Gradient of Linear Graphs
3.3 Gradient of Linear Graphs – Cheat Sheet
1. Gradient Basics
- Gradient (m) = measure of slope of a straight line
- Formula:
m = (change in y) / (change in x) = (y₂ − y₁) / (x₂ − x₁) - Positive m → line rises from left to right
- Negative m → line falls from left to right
- m = 0 → horizontal line
- Undefined (division by zero) → vertical line
2. Finding Gradient from a Graph
- Choose two points on the line.
- Find vertical change (rise) and horizontal change (run).
- Gradient = rise / run.
Example:
From (1, 2) to (4, 8): rise = 8 − 2 = 6, run = 4 − 1 = 3 → m = 6/3 = 2
3. Finding Gradient from Coordinates
Formula:
m = (y₂ − y₁) / (x₂ − x₁)
Example:
Points A(−3, 5) and B(4, −2)
m = (−2 − 5) / (4 − (−3)) = (−7) / (7) = −1
4. Special Gradients
| Line Type | Gradient | Example |
|---|---|---|
| Horizontal | 0 | y = 5 |
| Vertical | Undefined | x = −2 |
| Positive slope | m > 0 | y = 2x + 1 |
| Negative slope | m < 0 | y = −3x + 4 |
5. Examples Table for Quick Reference
| Points | Rise | Run | Gradient |
|---|---|---|---|
| (0, 0) to (3, 6) | 6 | 3 | 2 |
| (2, 4) to (5, 4) | 0 | 3 | 0 |
| (−1, 2) to (2, −4) | −6 | 3 | −2 |
| (4, 5) to (4, 9) | 4 | 0 | Undefined |