Coordinates in Four Quadrants

From one number line to two

You have probably used a single number line before, stretching from negative numbers on the left, through zero in the middle, to positive numbers on the right. A coordinate grid takes this idea further by using two number lines that cross each other at right angles.

• The horizontal number line is called the x-axis. Positive numbers go to the right, negative numbers to the left.
• The vertical number line is called the y-axis. Positive numbers go up, negative numbers go down.
• The point where the two axes cross is called the origin, and it has coordinates (0, 0).

This system is named after the mathematician René Descartes, which is why it is often called the Cartesian coordinate system. It lets us give every point on a flat surface a unique "address". For an introduction to the basic grid using only positive numbers, see our lesson on the coordinate plane.

How to read and write a coordinate

A coordinate is always written as a pair of numbers inside brackets, separated by a comma: (x, y).

• The first number is the x-coordinate — how far across (right if positive, left if negative).
• The second number is the y-coordinate — how far up or down (up if positive, down if negative).

The order matters a lot: (3, 5) is a completely different point from (5, 3). A reliable way to remember the order is the phrase "along the corridor, then up the stairs" — you always go across first, then up or down.

The four quadrants

The x-axis and y-axis divide the grid into four regions, and each region is called a quadrant. The quadrants are numbered with Roman numerals I, II, III and IV (1 to 4), starting from the top-right and going anticlockwise.

Each quadrant has its own fixed pattern of signs:

Quadrant	Position	Sign of x	Sign of y	Example
I (1)	top right	+	+	(3, 5)
II (2)	top left	−	+	(−3, 5)
III (3)	bottom left	−	−	(−3, −5)
IV (4)	bottom right	+	−	(3, −5)

A neat way to picture it: starting at the top right and spinning anticlockwise, the x-sign and then the y-sign flip on and off: (+,+) → (−,+) → (−,−) → (+,−).

A diagram in words

Imagine a large "plus sign" drawn in the middle of a page. The horizontal line is the x-axis; the vertical line is the y-axis. Where they cross, in the dead centre, is the origin (0, 0).

The top-right box is Quadrant I, where everything is positive — plot (3, 5) here by going 3 right and 5 up. The top-left box is Quadrant II; plot (−3, 5) by going 3 left and 5 up. The bottom-left box is Quadrant III; plot (−3, −5) by going 3 left and 5 down. The bottom-right box is Quadrant IV; plot (3, −5) by going 3 right and 5 down. The four points form a neat rectangle, one in each corner of the page, perfectly balanced around the origin.

Worked example 1: plotting a point

Plot the point A(−4, 2) and name its quadrant.

Start at the origin. The x-coordinate is −4, so move 4 squares left. The y-coordinate is +2, so move 2 squares up. Mark the point. Because x is negative and y is positive (−, +), point A lies in Quadrant II (top left).

Worked example 2: reading a point

A point B is found by starting at the origin, moving 5 right and 3 down. Write its coordinates and quadrant.

Moving 5 right gives an x-coordinate of +5. Moving 3 down gives a y-coordinate of −3. So B = (5, −3). With signs (+, −), point B is in Quadrant IV (bottom right).

Worked example 3: a point on an axis

Where does the point C(0, −6) lie?

The x-coordinate is 0, which means we do not move left or right at all — we stay on the y-axis. The y-coordinate is −6, so we move 6 down. Point C sits on the y-axis, 6 units below the origin. Because it is on an axis, it is not inside any quadrant.

Worked example 4: finding a missing corner

Three corners of a rectangle are (−2, 3), (4, 3) and (4, −1). Find the fourth corner.

A rectangle has opposite corners with matching coordinates. The fourth corner must share its x-coordinate with (−2, 3) (so x = −2) and its y-coordinate with (4, −1) (so y = −1). The missing corner is (−2, −1), which lies in Quadrant III.

Why this matters

Working in all four quadrants unlocks the whole coordinate plane, not just one corner of it. This is the language of graphs: when you plot a line, a curve, or the path of a moving object, the points can land anywhere — above or below, left or right of the origin — so you need negative coordinates to describe them. Scientists use coordinates to map temperatures below zero, navigators use them for positions east or west and north or south of a starting point, and every digital map, video game and computer screen locates things using a coordinate grid. Coordinates are also the bridge to algebra and geometry combined: equations like y = 2x − 3 become straight lines you can draw, and shapes can be transformed, measured and described using nothing but their coordinates. Master all four quadrants now and you are ready for graphs, functions and beyond.

Activity: coordinate battleships and treasure

1. Plot a picture: on squared paper, draw and label both axes from −6 to 6. Plot this set of points in order and join them as you go: (1, 1), (3, 1), (3, −2), (−2, −2), (−2, 1), then back to (1, 1). What shape appears, and which quadrants does it pass through?
2. Four-quadrant battleships: play with a partner. Each of you secretly marks 3 ships on a 12×12 grid using coordinates that include negatives. Take turns calling out coordinates like (−3, 4) to find your opponent's ships. The signs make it trickier — and more fun!
3. Treasure hunt: write directions to a treasure as a list of coordinates and challenge a friend to plot them. The treasure is wherever the path ends.
4. Challenge: plot any point, then write down where it lands after you make both its coordinates negative. What is the connection between the original point and the new one? (Hint: it is a 180° rotation about the origin.)