Overview
This lecture explains how to find the gradient (slope) between two points using their coordinates, with worked examples and tips for applying the formula.
Gradient Formula Using Coordinates
- The gradient (slope) between two points is calculated as the change in y divided by the change in x.
- Use the formula: gradient = (y₂ − y₁) / (x₂ − x₁).
- (x₁, y₁) are the coordinates of the first point; (x₂, y₂) are the coordinates of the second point.
- It doesn’t matter which point is labeled first or second, but using the larger numbers as the second point often makes calculations easier.
Example Calculations
- For points (2, 3) and (8, 12): gradient = (12 − 3) / (8 − 2) = 9 / 6 = 1.5.
- For points (8, –2) and (11, 7): gradient = (7 − (–2)) / (11 − 8) = 9 / 3 = 3.
Using Coordinates from a Graph
- When given a graph, choose any two points on the line and label their coordinates.
- Apply the same formula for the gradient using those coordinates.
- Example: Points (0, –1) and (4, –7): gradient = (–7 − (–1)) / (4 − 0) = (–6) / 4 = –1.5.
Key Terms & Definitions
- Gradient (Slope) — The rate of change of y with respect to x between two points.
- Coordinate — An (x, y) pair denoting a point on a graph.
- x₁, y₁, x₂, y₂ — The x and y values for the first and second points, respectively.
Action Items / Next Steps
- Practice finding gradients using the formula on additional coordinate pairs.
- Try identifying two points from a graph and calculating the gradient independently.