = 0). Vertical lines go straight up and down with no horizontal change. The equation is x = a (a constant). The y-values can be anything, but all x-values are the same.
How do you find slope from two points?
Use the slope formula: m = (y2 - y1) / (x2 - x1). Subtract the y-coordinates (rise) and divide by the difference in x-coordinates (run). For points (2,3) and (5,9): slope = (9-3)/(5-2) = 6/3 = 2. The order of points doesn't matter as long as you're consistent.
How are parallel and perpendicular line slopes related?
Parallel lines have identical slopes (m1 = m2). Perpendicular lines have slopes that are negative reciprocals - they multiply to -1. If one line has slope 2, a perpendicular line has slope -1/2 (flip and negate). This relationship is used in geometry and construction.