Gradient calculator from two points

• $(1, 2)$ and $(5, 10)$: $m = \dfrac{10 - 2}{5 - 1} = \dfrac{8}{4} = 2$.
• $(0, 6)$ and $(3, 0)$: $m = \dfrac{0 - 6}{3 - 0} = -2$ (the line slopes down).
• $(2, 4)$ and $(7, 4)$: $m = 0$ (a horizontal line).

• Positive gradient: the line goes up from left to right.
• Negative gradient: it goes down.
• Zero: horizontal, like $y = 4$.
• Undefined: vertical, like $x = 3$ (change in $x$ is 0, and you can't divide by 0).

The bigger the number (ignoring the sign), the steeper the line.

Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to $-1$: if one is 2, the other is $-\tfrac{1}{2}$ (the negative reciprocal). The equation of a line calculator gives the perpendicular gradient too.

On a distance-time graph the gradient is the speed; on a velocity-time graph it's the acceleration. A road sign showing a 10% gradient means it rises 10 m for every 100 m across. In each case the gradient is a rate: how fast one quantity changes as the other increases.

• Putting change in $x$ on top.
• Subtracting the coordinates in a different order on top and bottom (always $y_2 - y_1$ with $x_2 - x_1$).
• Reading a graph's scale wrongly: count the units, not the squares.

Practise with our straight-line graphs questions.

• From a graph, pick two points far apart where the line crosses grid corners exactly.
• Draw the right-angled triangle on the graph and label the rise and run.
• Give gradients as fractions or decimals as the question asks; $\tfrac{2}{3}$ is better than 0.667 when exact.

Gradients describe how fast something changes. On a graph of the cost of a taxi ride against distance, the gradient is the cost per mile. If a plant is 12 cm tall after 2 weeks and 30 cm after 8 weeks, its average growth rate is $(30 - 12) \div (8 - 2) = 3$ cm per week. Always include the units of the gradient: "units of $y$ per unit of $x$".

For curved graphs, the gradient changes from point to point. To estimate it at a point, draw a tangent (a straight line touching the curve there) and find the gradient of the tangent.