Points and Quadrants on the Coordinate Plane | Grade 7 Math | LevelKite

A coordinate plane is a flat grid made from two number lines that cross at right angles. The horizontal number line is the x-axis. The vertical number line is the y-axis. They cross at the origin, the place where both number lines read $0$ .

You name every location on the plane with an ordered pair $(x, y)$ . The first number, $x$ , is the x-coordinate. It tells how far the point is to the right or left of the origin. The second number, $y$ , is the y-coordinate. It tells how far the point is above or below the origin. In this lesson, one unit is one grid step, and the ticks on both axes are numbered from $-5$ to $5$ . The origin is the ordered pair $(0, 0)$ .

The two axes split the plane into four regions called quadrants. You number them with Roman numerals. Start in the upper right and turn counterclockwise:

Quadrant I (upper right): signs $(+, +)$ . Point A at $(3, 2)$ is here.
Quadrant II (upper left): signs $(-, +)$ . Point B at $(-4, 1)$ is here.
Quadrant III (lower left): signs $(-, -)$ . Point C at $(-2, -3)$ is here.
Quadrant IV (lower right): signs $(+, -)$ . Point D at $(1, -4)$ is here.

The signs come from the number lines. To the right of the origin $x > 0$ , and to the left $x < 0$ . Above the origin $y > 0$ , and below it $y < 0$ . Point B is farther left than Point C because $|-4| = 4$ is greater than $|-2| = 2$ .

Point A

(3, 2)

is in Quadrant I, B

(-4, 1)

in II, C

(-2, -3)

in III and D

(1, -4)

in IV. The origin is

(0, 0)

.

Key concept

Plotting a point from the origin

To plot $(x, y)$ , start at the origin.

Move along the x-axis. Go right if $x$ is positive and left if $x$ is negative. Move $|x|$ units.
Then move up if $y$ is positive and down if $y$ is negative. Move $|y|$ units.

A coordinate of $0$ means you make no move in that direction. You move the absolute value because absolute value is a number's distance from $0$ . The sign only tells you the direction.

Worked example

Plot a point

Plot the point $(-3, 4)$ and name its quadrant.

Read the pair in order. The first number is always the x-coordinate, and the second number is always the y-coordinate.
$x = -3,\quad y = 4$
Start at the origin, because both coordinates measure a distance from it.
$(0, 0)$
The x-coordinate is negative, so move left. Move $|-3| = 3$ units, because absolute value gives the distance from $0$ . You are now at the tick $-3$ on the x-axis.
$|-3| = 3 \text{ units left}$
The y-coordinate is positive, so from there move up $|4| = 4$ units. Mark the point where you stop.
$|4| = 4 \text{ units up}$
Use the signs to name the quadrant. Here $x$ is negative and $y$ is positive. That is the pattern $(-, +)$ , which belongs to Quadrant II.
$(-,\ +) \Rightarrow \text{Quadrant II}$

Answer

The point $(-3, 4)$ is 3 units left of the origin and 4 units up, and it lies in Quadrant II.

The finished point

(-3, 4)

sits 3 units left of the origin and 4 units up, in the upper-left region, Quadrant II.

To identify a point, run the same moves backward. Look straight up or down from the point to the x-axis and read the numbered tick there. That number is $x$ . Then look straight across from the point to the y-axis and read that tick. That number is $y$ . Write the two numbers in order as $(x, y)$ . Then use their signs to name the quadrant, or use a $0$ coordinate to name the axis.

Point E is 4 units left of the origin and 2 units down. Point F is 3 units straight down from the origin.

Worked example

Identify points

Write the ordered pair for Point E and for Point F. Then name the quadrant or axis where each point lies.

Point E is directly below the tick $-4$ on the x-axis. That puts it 4 units left of the origin, and left means negative.
$x = -4$
Point E is level with the tick $-2$ on the y-axis. That puts it 2 units below the origin, and below means negative.
$y = -2$
Write the pair in the order $(x, y)$ . Both coordinates are negative, which is the pattern $(-, -)$ , so Point E is in Quadrant III.
$E = (-4, -2)$
Point F is straight below the origin, so it has no move left or right. It is level with the tick $-3$ on the y-axis.
$x = 0,\quad y = -3$
Write the pair. Because $x = 0$ , Point F lies on the y-axis. Zero is neither positive nor negative, so Point F is in no quadrant.
$F = (0, -3)$

Answer

Point E is $(-4, -2)$ in Quadrant III, and Point F is $(0, -3)$ on the y-axis, in no quadrant.

The correct point

(2, -5)

is in Quadrant IV. The swapped point

(-5, 2)

is in Quadrant II.

Takeaway: Every point on the coordinate plane is one ordered pair $(x, y)$ . To plot it, start at the origin, move $x$ units right or left, and then move $y$ units up or down. The signs of the two coordinates tell you the quadrant, and a $0$ coordinate puts the point on an axis. Next, you will use coordinates to find the distance between two points on the coordinate plane.