Library / Foundations: Maths / FMA-03
Foundations: Maths
Gradient of a straight line
Rise over run between two points, and the line y = mx + c.
- Ref
- FMA-03
- Rev
- v1
- Field
- Foundations: Maths
- Result
- m
m is the gradient (rise ÷ run) and c the intercept on the y axis. Vertical lines have undefined gradient.
The gradient says how steeply a line rises: how much y changes for each unit of x. It is the rise divided by the run between any two points.
Calculator
- Intercept c = y₁ − m x₁−1
- Angle to the x axis63.43 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Gradient has units: it is (units of y) per (unit of x), such as m/s on a distance–time graph.
- Steeper lines have bigger gradients; a negative gradient means y falls as x rises.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| y1 First y | −10 to 20 | −2.75 | 4.75 | 7.5 | +10% | −1.25 % |
| y2 Second y | 0 to 30 | −0.25 | 7.25 | 7.5 | +10% | +11.3 % |
| x1 First x | 0 to 3 | 1.6 | 4 | 2.4 | +10% | +2.56 % |
| x2 Second x | 4 to 10 | 0.8889 | 2.667 | 1.778 | +10% | −11.1 % |
Sweep of Second y
| Second y | Gradient |
|---|---|
| 0 | −0.25 |
| 3.75 | 0.6875 |
| 7.5 | 1.625 |
| 11.25 | 2.563 |
| 15 | 3.5 |
| 18.75 | 4.438 |
| 22.5 | 5.375 |
| 26.25 | 6.313 |
| 30 | 7.25 |
Sweep one variable
Variables and how each one contributes
Gradient
What the equation gives you. Everything else feeds it.
First y
y value of the first point. Lowers the gradient as it rises.
Second y
y value of the second point. Raises the gradient.
First x
x value of the first point. Shortens the run, so a larger x₁ steepens the line.
Second x
x value of the second point. Lengthens the run and lowers the gradient.
Worked example with the values above
- Rise: y₂ − y₁ = 9 − 1 = 8. Run: x₂ − x₁ = 4.
- m = rise/run = 8 / 4.
- Gradient 2.
Assumptions and limits
- The graph is a straight line.
- The scales on both axes are read in their stated units.
Common mistakes
- Reading run as x₁ − x₂ with rise as y₂ − y₁.
- Ignoring the units of the axes.
Statistics and uncertainty
When the two points are close together, small errors in the readings are divided by a small run and the gradient becomes unreliable; use points far apart on the line.
Where it is used
- Speed from a distance–time graph.
- Calibration lines for sensors.
Maths you need
- Coordinates: points on a graph
- Division: rise divided by run