The Number System

The Coordinate Plane with Negative Numbers

Plotting points in all four quadrants of the coordinate plane.

Intermediate15 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Absolute Value

All Four Directions at Once

In Grade 5, every point plotted on the coordinate plane sat to the right of and above the origin. But what about a location that's west of a landmark as well as south of it? Before reading on, predict what kind of coordinates such a point would need.

A location west and south of the origin needs negative coordinates in both directions — negative x for west, negative y for south. Extending both axes to include negative numbers divides the coordinate plane into four regions, called quadrants, each with its own consistent pattern of positive and negative coordinates.

Definition — Quadrant

One of four regions the coordinate plane is divided into by the x- and y-axes. Quadrant I has both coordinates positive; Quadrant II has a negative x and positive y; Quadrant III has both coordinates negative; Quadrant IV has a positive x and negative y.

Worked Example — Plotting a Point with Negative Coordinates

Plot the point (−4, 3). Move 4 units to the left of the origin (since x is negative), then 3 units up (since y is positive). This point lands in Quadrant II.

Worked Example — Identifying a Quadrant from Coordinates

Without plotting it, which quadrant contains the point (−6, −2)? Both coordinates are negative, matching the pattern for Quadrant III.

Graph Visualizer

Trace modePoint mode (click graph to plot)

Show x^2

Color

Expression

Domain & range

x min

x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

The quadrants are numbered counterclockwise starting from the upper right — I, II, III, IV — the same order you'd travel if you started in the all-positive region and rotated around the origin.

Common Mistakes

Moving in the wrong direction for a negative coordinate, such as moving right instead of left for a negative x-value.
A negative x-coordinate always means moving left of the origin, and a negative y-coordinate always means moving down — the sign directly controls the direction.
Mixing up which quadrant corresponds to which combination of signs.
Quadrant I is both positive; moving counterclockwise, each quadrant flips one more sign to negative — II flips x, III flips both, IV flips only y.

Key Takeaways

Extending the coordinate plane with negative numbers creates four quadrants, each with a fixed pattern of coordinate signs.
The sign of each coordinate directly determines which direction to move from the origin.
Quadrants are numbered I through IV, counterclockwise starting from the all-positive region.

Summary

The full four-quadrant coordinate plane locates any point in any direction from the origin. The next unit turns from plotting numbers to representing unknown quantities symbolically, beginning with a shorthand for repeated multiplication.

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