Quadrant — Definition, Formula & Examples

A quadrant is one of the four sections created when the x-axis and y-axis divide the coordinate plane. The quadrants are numbered I through IV using Roman numerals, starting in the upper right and moving counterclockwise.

In a two-dimensional Cartesian coordinate system, the axes partition the plane into four regions called quadrants. Quadrant I contains all points where both coordinates are positive, Quadrant II where $x < 0$ and $y > 0$ , Quadrant III where both coordinates are negative, and Quadrant IV where $x > 0$ and $y < 0$ .

How It Works

To identify which quadrant a point is in, look at the signs of its coordinates. If both are positive, the point is in Quadrant I. If

x

is negative and

y

is positive, it is in Quadrant II. If both are negative, Quadrant III. If

x

is positive and

y

is negative, Quadrant IV. Points that lie exactly on the x-axis or y-axis are not in any quadrant — they sit on the boundary between quadrants.

Worked Example

Problem: Identify the quadrant of each point: A(3, 5), B(−4, 2), and C(−1, −6).

Point A: Both coordinates are positive (3 > 0 and 5 > 0).

A(3, 5) \text{ is in Quadrant I}

Point B: The x-coordinate is negative and the y-coordinate is positive.

B(-4, 2) \text{ is in Quadrant II}

Point C: Both coordinates are negative (−1 < 0 and −6 < 0).

C(-1, -6) \text{ is in Quadrant III}

Answer: A is in Quadrant I, B is in Quadrant II, and C is in Quadrant III.

Why It Matters

Identifying quadrants helps you quickly predict the signs of coordinates, which is essential when graphing equations or analyzing data on scatter plots. In algebra and precalculus, knowing which quadrant an angle's terminal side lies in determines the signs of trigonometric functions.

Common Mistakes

Mistake: Numbering the quadrants clockwise (I upper-right, II lower-right, etc.).

Correction: Quadrants are numbered counterclockwise: I is upper-right, II is upper-left, III is lower-left, and IV is lower-right.