Skip to content
MathAnvil
§ Geometry

Coordinates (Four Quadrants)

§ Geometry

Coordinates (Four Quadrants)

CCSS.6.NSCCSS.6.G3 min read

Coordinates in four quadrants extend the basic coordinate system by introducing negative values for both x and y axes. The coordinate plane divides into four regions: Quadrant I contains points with positive x and y values, Quadrant II has negative x and positive y, Quadrant III has negative x and negative y, and Quadrant IV has positive x and negative y. This system appears in Year 6 of the UK National Curriculum, where pupils describe positions on the full coordinate grid.

§ 01

Why it matters

Four-quadrant coordinates appear throughout GCSE mathematics, particularly in algebraic graphs where functions like y = x² - 4 cross into negative regions. Navigation systems use similar principles — GPS coordinates include negative values for locations south of the equator or west of the prime meridian. Temperature graphs plotting data below 0°C require negative y-values, whilst profit-loss charts in business studies use negative coordinates to show losses. Computer graphics and game programming rely heavily on four-quadrant systems for positioning objects. Weather maps showing temperature changes across regions frequently display data points in multiple quadrants. The concept prepares students for advanced topics like transformations, where reflections across axes change coordinate signs, and gradient calculations between points in different quadrants.

§ 02

How to solve coordinates (four quadrants)

Coordinates — Four Quadrants

  • Quadrant I: (+, +). Quadrant II: (−, +).
  • Quadrant III: (−, −). Quadrant IV: (+, −).
  • Negative x = left of origin; negative y = below origin.
  • Plot points by moving along x first, then y.

Example: (−2, 3) is in Quadrant II: 2 left, 3 up.

§ 03

Worked examples

Beginner§ 01

In which quadrant is the point (-4, 6)?

Answer: Quadrant II

  1. Check signs of x and y → x = -4 (negative), y = 6 (positive) — Quadrant II: x is negative, y is positive.
Easy§ 02

What are the coordinates after reflecting (4, 5) in the x-axis?

Answer: (4, -5)

  1. Reflect in the x-axis → (4, -5) — Reflecting in the x-axis negates the y-coordinate.
Medium§ 03

Find the distance between (-8, 7) and (7, 7).

Answer: 15

  1. Subtract x-coordinates (same y) → |7 - (-8)| = |15| = 15 — Distance on a horizontal line = absolute difference of x-coordinates.
§ 04

Common mistakes

  • Confusing quadrant numbers with coordinate signs — writing that (-3, 2) is in Quadrant III instead of Quadrant II because the x-coordinate is negative
  • Mixing up reflection rules — reflecting (4, 5) across the y-axis as (4, -5) instead of (-4, 5), incorrectly changing the y-coordinate rather than the x-coordinate
  • Calculating distance between (-6, 3) and (2, 3) as 4 instead of 8, forgetting that movement from negative to positive coordinates requires adding the absolute values
§ 05

Frequently asked questions

How do you remember which quadrant is which?
Start from Quadrant I in the top right (both coordinates positive) and move anticlockwise. Quadrant II is top left (negative x, positive y), Quadrant III is bottom left (both negative), and Quadrant IV is bottom right (positive x, negative y). Many students remember this as 'positive, negative, negative, positive' for the x-coordinates.
What happens to coordinates when you reflect across an axis?
Reflecting across the x-axis changes the sign of the y-coordinate only: (3, 4) becomes (3, -4). Reflecting across the y-axis changes the sign of the x-coordinate only: (3, 4) becomes (-3, 4). The point stays the same distance from the axis of reflection but moves to the opposite side.
How do you find distance between points in different quadrants?
Use the same method as with positive coordinates. For horizontal distance (same y-value), subtract x-coordinates and take the absolute value. For (-5, 2) and (3, 2), the distance is |3 - (-5)| = 8. For vertical distance, do the same with y-coordinates. The negative signs automatically account for crossing quadrants.
Why are coordinates sometimes written with brackets and sometimes without?
Brackets help distinguish coordinates from other mathematical expressions. The point with x-coordinate 3 and y-coordinate -2 is written as (3, -2) to show these are paired values representing one location. Without brackets, 3, -2 might be confused with two separate numbers in a list or calculation.
What's the difference between the origin and a point at (0, 0)?
These refer to the same location — the origin is the point where the x and y axes intersect, which has coordinates (0, 0). The origin serves as the reference point for all other coordinates. Points in different quadrants are described by their position relative to this central point on the coordinate plane.
§ 06

See also

MultipleGradientAbsolute value
§ 06

Where to next?

Share this article