📌 Purpose: Learn the common forms of a straight line, compute gradients, work with intercepts, test parallelism and perpendicularity, and apply these ideas to real-world inclines and simple modelling.
| Term | Definition / Note |
|---|---|
| Gradient (m) | Slope of the line: m = (y2 − y1) ÷ (x2 − x1). Rise over run; positive → uphill to the right, negative → downhill to the right. |
| y = mx + c | Gradient-intercept form: m = gradient, c = y-intercept (value where x = 0). |
| ax + by + d = 0 | General form. Convert to y = mx + c by isolating y: y = (−a/b)x − d/b (if b ≠ 0). |
| Point-gradient form | y − y1 = m(x − x1). Use when a point and gradient are known. |
| Parallel lines | Two lines are parallel ⇔ their gradients are equal: m1 = m2. |
| Perpendicular lines | Two lines are perpendicular ⇔ m1 × m2 = −1 (i.e., m2 = −1/m1, assuming neither is vertical). |
| Vertical / horizontal lines | Vertical: x = k (undefined gradient). Horizontal: y = k (m = 0). |
📌 How to compute gradient (3 quick ways)
- From two points (x1, y1) and (x2, y2):
m = (y2 − y1)/(x2 − x1). Always subtract in the same order. - From y = mx + c: read off m directly.
- From ax + by + d = 0 (b ≠ 0): rearrange:
y = (−a/b)x − d/b→m = −a/b.
📌 Common forms & quick conversions
- Gradient–intercept:
y = mx + c— easiest to interpret (m slope, c y-intercept). - Point–gradient:
y − y1 = m(x − x1)— ideal when you know one point and slope. - General:
ax + by + d = 0— convert to y = mx + c by isolating y (if possible). - Two-point to equation: Given two points, compute m then use point-gradient to get equation.
📌 Short worked examples
Example 1 (gradient from points): Points (1,2) and (4,8): m = (8 − 2)/(4 − 1) = 6/3 = 2. Equation using (1,2): y − 2 = 2(x − 1) ⇒ y = 2x (c = 0).
Example 2 (perpendicular): Line L has m = 3. A perpendicular line has m = −1/3. Use point-gradient to form equation if a point is given.
🧮 GDC Tips
- Use two-point input or line-fit routines (if available) to get equation quickly from data points.
- Use matrix or function tools to convert between forms (solve for y to get m and c).
- Check gradient numerically by computing (y2 − y1)/(x2 − x1) on the calculator to avoid arithmetic mistakes.
🧠 Examiner Tip
Write clean steps: show how you computed m, which form you used, and how you rearranged to the requested form. For perpendicular questions explicitly show the negative reciprocal step.
📝 Paper tips
- Paper 1: quick calculations — label points and show subtraction order for slope.
- Paper 2: interpret gradient in context (e.g., incline steepness, rate of change).
⚠️ Common pitfalls
- Mixing up the order of subtraction in slope formula — always (y2 − y1)/(x2 − x1).
- For vertical lines (x = k), do not attempt to compute m (it is undefined).
- When testing perpendicularity ensure neither line is vertical before using negative reciprocal; if one is vertical, the other must be horizontal (m = 0).
📌 Quick checklist
- Identify knowns: two points / point & gradient / equation coefficients.
- Compute gradient using correct formula and order.
- Write equation in requested form (y = mx + c / ax + by + d = 0 / point-gradient).
- For parallel/perpendicular: compare gradients (equal / negative reciprocal).