The Coordinate Plane

Learning goals

Name, plot, and read a point as an ordered pair $(x, y)$ , across then up
Say why order matters, so $(3, 2)$ differs from $(2, 3)$
Number the four quadrants counterclockwise, and give each its sign pattern
Measure horizontal and vertical distance with absolute differences
Read a rectangle's perimeter and area off its four corners

Two number lines, one plane

You already know the number line from the integers chapter. It is a straight line with $0$ in the middle, positive numbers to the right and negative numbers to the left. The coordinate plane is built from two of them.

Draw one number line horizontally across the page. Call it the x-axis; it measures position left and right. Draw a second number line vertically up the page, crossing the first at a right angle. Call it the y-axis; it measures position up and down. The point where the two axes cross, where both readings are $0$ , is the origin.

Together these two lines turn the whole flat page into a grid where any point can be located. To reach a point you read how far across it is along the x-axis, then how far up or down it is along the y-axis. Two readings, one for each direction, are exactly enough to name any point in the plane. That matches how one reading was enough on a single line.

The coordinate plane: a horizontal x-axis and a vertical y-axis crossing at the origin (0, 0). Positive x is to the right and positive y is up.

Naming a point: the ordered pair (x, y)

A point’s location is written as an ordered pair of numbers inside parentheses, separated by a comma:

(x, y).

The first number, $x$ , is the x-coordinate. It says how far the point is to the right (positive) or left (negative) of the origin, measured along the x-axis. The second number, $y$ , is the y-coordinate: how far the point is up (positive) or down (negative) from the origin, measured along the y-axis. The word “ordered” is the heart of it: the order is fixed as across first, then up. Because that order is fixed, you always read and write the horizontal number before the vertical one.

That fixed order is exactly why the order matters. The pair $(3, 2)$ means go $3$ right and then $2$ up. The pair $(2, 3)$ means go $2$ right and then $3$ up. Those are two different places on the page, so $(3, 2)$ and $(2, 3)$ are different points even though they use the same two numbers. Swapping the coordinates moves the point, which is why $(x, y)$ is an ordered pair and not just a pair of two numbers in any order.

Here is a short way to remember the order. You walk into a room before you climb a ladder, so go across the hall, then up the stairs. The x-coordinate (across) always comes first.

Plotting a point from its coordinates

To plot a point means to mark its location given the pair $(x, y)$ . The recipe follows straight from the meaning of the two numbers:

Start at the origin, $(0, 0)$ .
Move along the x-axis by the first number: right if it is positive, left if it is negative.
From there move parallel to the y-axis by the second number: up if it is positive, down if it is negative.
Mark the point where you land.

To plot $(3, 2)$ , start at the origin, move $3$ units right, then $2$ units up, and mark the spot. To plot $(-4, 1)$ , move $4$ units left (because the x-coordinate is negative), then $1$ unit up. To plot $(0, -3)$ , do not move across at all (the x-coordinate is $0$ ), then move $3$ units down, which lands you right on the y-axis.

Four plotted points, one in each quadrant. The dashed path shows how A at (3, 2) was reached: 3 right, then 2 up. B is at (-4, 1), C is at (-2, -3), and D is at (1, -2), each reached the same way.

Check your understanding

To plot the point $(-5, 4)$ , which way do you move from the origin?

5

left, then

4

up B

5

right, then

4

up C

5

up, then

4

left D

4

left, then

5

Reading the coordinates of a plotted point

Reading runs the recipe in reverse. Given a point already marked on the grid, you find its coordinates by measuring back to the axes:

Drop straight down (or up) from the point to the x-axis and read the number you land on. That is the x-coordinate.
Go straight across (left or right) from the point to the y-axis and read the number there. That is the y-coordinate.
Write them in order as $(x, y)$ .

Three points to read. Dashed guide lines show how K and L were read: trace straight to the x-axis, then straight to the y-axis. Try reading M the same way before checking below.

Take point $K$ in the figure above. $K$ sits above the x-axis, so trace straight down to it and you land on $3$ : that is the x-coordinate. Slide across from $K$ to the y-axis and you land on $3$ , so its y-coordinate is $3$ too. Written as an ordered pair, $K$ is at $(3, 3)$ .

Point $L$ sits below the x-axis, so trace straight up to it instead: you land on $-3$ , the x-coordinate. Slide across from $L$ to the y-axis and you land on $-2$ , the y-coordinate. So point $L$ is at $(-3, -2)$ .

Check your understanding

What are the coordinates of point $M$ in the figure above?

(1, -4)

(-4, 1)

(1, 4)

(4, -1)

The four quadrants

The two axes cut the plane into four regions, called quadrants. They are numbered with Roman numerals I, II, III, IV, starting in the upper-right region and going counterclockwise:

Quadrant I: upper right.
Quadrant II: upper left.
Quadrant III: lower left.
Quadrant IV: lower right.

Each quadrant has its own pattern of signs. You do not have to memorize those patterns, because each one follows directly from which way you move along each axis.

