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Cambridge IGCSE Mathematics — 0580 Extended

Topic 3.3: Coordinate geometry — Gradient

Gradient from a graph

The gradient \(m\) of a straight line measures how steep it is:

\[ m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x} \]

Pick two clear lattice points on the line. Draw a right-angled triangle: horizontal side = run, vertical side = rise.

Method

  1. Choose two points on the line with integer coordinates if possible.
  2. Run = difference in \(x\) (positive when moving right).
  3. Rise = difference in \(y\) (positive up, negative down).
  4. Write \(m = \dfrac{\text{rise}}{\text{run}}\) and simplify.

Find the gradient of the line shown on the grid.

Positive gradient line with rise 2 and run 3 marked, so m equals 2/3
\(m = \dfrac{2}{3}\).
Negative gradient line with rise negative 3 and run 3, so m equals negative 1
When \(y\) falls as \(x\) increases, the rise is negative and \(m\) is negative.

Gradient from two points

For points \(A(x_1, y_1)\) and \(B(x_2, y_2)\):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

The order of the points does not matter, but you must subtract in the same order in the numerator and the denominator.

Find the gradient of the line through \(A(1, 2)\) and \(B(4, -1)\).

Points A(1,2) and B(4,-1) with triangle showing m equals negative 1
\(m = \dfrac{-1 - 2}{4 - 1} = \dfrac{-3}{3} = -1\).

Try this

Find the gradient of the line through \(P(-2, 5)\) and \(Q(4, 2)\).

Show answer
Answer

  1. Substitute into the formula

    \[ m = \frac{2 - 5}{4 - (-2)} \]

  2. Simplify

    \[ m = \frac{-3}{6} = -\frac{1}{2} \]

  3. Gradient

    \[ m = -\dfrac{1}{2} \]

Types of gradient

TypeWhat the line doesValue of \(m\)
PositiveSlopes up left \(\to\) right\(m \gt 0\)
NegativeSlopes down left \(\to\) right\(m \lt 0\)
ZeroHorizontal (\(y = k\))\(m = 0\)
UndefinedVertical (\(x = k\))undefined (not \(0\))
Horizontal line y equals 2 with gradient 0 and vertical line x equals 3 with undefined gradient
Horizontal \(\Rightarrow m = 0\); vertical \(\Rightarrow\) gradient undefined.

Exam Traps

  • Do not swap rise and run — \(\dfrac{\text{run}}{\text{rise}}\) is not the gradient.
  • A vertical line has undefined gradient; do not write \(m = 0\).

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