Coordinate Plane

The coordinate plane is a grid made by two number lines that cross at 0. The flat one is the x-axis, and the up-and-down one is the y-axis. Every point has an ordered pair (x, y). To plot (3, −2), start at 0, move 3 right, then move 2 down.

Before this lesson: You should know how to use negative numbers on a number line and how to find absolute value.

Parts of the Coordinate Plane

Think of a city map with streets laid out in a grid. To tell a friend where to meet, you give two directions: how far across and how far up. The coordinate plane works the same way.

• The x-axis is the horizontal number line. It runs left and right.
• The y-axis is the vertical number line. It runs up and down.
• The origin is the point where the two axes cross. Its ordered pair is (0, 0).
• An ordered pair (x, y) names a point. The first number is the x-coordinate, and the second is the y-coordinate.

The four quadrants

The axes split the plane into four parts called quadrants. They are numbered I, II, III and IV. Start at the top right and go counterclockwise, the opposite way a clock’s hands turn.

Quadrant	Where it is	Sign of x	Sign of y	Example
I	Top right	+	+	(3, 5)
II	Top left	−	+	(−3, 5)
III	Bottom left	−	−	(−3, −5)
IV	Bottom right	+	−	(3, −5)

A point on an axis, like (0, 4) or (−2, 0), is not in any quadrant.

How to Plot a Point on the Coordinate Plane

1. Start at the origin, (0, 0).
2. Look at the x-coordinate. Move right if it is positive. Move left if it is negative.
3. Look at the y-coordinate. From where you stopped, move up if it is positive. Move down if it is negative.
4. Draw a dot and label it with its ordered pair.

If a coordinate is 0, don’t move in that direction. A memory trick: x comes before y in the alphabet, so you always move across before you move up or down.

Worked Examples

The graph below shows the points from Examples 1 to 3.

Example 1: Plot a point in Quadrant I

Plot (4, 2).

Step 1. Start at the origin.

Step 2. The x-coordinate is 4, so move 4 units right.

Step 3. The y-coordinate is 2, so move 2 units up. Draw the dot.

Answer: The point (4, 2) is in Quadrant I, because both coordinates are positive.

Example 2: Plot a point with negative coordinates

Plot (−5, −3).

Step 1. Start at the origin. The x-coordinate is −5, so move 5 units left.

Step 2. The y-coordinate is −3, so move 3 units down. Draw the dot.

Answer: The point (−5, −3) is in Quadrant III, because both coordinates are negative.

Example 3: Points on the axes

Plot (0, −4) and (5, 0).

Step 1. For (0, −4), the x-coordinate is 0, so don’t move across. Move 4 units down. The point is on the y-axis.

Step 2. For (5, 0), move 5 units right. The y-coordinate is 0, so don’t move up or down. The point is on the x-axis.

Answer: Both points sit on an axis, so neither one is in a quadrant.

Example 4: Read points from a graph

Write the ordered pair for each point, and name its quadrant.

Step 1. For P, count from the origin: 4 units left, then 3 units up. So P is (−4, 3).

Step 2. For Q, count 3 units right, then 5 units down. So Q is (3, −5).

Step 3. For R, count 6 units left, then 2 units down. So R is (−6, −2).

Answer: P(−4, 3) is in Quadrant II, Q(3, −5) is in Quadrant IV and R(−6, −2) is in Quadrant III.

Example 5: Word problem

A town map uses a coordinate plane, and each unit is one block. The library is at (−3, 2), the school is at (4, 2), and the park is at (4, −3). How many blocks is it from the library to the school? From the school to the park?

Step 1. The library and the school both have a y-coordinate of 2, so they are on the same horizontal line.

Step 2. The library is |-3| = 3 blocks left of the y-axis. The school is |4| = 4 blocks right of it. They are on opposite sides, so add: 3 + 4 = 7 blocks.

Step 3. The school and the park both have an x-coordinate of 4. The school is 2 blocks above the x-axis, and the park is |-3| = 3 blocks below it. Add: 2 + 3 = 5 blocks.

Answer: It is 7 blocks from the library to the school, and 5 blocks from the school to the park.

