How steep is it?
Slope measures how steep a line is:
slope (m) = rise ÷ run
- Rise = vertical change (up is positive, down is negative)
- Run = horizontal change (right is positive)
Slope from two points
m = (y₂ − y₁) / (x₂ − x₁)
Example: (1, 3) and (5, 11)
m = (11 − 3)/(5 − 1) = 8/4 = 2
The line rises 2 units for every 1 unit to the right.
Example: (−2, 7) and (4, −5)
m = (−5 − 7)/(4 − (−2)) = −12/6 = −2
Four types of slope
| Slope | Line | Picture |
|---|---|---|
| positive | rises left to right | uphill |
| negative | falls left to right | downhill |
| zero slope | horizontal | flat road |
| undefined slope | vertical | a wall (run = 0, can’t divide) |
Bigger number, steeper line
A slope of 5 is steeper than a slope of 2. A slope of −5 is steeper than −2 (just going down). Slopes between −1 and 1 are fairly gentle.
Slope as a rate of change
In real situations, slope is a rate of change with units.
| Graph | Slope means |
|---|---|
| distance (km) vs time (h) | speed in km/h |
| cost ($) vs items | price per item |
| water (L) vs time (min) | flow rate in L/min |
| height (m) vs horizontal distance (m) | steepness of a ramp or road |
Example: A car travels from 60 km at 1 h to 300 km at 4 h.
m = (300 − 60)/(4 − 1) = 240/3 = 80 km/h
Slopes in the real world
- Wheelchair ramps: building codes often require a gentle slope, around 1:12 (rise 1, run 12).
- Roads: a 6% grade means a rise of 6 m per 100 m.
- Roofs: pitch is described as rise over run, like 4/12.
Worked examples
Example 1: Slope through (0, 0) and (3, 9)? 3.
Example 2: Slope through (2, 5) and (2, −1)? Run = 0 → undefined.
Example 3: Slope through (−1, 4) and (3, 4)? Rise = 0 → 0.
Common mistakes
- Run over rise instead of rise over run.
- Mixing up the order: (y₂ − y₁)/(x₁ − x₂) gives the wrong sign.
- Calling a vertical line’s slope zero. It’s undefined.
Try it yourself
- Slope through (2, 1) and (6, 9)
- Slope through (−3, 5) and (1, −3)
- Slope through (4, −2) and (4, 6)
- A tank goes from 100 L to 40 L in 12 minutes. Rate of change?
- Which is steeper: m = −4 or m = 3?
Answers: 1) 2 2) −2 3) undefined 4) −5 L/min 5) m = −4
Quick recap
- Slope = rise ÷ run = (y₂ − y₁)/(x₂ − x₁).
- Positive rises, negative falls; horizontal = zero slope, vertical = undefined slope.
- In context, slope is a rate of change.