Gradient of linear graphs
Section: Coordinate geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Finding the Gradient from a Graph
The gradient of a straight line measures how steep it is - how much y changes for every unit increase in x. It can be found directly from a graph by comparing the vertical and horizontal distance between two points on the line.
- Gradient = rise ÷ run, i.e. the vertical change divided by the horizontal change between two points on the line
- Choose two points on the line that sit exactly on grid intersections, to make the rise and run easy to count accurately
- A line sloping upward from left to right has a positive gradient; a line sloping downward has a negative gradient
- The steeper the line, the larger the size (magnitude) of its gradient
Worked Example: Reading a Gradient from a Graph
- Question: A straight line passes through the grid points (2, 3) and (6, 11). Find the gradient of the line.
- Step 1: Rise (vertical change): 11-3=8
- Step 2: Run (horizontal change): 6-2=4
- Step 3: Gradient = rise ÷ run: 84=2
- Answer: gradient = 2
Common Mistakes
MistakeDividing the run by the rise instead of the rise by the run, giving the reciprocal of the correct gradient
Fixgradient is always vertical change ÷ horizontal change - rise over run, not the other way round
MistakeCounting grid squares inaccurately when the two chosen points don't sit exactly on grid line intersections
Fixpick two points that lie precisely on grid intersections, so both the rise and run can be counted as whole numbers
Calculating the Gradient from Two Points
The gradient between two points can also be calculated directly from their coordinates, without needing a graph at all.
- For two points (x_1,y_1) and (x_2,y_2), the gradient is (y_2-y_1/x_2-x_1)
- It does not matter which point is labelled 1 and which is labelled 2, as long as the same order is used consistently in both the numerator and denominator
- A gradient can be a whole number, a fraction, or negative - simplify a fractional answer where possible
Worked Example: Calculating a Gradient from Two Coordinates
- Question: Point P has coordinates (-3, 5) and point Q has coordinates (5, 1). Find the gradient of the line PQ.
- Step 1: Gradient =(y_2-y_1/x_2-x_1)=(1-5/5-(-3))
- Step 2: =(-4/8)
- Step 3: Simplify: -(1/2)
- Answer: gradient of PQ = -1/2
Common Mistakes
MistakeSubtracting the coordinates in a different order for x and y, e.g. y₂-y₁ in the numerator but x₁-x₂ in the denominator
Fixuse the same order consistently - if point 2's y-coordinate is used first on top, point 2's x-coordinate must be used first on the bottom too
MistakeLosing a negative sign when one or both coordinates are negative
Fixwrite each subtraction out in full using brackets around negative values before simplifying, e.g. (1-5) and (5-(-3))
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