Key takeaway: The quadrants are the four regions of the coordinate plane cut by the axes, numbered I to IV counterclockwise from the upper right, each with its own pattern of signs.
Grade 66.NS.C.6bGeometry and Measurement
Definition of Quadrants
The two axes of the coordinate plane split it into four quadrants. Each region contains the points whose coordinates have a particular pair of signs, so the signs alone tell you where to look before you plot anything.
The quadrants are numbered with Roman numerals, starting at the upper right and moving counterclockwise: I, II, III, IV. A point lying exactly on an axis belongs to none of the quadrants, because one of its coordinates is zero and zero is neither positive nor negative.
Surveyors, game designers and map makers use this sign test constantly: a character moving left and down on a screen is in quadrant III relative to its starting point.
Notation
The sign pattern of \((x,\, y)\) identifies the region:
| Quadrant | \(x\) | \(y\) | Location |
|---|---|---|---|
| I | + | + | upper right |
| II | − | + | upper left |
| III | − | − | lower left |
| IV | + | − | lower right |
Key facts
- Numbering of the quadrants runs counterclockwise, opposite to the direction a clock hand moves.
- A point with a zero in either coordinate lies on an axis, not inside a region.
- Reflecting a point across the y-axis moves it between the two upper regions or the two lower ones.
- Only the first of the four quadrants has two positive coordinates.
Quadrants example
Name the quadrant of \((-7,\, -2.5)\), \((6,\, -8)\) and \((0,\, 5)\).
-
\((-7,\, -2.5)\): both numbers are negative, so it is in the lower left region.
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\((6,\, -8)\): positive \(x\), negative \(y\), so it is in the lower right region.
-
\((0,\, 5)\): the \(x\)-value is 0, so the point is on the y-axis.
The points are in Quadrant III, Quadrant IV, and on the y-axis (no quadrant).
Common mistake
Watch out
Sofia numbers the regions clockwise and says \((-4,\, 3)\) is in Quadrant IV.
Why it is wrong
The numbering is counterclockwise from the upper right. Negative \(x\) with positive \(y\) is the upper left region, which is Quadrant II. Sketch the signs to check.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Geometry and Measurement