The four quadrants and their sign patterns. The point (2, 0) sits on the x-axis, so it belongs to no quadrant.

A point’s x-coordinate is positive to the right of the y-axis and negative to the left. Its y-coordinate is positive above the x-axis and negative below. Combine those two facts and every quadrant’s sign pattern follows for free, matching the diagram above: Quadrant I is $(+, +)$ .

Moving counterclockwise to the next quadrant crosses exactly one axis, so exactly one sign flips. Crossing the y-axis flips the x-sign; crossing the x-axis flips the y-sign. That single rule produces all four patterns in order: $(+,+)$ , $(-,+)$ , $(-,-)$ , $(+,-)$ . It is one idea applied four times, not four separate rules to memorize.

A point that lands on an axis belongs to no quadrant, like $(2, 0)$ in the diagram above. If a point sits on the x-axis its y-coordinate is $0$ , and if it sits on the y-axis its x-coordinate is $0$ . The axes are the boundaries of the quadrants, not part of any of them, and the origin sits on both.

Worked example 1 Plot a point and name its quadrant

Plot the point $(-3, 5)$ and state which quadrant it lies in.

Start at the origin. The x-coordinate is $-3$ , so move $3$ units to the left (negative means left). The y-coordinate is $5$ , so from there move $5$ units up (positive means up). Mark the point where you land.

Now read off the signs to find the quadrant. The x-coordinate is negative and the y-coordinate is positive, which is the upper-left region:

(-, +) \;\Rightarrow\; \text{Quadrant II}.

So $(-3, 5)$ sits in Quadrant II, $3$ units left of the y-axis and $5$ units above the x-axis.

Check your understanding

In which quadrant does the point $(-6, -1)$ lie?

A Quadrant III B Quadrant II C Quadrant IV D Quadrant I

Try it yourself. Drag the point in the figure below, or step it with the buttons, and watch the two moves from the origin and the quadrant change together. Find every position the figure says belongs to no quadrant, and look at what those coordinates have in common.

Where do the two moves land you?

Across Up

The point is at (3, -2). From the origin: 3 right, then 2 down. Right of the vertical axis and below the horizontal one: quadrant IV.

A coordinate plane with one point on it and dashed guides from the point to each axis. The point can be dragged anywhere on the grid or stepped one unit at a time, and the sentence below names its ordered pair, the two moves from the origin that reach it, and the quadrant those moves land in.

Horizontal and vertical distance on the grid

When two points share a coordinate, the distance between them is easy to read straight off the grid. That is because the segment joining them runs along a gridline.

If two points have the same y-coordinate, they sit at the same height, so the segment between them is horizontal. The distance is just how far apart their x-coordinates are. If two points have the same x-coordinate, they sit in the same column, so the segment between them is vertical. The distance is then how far apart their y-coordinates are. In both cases you find the gap by subtracting the smaller value from the larger one. That subtraction is the same as taking the absolute value of their difference, so the distance never comes out negative.

For example, the points $(2, 5)$ and $(7, 5)$ share the height $y = 5$ , so they are a horizontal distance

|7 - 2| = 5

apart. The points $(-3, 1)$ and $(-3, 6)$ share the column $x = -3$ , so they are a vertical distance

|6 - 1| = 5

apart. You can also simply count the unit spaces between the two points and get the same number.

Why horizontal distance is the difference of the x-coordinates#

Take two points on the same horizontal line, say $(x_1, y)$ and $(x_2, y)$ , with the same height $y$ , matching the $(2, 5)$ and $(7, 5)$ example above. The y-coordinate does not change between them, so moving from one to the other is pure left-right motion along that height, exactly like moving along a single number line.

On a number line the distance between two positions is how many units separate them, which you get by subtracting the smaller from the larger. So the distance here is the gap between $x_1$ and $x_2$ :

d = |x_2 - x_1|.

The absolute value bars just make sure the distance is never reported as negative. The reported length is the same no matter which point you list first. The identical argument with the roles of $x$ and $y$ swapped shows that two points sharing an x-coordinate are a vertical distance $|y_2 - y_1|$ apart. In this lesson, a single coordinate difference gives the full distance only for a horizontal or vertical segment; a slanted segment needs a different method, which is not covered here.

Worked example 2 Read coordinates and measure a horizontal distance

A point $P$ sits $4$ units to the left of the origin and $2$ units up. A point $Q$ sits $3$ units to the right of the origin and $2$ units up. Find the coordinates of each point and the distance between them.

Translate each description into an ordered pair, across first and then up. Point $P$ is $4$ left and $2$ up, so $P = (-4, 2)$ . Point $Q$ is $3$ right and $2$ up, so $Q = (3, 2)$ .

Both points have the same y-coordinate, $2$ , so they lie on the same horizontal line and the distance between them is horizontal. Subtract the x-coordinates and take the absolute value:

d = |3 - (-4)| = |3 + 4| = 7.

So $P$ and $Q$ are $7$ units apart. Counting unit spaces from $x = -4$ across to $x = 3$ gives the same $7$ units, which is a good way to check the subtraction.