Common Mistakes

Mistake	What it looks like	How to fix it
Switching x and y	Plotting (2, 5) by going up 2 and right 5	x always comes first: go right 2, then up 5.
Moving the wrong way	Plotting (−3, 1) to the right of the y-axis	A negative x-coordinate means move left. A negative y-coordinate means move down.
Putting an axis point in a quadrant	“(0, −6) is in Quadrant IV”	A point with a 0 coordinate sits on an axis, not in a quadrant.
Numbering quadrants clockwise	Calling the bottom right Quadrant II	Count counterclockwise from the top right: I, II, III, IV.
Counting the origin as 1	Plotting (3, 2) one unit short	Start counting at 0. The first grid line away from the origin is 1.

Distance and Reflections on the Coordinate Plane

Distance between two points on a grid line

When two points have the same x-coordinate or the same y-coordinate, they are on one grid line. You can find the distance between them with absolute values.

• If the points are on opposite sides of an axis, add the absolute values. From (−3, 2) to (4, 2) is 3 + 4 = 7 units.
• If the points are on the same side, subtract the absolute values. From (2, 6) to (2, 1) is 6 − 1 = 5 units.

To find the distance between points that don’t line up, you’ll use the distance formula in Geometry (coming soon).

Reflections across an axis

If you change the sign of one coordinate, the point flips across an axis like a mirror image.

• (3, 4) and (3, −4) are reflections across the x-axis. Only the y-coordinate changes sign.
• (3, 4) and (−3, 4) are reflections across the y-axis. Only the x-coordinate changes sign.

Later you’ll graph whole lines on the coordinate plane. Next steps include two-variable linear equations, x-intercepts and y-intercepts and slope.

Practice Problems

For problems 1–4, name the quadrant or axis where each point lies.

1. (−6, 2)
2. (7, −1)
3. (−2, −9)
4. (0, 3)
5. Describe how to move from the origin to plot (−4, 6).
6. A point is 5 units right of the origin and 2 units below it. What is its ordered pair?
7. The point (−2, 5) is reflected across the x-axis. What is the new point?
8. Find the distance between (1, 7) and (1, −2).
9. Find the distance between (−8, −3) and (−2, −3).
10. On a town map, each unit is one block. Home is at (−5, −1) and the store is at (3, −1). How many blocks apart are they?

Show answers

1. Quadrant II. x is negative and y is positive.
2. Quadrant IV. x is positive and y is negative.
3. Quadrant III. Both coordinates are negative.
4. On the y-axis. The x-coordinate is 0, so it is not in a quadrant.
5. Move 4 units left, then 6 units up.
6. (5, −2). Right means a positive x, and below means a negative y.
7. (−2, −5). A reflection across the x-axis changes the sign of the y-coordinate.
8. The x-coordinates match. The points are on opposite sides of the x-axis, so add: |7| + |-2| = 7 + 2 = 9 units.
9. The y-coordinates match. Both points are left of the y-axis, so subtract: |-8| - |-2| = 8 - 2 = 6 units.
10. The y-coordinates match, and the points are on opposite sides of the y-axis: |-5| + |3| = 5 + 3 = 8 blocks.

Coordinate Plane FAQ

What are the 4 quadrants of the coordinate plane?

Quadrant I is the top right, where x and y are both positive. Going counterclockwise, Quadrant II is the top left (−, +), Quadrant III is the bottom left (−, −) and Quadrant IV is the bottom right (+, −).

Which comes first in an ordered pair, x or y?

The x-coordinate always comes first. In (2, 7), you move 2 across and then 7 up.

What quadrant is the origin in?

None. The origin, (0, 0), sits on both axes, so it is not in any quadrant.

Why is it called the Cartesian plane?

It is named after René Descartes, a French mathematician who wrote about it in 1637. “Coordinate plane” and “Cartesian plane” mean the same thing.

Summary

• The coordinate plane has a horizontal x-axis and a vertical y-axis that cross at the origin, (0, 0).
• An ordered pair (x, y) gives the x-coordinate first, then the y-coordinate.
• To plot a point, start at the origin, move left or right for x, then up or down for y.
• The quadrants are numbered I to IV counterclockwise from the top right.
• Points on an axis are not in any quadrant.
• For points on the same grid line, use absolute values to find the distance.

Previous lesson: Multiplying and Dividing Integers

Next lesson: Prime Factorization

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