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

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Tier 1: basic skills

Fluency

Quadrant 1
Quadrant 2
Quadrant 3
Quadrant 4
On the $y$ -axis (not in a quadrant)
On the $x$ -axis
$A$ in Q1; $B$ in Q2; $C$ in Q3; $D$ in Q4. (Check positions on a plotted grid.)
$(5, -2)$
$(-3, 0)$
$(0, 0)$
$(4, -4)$
Quadrant 2 (the point becomes $(-2, 3)$ )
Quadrant 3 (the point becomes $(-4, -5)$ )
Any point of the form $(0, n)$ with $n < 0$ , e.g. $(0, -3)$
Any point with both coordinates negative, e.g. $(-1, -1)$

Tier 2: mixed practice

Reasoning

Mixed practice

$y = 2x$ . Each $y$ -value is double its $x$ -value; the points lie on the line $y = 2x$ .
They all lie on the $x$ -axis (every $y$ -coordinate is $0$ ).
$6$ units $^2$ . Method: right-angled triangle with legs $4$ and $3$ ; area $= \dfrac{1}{2}(4)(3)$ .
Perimeter $16$ units; area $15$ units $^2$ . Method: length $= 6 - 1 = 5$ ; height $= 4 - 1 = 3$ ; $P = 2(5 + 3) = 16$ ; $A = 5 \times 3 = 15$ .
$y$ -values: $-5,\ -3,\ -1,\ 1,\ 3$ .
Yes. When $x = 3$ , $y = 2(3) - 1 = 5$ , matching.
The $y$ -coordinate changes sign.
$(-2, 1)$ .

Tier 3: explain and spot the mistake

Reasoning

Explain and spot the mistake

Ravi treated the $x$ -coordinate as positive. For $(-2, 3)$ you move $2$ units left (because $x$ is negative), then $3$ up. The point belongs in quadrant 2, not quadrant 1.
The four quadrants are the open regions between the axes - they exclude the axes themselves. Since $(0, 5)$ has $x = 0$ , it lies on the $y$ -axis, not inside any quadrant.
Almost. If both coordinates are strictly positive, the point is in quadrant 1. But if one of them is $0$ (e.g. $(3, 0)$ or $(0, 3)$ ), the point sits on an axis, not in the quadrant. So the correct statement is “every point with strictly positive coordinates is in quadrant 1”.
Any points where the two coordinates are equal, e.g. $(0, 0)$ , $(1, 1)$ , $(-2, -2)$ .

Tier 4: real-world problems

Problem solving

Real-world problems

$300$ m east, $200$ m north; straight-line distance $\approx 360.6$ m. Method: Pythagoras $\sqrt{300^2 + 200^2}$ .
$(2, 3)$ ; distance $\sqrt{2^2 + 3^2} = \sqrt{13} \approx 3.6$ units. Method: $4 - 2 = 2$ east; $3$ north.
Fourth vertex $(1, 5)$ ; perimeter $20$ units. Method: width $6$ (from $1$ to $7$ ), height $4$ (from $1$ to $5$ ); $P = 2(6 + 4)$ .
Centre at $(4, 2.5)$ . Method: average opposite corners, e.g. $(0+8)/2, (0+5)/2$ .
Midpoint $(4, 5)$ .

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