The Coordinate Plane: Axes, Quadrants, and Graphing Points | Grade 6 Math | LevelKite

You already know how to place positive and negative numbers on a number line. On a horizontal number line, one number is enough to name a spot. For example, $-3$ is the spot 3 units to the left of $0$ .

A flat surface is different. A spot on it can be left or right, and it can also be up or down. One number cannot tell you both. You need two number lines that work together.

In this lesson you will learn the parts of that two-line system. You will see what the signs of a point's two numbers tell you. Then you will graph points, which means marking exactly where they are. Some people say plot instead of graph.

A coordinate plane is a flat surface with two number lines drawn on it. Each number line is called an axis. The plural of axis is axes. A standard coordinate plane is set up like this:

The x-axis is the horizontal number line. It runs left and right. Its positive numbers are to the right of $0$ and its negative numbers are to the left.
The y-axis is the vertical number line. It runs up and down. Its positive numbers are above $0$ and its negative numbers are below.
The origin is the point where the two axes cross. Both axes read $0$ there.

On each axis, one unit is the distance from $0$ to $1$ .

Every point on the coordinate plane has an address made of two numbers. The address is called an ordered pair. Call the first number $x$ and the second number $y$ . Then the ordered pair is written $(x, y)$ .

$x$ is the x-coordinate. It tells how far the point is to the right or left of the origin. Positive means right. Negative means left.
$y$ is the y-coordinate. It tells how far the point is above or below the origin. Positive means up. Negative means down.

The x-coordinate always comes first, just as x comes before y in the alphabet. That is why the pair is called ordered.

Look at the figure below. Its grid lines are 1 unit apart, and it shows the plane from $-6$ to $6$ on each axis. The x-axis is the horizontal line through the origin. The y-axis is the vertical line through the origin. The origin is labeled $(0, 0)$ . Both of its coordinates are $0$ , because the origin sits at $0$ on the x-axis and at $0$ on the y-axis.

Point A has the ordered pair $(4, 3)$ . The $4$ puts it 4 units to the right of the origin. The $3$ puts it 3 units up.

The origin is labeled

(0, 0)

. Point A is at

(4, 3)

, Point B is at

(-3, 5)

, Point C is at

(-4, -2)

and Point D is at

(2, -4)

. Grid lines are 1 unit apart.

The two axes cut the plane into four regions. Each region is called a quadrant. The quadrants are numbered with the Roman numerals I, II, III and IV, which mean 1, 2, 3 and 4. Start in the upper right. Then go counterclockwise, which is the opposite of the way clock hands turn.

Quadrant I is the upper right. Point A at $(4, 3)$ is in it.
Quadrant II is the upper left. Point B at $(-3, 5)$ is in it.
Quadrant III is the lower left. Point C at $(-4, -2)$ is in it.
Quadrant IV is the lower right. Point D at $(2, -4)$ is in it.

The axes are the borders between the quadrants. A point that is on an axis is not in any quadrant. The origin is on both axes, so it is in no quadrant.

You can name a point's quadrant without a picture, because each sign answers one question.

The sign of the x-coordinate tells which side of the y-axis the point is on. Positive is the right side. Negative is the left side.
The sign of the y-coordinate tells which side of the x-axis the point is on. Positive is above. Negative is below.

This works only when neither coordinate is $0$ . The number $0$ is neither positive nor negative, and a coordinate of $0$ puts the point on an axis, not in a quadrant.

When both coordinates do have a sign, there are two choices for the first sign and two choices for the second. That makes $2 \times 2 = 4$ sign patterns, one for each quadrant. The table lists all four.

Sign patterns for the four quadrants, when neither coordinate is 0

Quadrant	Where it is	Sign of x-coordinate	Sign of y-coordinate	Example point
I	right of the y-axis, above the x-axis	positive	positive	Point A, $(4, 3)$
II	left of the y-axis, above the x-axis	negative	positive	Point B, $(-3, 5)$
III	left of the y-axis, below the x-axis	negative	negative	Point C, $(-4, -2)$
IV	right of the y-axis, below the x-axis	positive	negative	Point D, $(2, -4)$

Key concept

An ordered pair is two moves from the origin

On a standard coordinate plane, the ordered pair $(x, y)$ tells you how to reach a point from the origin.

The x-coordinate comes first. It is the left-right move: right if it is positive, left if it is negative, no move if it is $0$ .
The y-coordinate comes second. It is the up-down move: up if it is positive, down if it is negative, no move if it is $0$ .

Each sign is a direction, so the two signs together name the quadrant. This holds only when neither coordinate is $0$ . If a coordinate is $0$ , the point is on an axis and is in no quadrant.

Worked example

Name the quadrant from the signs

Without graphing, decide where each point is: $(-12, 9)$ , $(8, -15)$ and $(0, -6)$ . Name the quadrant, or say that the point is on an axis. The first two points are too far out to fit on a grid that stops at $6$ , so use what the signs tell you.

Look for a 0 first. The sign patterns name a quadrant only when neither coordinate is 0. The points (-12, 9) and (8, -15) have no 0, so each one is in a quadrant. The point (0, -6) has an x-coordinate of 0, so save it for the last step.
Read the signs of (-12, 9). The x-coordinate, -12, is negative, so the point is left of the y-axis. The y-coordinate, 9, is positive, so the point is above the x-axis. Left and above is the upper left region, Quadrant II. Because neither coordinate is 0, only the signs matter here, not how big the numbers are.
$(-12, 9) \rightarrow (\text{negative}, \text{positive}) \rightarrow \text{Quadrant II}$
Read the signs of (8, -15). The x-coordinate, 8, is positive, so the point is right of the y-axis. The y-coordinate, -15, is negative, so the point is below the x-axis. Right and below is the lower right region, Quadrant IV.
$(8, -15) \rightarrow (\text{positive}, \text{negative}) \rightarrow \text{Quadrant IV}$
Now place (0, -6). An x-coordinate of 0 means no move left or right, so the point stays on the y-axis. The y-coordinate, -6, is negative, so the point is 6 units below the origin. It sits on a border between quadrants, not inside one.

