Slope
m=(yโโyโ)/(xโโxโ)=rise/run. Percent grade=100m. Angle ฮธ=arctan(m). Perpendicular mโฅ=โ1/m. Vertical: undefined. Horizontal: 0.
Why This Mathematical Concept Matters
Why: Slope measures steepness โ roads, roofs, ramps. Percent grade: 100รslope. Angle from horizontal. Perpendicular lines: slopes multiply to โ1. Used in construction and navigation.
How: Two points (xโ,yโ), (xโ,yโ): m=(yโโyโ)/(xโโxโ). Rise=ฮy, run=ฮx. Percent grade=100m. Angle: ฮธ=arctan(m) in degrees. Perpendicular: mโ=โ1/mโ.
- โRise over run: vertical change divided by horizontal change.
- โVertical line: ฮx=0, slope undefined (division by zero).
- โParallel lines have same slope. Perpendicular: product of slopes = โ1.
๐ Examples โ Click to Load
Point 1 (xโ, yโ)
Point 2 (xโ, yโ)
Rise, Run & Slope
Rise vs Run Proportion
โ ๏ธFor educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
(1,2) to (4,6): rise=4, run=3, m=4/3. Grade=133%.
โ Example
Perpendicular: m=2 and m=โ1/2. 2ร(โ1/2)=โ1.
โ Perpendicular
๐ Slope as Rate of Change
Slope m = (yโ - yโ)/(xโ - xโ) = rise/run. It measures how much y changes per unit change in x. Positive slope: line rises left-to-right. Negative: falls. Zero: horizontal. Undefined: vertical.
๐ Slope Types
| Type | Condition | Line |
|---|---|---|
| Positive | m > 0 | Rises left to right |
| Negative | m < 0 | Falls left to right |
| Zero | m = 0 | Horizontal |
| Undefined | run = 0 | Vertical |
โฅ Parallel & โฅ Perpendicular
Parallel lines have equal slopes: mโ = mโ. Perpendicular lines have slopes that are negative reciprocals: mโ ยท mโ = -1, so mโ = -1/mโ.
๐ฏ Applications
- Roof pitch: Rise/run (e.g., 4/12 = 4" rise per 12" run)
- Road grade: Percent = 100 ร rise/run (6% = 6 ft rise per 100 ft)
- Wheelchair ramp: ADA max 1:12 (8.33%)
- Angle of inclination: ฮธ = arctan(m) in degrees
โ FAQ
What is rise over run?
Rise = vertical change (yโ-yโ), Run = horizontal change (xโ-xโ). Slope m = rise/run.
When is slope undefined?
When run = 0 (vertical line). Division by zero.
How do I find a perpendicular slope?
m_perp = -1/m. Negative reciprocal. Example: slope 2 โ perpendicular -1/2.
What is percent grade?
Grade = 100 ร slope. A 6% road rises 6 ft per 100 ft horizontal.
What slope for ADA ramp?
Maximum 1:12 (8.33%). One unit rise per 12 units run.
๐ Worked Examples
โ ๏ธ Common Mistakes
- Mixing up (y2-y1)/(x2-x1) โ always subtract in the same order for both numerator and denominator.
- Vertical line: slope is undefined, not infinity โ we say undefined because division by zero is not defined.
- Perpendicular: mโยทmโ = -1, so mโ = -1/mโ. Not just the negative, but the negative reciprocal.
๐ Summary
Slope m = (yโ-yโ)/(xโ-xโ) = rise/run. Positive: line rises; negative: falls; zero: horizontal; undefined: vertical. Percent grade = 100m. Angle ฮธ = arctan(m). Perpendicular slope = -1/m. Applications: roof pitch, road grade, ramps, rate of change in physics and economics.