This chapter brings algebra and shapes together on a grid. Every map, screen and video game uses coordinates to pinpoint a spot — you'll learn to do the same, then meet functions — rules that turn an input into an output — and see what their graphs look like.
1 · The coordinate plane
Two number lines crossing at right angles make a grid. The horizontal one is the x-axis, the vertical one is the y-axis, and they meet at the origin, (0, 0).
A point's position is written as (x, y) — always across first, then up.
The axes split the grid into four quadrants. Anticlockwise from the top-right: 1st (+, +), 2nd (−, +), 3rd (−, −), 4th (+, −).
2 · Relations, lines and functions
A relation simply links inputs to outputs. A function is a special relation where each input gives exactly one output — like a vending machine: press one button and exactly one snack drops out, never two.
We write functions as f(x) ("f of x"). To find f(3), substitute 3 wherever you see x.
3 · Slope (gradient)
The slope — called the gradient in the UK — measures how steep a line is: how far it rises for each step across.
4 · Linear functions
A linear function has the form y = mx + c, and its graph is always a straight line (m is the slope, c is where it crosses the y-axis).
5 · Simultaneous linear equations, on a graph
Two straight lines usually cross at one point. That crossing point is the solution to both equations at once — the (x, y) that fits each one. (We solve these with algebra in the Simultaneous Equations chapter.)
6 · Nonlinear functions
Not every function is a straight line. A nonlinear function graphs as a curve. The most common is the quadratic y = x², whose graph is a U-shaped parabola.
Here is a quick guide to the shapes you'll meet, sorted by the highest power of x:
| Function | Highest power of x | Graph shape |
|---|---|---|
| y = 5 | 0 | flat horizontal line |
| y = 2x + 3 | 1 | straight sloping line |
| y = x² | 2 | U-shaped curve (parabola) |
| y = kx (proportion) | 1 | straight line through the origin |
7 · Shapes on the coordinate plane
You can plot the corners (vertices) of a polygon as coordinates, then read off side lengths by counting squares — or use Pythagoras for slanted sides.
8 · Transformations
Shapes can be moved around the grid — slid (translation), flipped (reflection), turned (rotation) or resized (enlargement). These are covered in full in the Transformations chapter.
9 · Proportional relationships & graphs
When two quantities are in direct proportion (y = kx), their graph is a straight line through the origin — double the input, double the output.