Answer

The point $(-12, 9)$ is in Quadrant II. The point $(8, -15)$ is in Quadrant IV. The point $(0, -6)$ is on the y-axis, so it is in no quadrant.

Now use the two moves to graph a point. Follow the ordered pair from left to right.

First, start at the origin, $(0, 0)$ .
Next, read the x-coordinate. Slide along the x-axis to that number, just as you would find it on a number line. Go right for a positive number and left for a negative number. For $0$ , stay at the origin.
Then, read the y-coordinate. From where you stopped, go straight up for a positive number or straight down for a negative number. Count as many units as the number is from $0$ . For $0$ , stay where you are.
Last, mark a dot where you finish.

Counting grid lines is the same as counting units only when the grid lines are 1 unit apart. That is true in every figure in this lesson. Each figure shows the plane from $-6$ to $6$ on both axes. Every point you graph here has integer coordinates in that range, with no fractions or decimals. So each dot fits on the grid and sits exactly where two grid lines cross.

Worked example

Graph one point step by step

A coordinate plane shows $-6$ to $6$ on each axis, with grid lines 1 unit apart. Graph Point E at $(-5, 2)$ . Then name its quadrant.

First check that the point fits on this grid. Both -5 and 2 are between -6 and 6, so it does. Now start at the origin, (0, 0). Both coordinates are measured from the origin, because both axes read 0 there.
Read the x-coordinate, -5. It comes first, so it is the left-right move. On a number line, -5 is 5 units to the left of 0. So move 5 units left along the x-axis. You stop at -5 on the x-axis, which is the point (-5, 0). The figure below marks this stop with the label -5.
$(0, 0) \rightarrow (-5, 0)$
Read the y-coordinate, 2. It comes second, so it is the up-down move. It is positive, so move 2 units straight up from your stop. Moving straight up keeps you 5 units left of the y-axis, so the x-coordinate is still -5.
$(-5, 0) \rightarrow (-5, 2)$
Mark a dot there and label it Point E. Now check the quadrant. Neither coordinate is 0, so the signs name it. The negative x-coordinate means left of the y-axis. The positive y-coordinate means above the x-axis. That is the upper left region, Quadrant II.
$(\text{negative}, \text{positive}) \rightarrow \text{Quadrant II}$

Answer

Point E is 5 units left of the origin and 2 units up. It is at $(-5, 2)$ , in Quadrant II.

Start at the origin, labeled

(0, 0)

. The point labeled

-5

is the stop on the x-axis, at

(-5, 0)

. Point E is 2 units above that stop, at

(-5, 2)

. Grid lines are 1 unit apart.

Worked example

Graph points in the other three quadrants

Point E is already at $(-5, 2)$ on a plane that shows $-6$ to $6$ on each axis, with grid lines 1 unit apart. Graph Point F at $(4, 1)$ , Point G at $(-3, -4)$ and Point H at $(2, -5)$ . Name the quadrant of each new point.

Graph Point F, (4, 1). Start at the origin. The x-coordinate, 4, is positive, so move 4 units right to (4, 0). The y-coordinate, 1, is positive, so move 1 unit up. Right and up is the upper right region, Quadrant I.
$(0, 0) \rightarrow (4, 0) \rightarrow (4, 1)$
Graph Point G, (-3, -4). Go back to the origin, because each ordered pair is measured from there. The x-coordinate, -3, is negative, so move 3 units left to (-3, 0). The y-coordinate, -4, is negative, so move 4 units down. Left and down is the lower left region, Quadrant III.
$(0, 0) \rightarrow (-3, 0) \rightarrow (-3, -4)$
Graph Point H, (2, -5). Start at the origin again. The x-coordinate, 2, is positive, so move 2 units right to (2, 0). The y-coordinate, -5, is negative, so move 5 units down. Right and down is the lower right region, Quadrant IV.
$(0, 0) \rightarrow (2, 0) \rightarrow (2, -5)$
Check the three new points. Every coordinate is an integer from -6 to 6, so each dot fits on the grid where two grid lines cross. No coordinate is 0, so each point is inside a quadrant and its signs name that quadrant. The figure below shows the result.

Answer

Point F at $(4, 1)$ is in Quadrant I, Point G at $(-3, -4)$ is in Quadrant III, and Point H at $(2, -5)$ is in Quadrant IV. With Point E in Quadrant II, each quadrant now holds one point.

Point E is at

(-5, 2)

, Point F is at

(4, 1)

, Point G is at

(-3, -4)

and Point H is at

(2, -5)

. Each quadrant holds one point. Grid lines are 1 unit apart.

Here is the idea to keep. A coordinate plane is two number lines, the x-axis and the y-axis, that cross at the origin, $(0, 0)$ . An ordered pair $(x, y)$ is a point's address. The first number, $x$ , is the left-right move from the origin. The second number, $y$ , is the up-down move. On a standard coordinate plane, positive means right or up, and negative means left or down. When neither coordinate is $0$ , the two signs name one of the four quadrants. When a coordinate is $0$ , the point is on an axis.

Next, you will work in the other direction and read the coordinates of a point that is already graphed. You will also graph points with fraction and decimal coordinates, and you will reflect (flip) a point across an axis.