The Coordinate Plane

Learning goals

Name a point with an ordered pair, across then up
Plot $(x, y)$ from the origin, and read a plotted point back
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). The axes split the plane into four regions called quadrants, numbered I to IV counterclockwise starting from the upper right.

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. A at (3, 2) is 3 right and 2 up. B at (-4, 1) is 4 left and 1 up. C at (-2, -3) is 2 left and 3 down. D at (1, -2) is 1 right and 2 down.

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 $(x, y)$ .

Take point $A$ in the figure above. Drop from $A$ to the x-axis and you land on $3$ , so its x-coordinate is $3$ . Slide across from $A$ to the y-axis and you land on $2$ , so its y-coordinate is $2$ . Written as an ordered pair, $A$ is at $(3, 2)$ . Read point $C$ the same way: it lines up with $-2$ above it on the x-axis and $-3$ across on the y-axis. So point $C$ is at $(-2, -3)$ .

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

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.

Why each quadrant has its own sign pattern#

A point’s x-coordinate is positive when the point is to the right of the y-axis and negative when it is to the left. Its y-coordinate is positive when the point is above the x-axis and negative when it is below. The quadrant a point lands in is decided by those same two facts, namely which side of each axis the point falls on. So the signs and the quadrant carry exactly the same information.

In Quadrant I, the upper right, a point is to the right of the y-axis and above the x-axis. So $x$ is positive and $y$ is positive, and the pattern is $(+, +)$ . Now move across to Quadrant II, the upper left. The point is now to the left of the y-axis but still above the x-axis. So $x$ turns negative while $y$ stays positive: $(-, +)$ . In Quadrant III, the lower left, the point is left of the y-axis and below the x-axis, making both coordinates negative: $(-, -)$ . In Quadrant IV, the lower right, the point is right of the y-axis and below the x-axis. So here $x$ is positive and $y$ is negative: $(+, -)$ .

Read the four patterns in order: $(+,+)$ , $(-,+)$ , $(-,-)$ , $(+,-)$ . You can see the counterclockwise march around the origin. Only the x-sign flips as you cross the y-axis, and only the y-sign flips as you cross the x-axis. The signs are not a rule to memorize but a direct record of which side of each axis the point sits on.

A point that lands on an axis belongs to no quadrant. 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 quadrants are the open regions strictly between the axes. So the axes themselves are not counted as part of any quadrant, and neither is the origin, which sits on both.

Check your understanding

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

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

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 comes out positive.

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 $y$ . The y-coordinate does not change between them. So moving from one to the other is pure left-right motion along that height. That motion is 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 reported as a positive length. The reported length is the same no matter which point you list first, since a distance can never be negative. 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. This is the only kind of distance we measure on the grid: along a row or along a column, never along a slanted line.

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 gridlines between the two points and get the same number.

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 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.

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 gridlines from $x = -4$ across to $x = 3$ gives the same $7$ units, which is a good way to check the subtraction.

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.

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.

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, so the area is 5 times 3 = 15 square units.

A bit further: reflecting a point across an axis

Here is a pattern worth noticing once you are comfortable plotting. Reflecting a point across an axis means flipping it to the mirror-image position on the other side of that axis, the same distance away.

Reflect a point across the x-axis and it keeps the same x-coordinate, so it stays in the same column. But its y-coordinate changes sign, so the point flips from above the axis to below, or below to above. So $(3, 2)$ reflected across the x-axis becomes $(3, -2)$ . Reflect a point across the y-axis and the opposite happens: the y-coordinate stays the same while the x-coordinate changes sign. So the point $(3, 2)$ , reflected across the y-axis, becomes $(-3, 2)$ . Each reflection flips exactly one sign, the sign belonging to the axis you crossed.

This is the same sign behavior you saw in the quadrants, now viewed in motion. Crossing the x-axis flips the y-sign, and crossing the y-axis flips the x-sign.

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, drop to the x-axis for the first number and slide 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 a distance only when the two points share a coordinate (a horizontal or vertical segment). For points that are diagonal to each other, counting across or up does not give the straight-line distance.
Forgetting the absolute value. A distance is a positive length. Take the positive difference of the coordinates, for example $|3 - (-4)| = 7$ , 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, drop to the x-axis and slide 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 $A = l \times w$ .