Coordinates & the Cartesian plane

What you will learn

read the coordinates of a point from a graph,
plot a point given its coordinates,
identify which of the four quadrants a point lies in,
use coordinates to describe simple patterns, like “the $y$ -value is always double the $x$ -value”.

1. The plane

The Cartesian plane is formed by two number lines that cross at right angles. The horizontal line is the $x$ -axis; the vertical line is the $y$ -axis. Their crossing point is the origin, $O = (0, 0)$ .

The Cartesian plane with its four quadrants, each labelled with the signs of (x, y).

A point is described by an ordered pair $(x, y)$ : its horizontal distance from the origin first, vertical distance second.

2. Plotting and reading points

Worked example 1 Plotting

Plot the point $A = (-3, 2)$ .

Start at the origin. Move $3$ units to the left (because $x$ is negative), then $2$ units up (because $y$ is positive). The point sits in quadrant 2.

Plotting A = (-3, 2): three units left, then two units up. The point lands in quadrant 2.

Worked example 2 Reading coordinates

If a grid shows a point $3$ units right of the origin and $4$ units below, its coordinates are $(3, -4)$ (quadrant 4).

3. Quadrants

The axes divide the plane into four quadrants, numbered anticlockwise starting from the top right.

Quadrant	$x$	$y$
Q1	positive	positive
Q2	negative	positive
Q3	negative	negative
Q4	positive	negative

4. Simple rules between x and y

If a table pairs $x$ -values with $y$ -values according to a rule, you can plot the points and see a pattern.

Worked example 3 A simple rule

Plot points with rule $y = x + 2$ : $(-1, 1), (0, 2), (1, 3), (2, 4)$ .

These four points line up along a straight line. Every unit you move right, the $y$ -value goes up by $1$ .

The four points from the rule y = x + 2 all sit on a single straight line. Right 1, up 1 each time.

Practice

Fluency

Tier 1: basic skills

Which quadrant contains $(5, 3)$ ?
Which quadrant contains $(-4, 2)$ ?
Which quadrant contains $(-1, -7)$ ?
Which quadrant contains $(6, -2)$ ?
Where does $(0, 4)$ lie?
Where does $(-3, 0)$ lie?
Plot $A = (2, 3)$ , $B = (-1, 4)$ , $C = (-2, -5)$ , $D = (4, -3)$ .
State the coordinates of a point $5$ units right of the origin and $2$ units below.
State the coordinates of a point $3$ units left of the origin and on the $x$ -axis.
What are the coordinates of the origin?
Find the coordinates reached by starting at $(1, 1)$ and moving $3$ right and $5$ down.
What quadrant do you enter if you reflect $(2, 3)$ in the $y$ -axis?
What quadrant do you enter if you reflect $(-4, 5)$ in the $x$ -axis?
Give any point on the $y$ -axis with $y$ negative.
Give any point in quadrant 3.

Reasoning

Tier 2: mixed practice

Plot the points $(1, 2), (2, 4), (3, 6), (4, 8)$ . Describe the pattern between $x$ and $y$ .
Plot $(-3, 0), (-1, 0), (1, 0), (3, 0)$ . What do all these points have in common?
A triangle has vertices $(0, 0), (4, 0), (0, 3)$ . Find its area.
A rectangle has opposite corners at $(1, 1)$ and $(6, 4)$ . Find its perimeter and area.
Complete the table for $y = 2x - 1$ and then plot the points:

$x$ $-2$ $-1$ $0$ $1$ $2$
$y$ ? ? ? ? ?
Check whether the point $(3, 5)$ lies on the line described by $y = 2x - 1$ .
A point is reflected in the $x$ -axis. Which coordinate changes sign?
Translate $(4, -2)$ by $\binom{-6}{3}$ . Find the image.

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	?	?	?	?	?

Reasoning

Tier 3: explain and spot the mistake

Ravi plots $(-2, 3)$ by going right $2$ and up $3$ . What has Ravi done wrong?
Explain why $(0, 5)$ is not in any quadrant.
A student says “every point with positive coordinates is in quadrant 1”. Is that correct? Explain.
Write three different points on the line $y = x$ .

Problem solving

Tier 4: real-world problems

A town map uses a Cartesian system with a school at the origin. The library is at $(3, 2)$ (each unit is $100$ m east/north). How far east and how far north of the school is the library? How far in a straight line? (Hint: use Pythagoras.)
A boat leaves a harbour $(0, 0)$ and sails $4$ units east, then $3$ units north, then $2$ units west. What are its current coordinates? How far is it from the harbour in a straight line?
Three vertices of a rectangle are $(1, 1), (7, 1), (7, 5)$ . Find the fourth vertex and the rectangle’s perimeter.
A park has corners at $(0, 0), (8, 0), (8, 5), (0, 5)$ . Where is the centre of the park? (Hint: average the coordinates of opposite corners.)
The midpoint between the points $(2, 3)$ and $(6, 7)$ lies at what coordinates? (Hint: average of the $x$ s and average of the $y$ s.)

Interactive

Try it yourself: drag and place

Work through 3 examples on one graph. Drag, check, then move to the next.

Example 1 (easy). Drag point P so it lies in the third quadrant (where both x and y are negative).

P: (2, 2)
Quadrant: 1