Knowledge Organiser: Gradient and Intercepts
Key Terms
- Gradient: The steepness of a line: rise divided by run between two points
- Rise: The vertical change between two points on a line
- Run: The horizontal change between two points on a line
- y-intercept: The point where the line crosses the y-axis (x = 0)
- x-intercept: The point where the line crosses the x-axis (y = 0)
- Undefined gradient: A vertical line (x = constant) has no defined gradient
Must-Know Facts
- Gradient = rise ÷ run; use two clearly-readable points from the graph
- Positive gradient: line goes up left to right; negative gradient: line goes down
- Zero gradient means a horizontal line (y = constant)
- To find the y-intercept from an equation, set x = 0
- To find the x-intercept from an equation, set y = 0 and solve
- Always check the sign of the gradient matches the direction of the line on the graph
Key Formulas
- Gradient = (y₂ − y₁) ÷ (x₂ − x₁)
- y-intercept: set x = 0 in the equation, read c from y = mx + c
- x-intercept: set y = 0 in the equation, solve for x
- Gradient from a graph: draw a right-angled triangle and count squares
Common Mistakes
- Reading c from non-y-axis crossing: c is only readable directly when the line crosses the y-axis: for other forms, substitute x = 0
- Sign of gradient: Downward slope (top-left to bottom-right) = negative gradient: always check direction
- Using non-grid points: When calculating gradient from a graph, use points that lie exactly on grid intersections for accuracy
- Triangle too small: Use a large right-angled triangle (spanning several squares) to reduce reading errors