A quadrant is one of the four regions that the x-axis and y-axis split the coordinate plane into. They are numbered I, II, III and IV counterclockwise, starting at the top right. The signs of a point’s coordinates tell its quadrant: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV.
In Spanish
cuadrante
Also called
quadrants of a graph, quadrants of the coordinate plane, Quadrant I, II, III, IV
Each quadrant has its own sign pattern. To place a point, check the sign of x (right or left) and the sign of y (up or down).
Formal definition and notation
The coordinate axes divide the plane into four open regions. Quadrant I is the set of points (x, y) with x > 0 and y > 0; Quadrant II has x < 0, y > 0; Quadrant III has x < 0, y < 0; Quadrant IV has x > 0, y < 0. Points with x = 0 or y = 0 lie on an axis and belong to no quadrant.
Symbol
Read it as
Meaning
Quadrant I
quadrant one
top right: x > 0, y > 0
Quadrant II
quadrant two
top left: x < 0, y > 0
Quadrant III
quadrant three
bottom left: x < 0, y < 0
Quadrant IV
quadrant four
bottom right: x > 0, y < 0
Key facts
Quadrants are numbered counterclockwise with Roman numerals: I top right, II top left, III bottom left, IV bottom right.
Look only at the signs of (x, y): (7, −3) is (+, −), so it is in Quadrant IV.
The origin(0, 0) and every point on the x-axis or y-axis are in no quadrant.
Changing the sign of x reflects a point across the y-axis into the neighboring quadrant: (3, 2) in I becomes (−3, 2) in II.
Changing both signs moves a point to the opposite quadrant: (3, 2) in I and (−3, −2) in III.
Many real graphs use only Quadrant I, because time, length and price are not negative.
Try it: find the quadrant
Interactive
Drag the points, or select one and use the arrow keys. The readout names the quadrant of each point, or the axis it sits on. Try moving a point onto an axis.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Quadrant examples, step by step
Example 1: Name the quadrant
Warm-up
In which quadrant is (−4, 7)?
x is −4, negative: the point is left of the y-axis.
y is 7, positive: the point is above the x-axis.
Left and up is the top left region: (−, +).
Answer:(−4, 7) is in Quadrant II.
Example 2: Points on the axes
Core
Which of these points are in a quadrant? (0, −5), (6, 0), (−1, −1), (0, 0).
A point is in a quadrant only if neither coordinate is 0.
(0, −5) is on the y-axis and (6, 0) is on the x-axis.
(0, 0) is the origin, where the axes cross.
Only (−1, −1) qualifies: both signs are negative, so Quadrant III.
Answer: Only (−1, −1), which is in Quadrant III. The other three lie on an axis.
Example 3: Signs with variables
Test-style
If a < 0 and b > 0, in which quadrant is the point (b, a)?
The first coordinate is b, which is positive.
The second coordinate is a, which is negative.
The pattern (+, −) is the bottom right region.
Watch the order: (a, b) would be in Quadrant II, but this point is (b, a).
Answer:(b, a) is in Quadrant IV.
Common mistakes
✗ MistakeNumbering clockwise: top right I, bottom right II.
✓ InsteadCounterclockwise: top right I, top left II, bottom left III, bottom right IV.
The quadrants follow the direction angles turn in math, counterclockwise from the positive x-axis.
✗ Mistake(0, 4) is in Quadrant I.
✓ Instead(0, 4) is on the y-axis, in no quadrant.
A quadrant needs both coordinates to be nonzero. Zero is neither positive nor negative, so the point sits on a boundary.
✗ Mistake(2, −6) is in Quadrant II because one sign is negative.
✓ Instead(2, −6) is (+, −): Quadrant IV.
Check which coordinate is negative. A negative y (down) with a positive x (right) is IV; a negative x with a positive y is II.
An ordered pair like (3, −2) names one point; the quadrant names the region that point is in.
Quadrantvs.Quadrilateral
A quadrilateral is a four-sided polygon. Both words come from the root for four, but a quadrant is a region of a graph, not a shape you draw.
Quick check: 3 questions
Question 1 of 3
In which quadrant is (5, −8)?
Show answer and explanation
Answer: D. x is positive (right) and y is negative (down): (+, −) is Quadrant IV. Quadrant II would need a negative x and a positive y.
Question 2 of 3
Which point is in Quadrant III?
Show answer and explanation
Answer: B. Quadrant III needs both coordinates negative, which only (−3, −1) has. (−2, 6) is in II, (4, −4) is in IV, and (0, −4) is on the y-axis.
Question 3 of 3
The point (7, 3) is reflected across the x-axis. Which quadrant is its image in?
Show answer and explanation
Answer: A. Reflecting across the x-axis changes the sign of y: (7, 3) becomes (7, −3), which is (+, −), Quadrant IV. Reflecting across the y-axis instead would give Quadrant II.
Quadrant: frequently asked questions
Which quadrant is positive?
Quadrant I, the top right, is the only quadrant where both coordinates are positive. In Quadrant II only y is positive, in Quadrant IV only x is positive, and in Quadrant III both are negative. Many real-world graphs, such as distance over time, stay entirely in Quadrant I.
Is the origin in a quadrant?
No. The origin (0, 0) is where the axes cross, and points on either axis do not belong to any quadrant. A quadrant needs both coordinates to be nonzero. So (0, 0), (0, 3) and (−5, 0) are all on the axes, not in quadrants.
Why are quadrants numbered counterclockwise?
Math measures angles counterclockwise from the positive x-axis, and the quadrant numbers follow the same direction. Start at the top right, where angles from 0 to 90 degrees point, and turn left: II covers 90 to 180 degrees, III covers 180 to 270, and IV covers 270 to 360.
Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7QuadrilateralA quadrilateral is a polygon with four straight sides, four vertices and four angles. Its four interior angles always add to 360 degrees. Squares, rectangles, rhombuses, parallelograms, trapezoids and kites are special quadrilaterals, each named for extra properties of its sides or angles.Geometry · Grade 2, Grade 3, Grade 5PolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1GraphA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6, Grade 7ReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, GeometryMidpointA midpoint is the point exactly halfway between the two endpoints of a line segment, so it splits the segment into two equal parts. On the coordinate plane, find it by averaging the coordinates: the midpoint of (x1, y1) and (x2, y2) is (x1 + x22, y1 + y22).Geometry · Grade 8, Algebra 1, Geometry