Gradient · ⁨斜率⁩

The steepest street on Earth

• Baldwin Street in Dunedin, New Zealand, climbs 35 m for every 100 m along the ground.
• That ratio — rise 上升量 ÷ run 水平量 — is what mathematicians call the gradient 斜率.
• A single number tells you exactly how steep something is, whether it's a street, a roof, or a line on a graph.

What gradient measures

• The gradient $m$ tells you two things about a straight line:
• How steep it is (larger absolute gradient = steeper).
• Which direction it goes (positive = up to the right, negative = down to the right).
• A gradient of $0$ means the line is perfectly horizontal — no rise at all.

A steep alpine pass: gradient measures steepness as rise over run

Gradient from a grid; coordinates in Extended

• Pick any two points on the line, $(x_1, y_1)$ and $(x_2, y_2)$:

• The rise is the vertical gap between the points; the run is the horizontal gap.

The gradient triangle: count the squares up (rise) and across (run), then divide.

Worked example

• Find the gradient of the line through $(1, 2)$ and $(4, 11)$.
• Rise $= 11 - 2 = 9$; run $= 4 - 1 = 3$.
• $m = \dfrac{9}{3} = 3$.
• The line climbs $3$ units up for every $1$ unit across — quite steep.

The subtraction trap

• You must subtract in the same order for both coordinates:
• $(y_2 - y_1)$ paired with $(x_2 - x_1)$, or $(y_1 - y_2)$ paired with $(x_1 - x_2)$.
• Mixing the order flips the sign and gives the wrong gradient.

Gradient in the real world

• Road signs show gradient as a percentage: a 10% hill rises 10 m for every 100 m along.
• Wheelchair ramps must have a gradient of at most $\dfrac{1}{12}$ (about 8%) — that's building code in many countries.
• Roof pitch is a gradient too — steeper roofs shed rain faster.

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