Check your understanding

Points $(-5, -2)$ and $(-5, 4)$ share an x-coordinate. What is the vertical distance between them?

6

2

-6

1

Perimeter and area of a rectangle on the grid

Once you can measure horizontal and vertical gaps, you can handle a rectangle whose corners are given as coordinates. That works as long as its sides run along the gridlines, which makes it an axis-aligned rectangle. Find the length of one horizontal side and one vertical side, then use the rectangle formulas you already know from the perimeter and area lesson:

P = 2(l + w), \qquad A = l \times w.

The two side lengths are just a horizontal distance and a vertical distance, each found by subtracting coordinates. Because the rectangle is axis-aligned, every side is either horizontal or vertical, so no slanted length is ever needed.

Worked example 3 Perimeter and area of a rectangle from its corners

A rectangle has corners at $(1, 1)$ , $(6, 1)$ , $(6, 4)$ , and $(1, 4)$ . Find its perimeter and area.

The rectangle with corners (1, 1), (6, 1), (6, 4), (1, 4). The bottom side is |6 - 1| = 5 units long and the left side is |4 - 1| = 3 units tall.

First find the length of a horizontal side. The bottom side runs from $(1, 1)$ to $(6, 1)$ , and these two corners share the height $y = 1$ . So that side is horizontal, and its length is the gap in the x-coordinates:

l = |6 - 1| = 5.

Now find a vertical side. The left side runs from $(1, 1)$ to $(1, 4)$ , and these two corners share the column $x = 1$ . So that side is vertical, and its length is the gap in the y-coordinates:

w = |4 - 1| = 3.

The rectangle is $5$ units long and $3$ units wide. Apply the perimeter and area formulas:

P = 2(l + w) = 2(5 + 3) = 16,

A = l \times w = 5 \times 3 = 15.

So the rectangle has a perimeter of $16$ units and an area of $15$ square units.

Check your understanding

A rectangle has corners at $(2, 1)$ , $(2, 5)$ , $(7, 5)$ , and $(7, 1)$ . What is its area?

20

9

18

40

Check your understanding

A rectangle has corners at $(0, 2)$ , $(0, 5)$ , $(7, 5)$ , and $(7, 2)$ . What is its perimeter?

20

10

21

14

Once you can plot a point, one more move is worth knowing: reflecting it to the mirror-image spot on the other side of an axis. Crossing the x-axis flips the sign of $y$ ; crossing the y-axis flips the sign of $x$ .

Common mistakes

Swapping x and y. The order is across first, then up: $(x, y)$ . Plotting $(2, 3)$ as ” $3$ right, $2$ up” lands on the wrong point. Remember $(3, 2)$ and $(2, 3)$ are different places.
Mixing up the direction of negatives. A negative x-coordinate moves left, a negative y-coordinate moves down. Watch the sign of each coordinate separately.
Reading the wrong axis for a coordinate. To read a plotted point, move straight to the x-axis (down if the point is above it, up if the point is below it) for the first number, and straight to the y-axis for the second. Reading height off the x-axis is a common slip.
Putting axis points in a quadrant. A point on an axis has a $0$ coordinate and belongs to no quadrant. For example $(0, -4)$ is on the y-axis, not in Quadrant III or IV.
Subtracting coordinates for a diagonal distance. Subtracting coordinates gives the straight-line distance only when the two points share a coordinate (a horizontal or vertical segment). For points that are diagonal to each other, one coordinate’s difference is only part of the movement, not the full distance between them.
Forgetting the absolute value. A distance is never negative. Subtracting in the wrong order can give a negative number, for example $-4 - 3 = -7$ , but the distance is $|-4 - 3| = 7$ , not $-7$ .

Key takeaways

The coordinate plane is two perpendicular number lines: a horizontal x-axis and a vertical y-axis crossing at the origin $(0, 0)$ .
A point is named by an ordered pair $(x, y)$ : the x-coordinate is the horizontal position (right if positive, left if negative). The y-coordinate is the vertical position (up if positive, down if negative). Order matters, so $(3, 2) \ne (2, 3)$ .
To plot $(x, y)$ : from the origin go across by $x$ , then up or down by $y$ . To read a point, move straight to the x-axis (down from above it, up from below it) and straight to the y-axis.
The axes make four quadrants, numbered I to IV counterclockwise from the upper right, with sign patterns $(+,+)$ , $(-,+)$ , $(-,-)$ , $(+,-)$ . A point on an axis is in no quadrant.
For points that share a coordinate, the horizontal distance is $|x_2 - x_1|$ and the vertical distance is $|y_2 - y_1|$ .
From the four corners of an axis-aligned rectangle, read a horizontal side and a vertical side, then use $P = 2(l + w)$ and $A = l \times w$ .

The Coordinate Plane

Learning goals

Two number lines, one plane

Naming a point: the ordered pair (x, y)

Plotting a point from its coordinates

Reading the coordinates of a plotted point

The four quadrants

Horizontal and vertical distance on the grid

Perimeter and area of a rectangle on the grid

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Core practice