The Coordinate Plane
Map the world of numbers in two dimensions
If the coordinate plane feels intimidating, take a deep breath. You have been using this concept your entire life without even realizing it. Every time you find a seat in a movie theater (“row G, seat 14”), locate a spot on a map, or tell someone to meet you at “the corner of 5th and Main,” you are using coordinate thinking. The coordinate plane is simply a way to describe exact locations using two numbers. That is it. No magic, no tricks - just a very clever system for pinpointing where things are.
Core Concepts
What is the Coordinate Plane?
The coordinate plane (also called the Cartesian plane, named after mathematician René Descartes) is a flat surface with two number lines that cross each other at right angles. Think of it as a perfectly organized map where every single point has its own unique address.
Imagine drawing a horizontal number line across a piece of paper. Now draw a vertical number line that crosses it right at zero. Congratulations - you have just created a coordinate plane.
The Axes
The two number lines that form the coordinate plane have specific names:
-
The x-axis is the horizontal line (going left and right, like the horizon). Numbers increase as you move right and decrease as you move left.
-
The y-axis is the vertical line (going up and down). Numbers increase as you move up and decrease as you move down.
An easy way to remember: “x” sounds like “across” if you say it fast, and the y-axis is the one that asks “why so high?” (pointing up).
The Origin
The point where the x-axis and y-axis cross is called the origin. This is the starting point of our coordinate system, and its address is $(0, 0)$. Everything else on the plane is measured from here.
Think of the origin like the “You Are Here” dot on a mall map. Every direction you could go - left, right, up, down - starts from that spot.
The Four Quadrants
The two axes divide the plane into four regions called quadrants. We number them using Roman numerals, starting from the upper right and going counterclockwise:
- Quadrant I (upper right): Both x and y are positive $(+, +)$
- Quadrant II (upper left): x is negative, y is positive $(-, +)$
- Quadrant III (lower left): Both x and y are negative $(-, -)$
- Quadrant IV (lower right): x is positive, y is negative $(+, -)$
$$\begin{array}{c|c} \text{Quadrant II} & \text{Quadrant I} \ (-, +) & (+, +) \ \hline \text{Quadrant III} & \text{Quadrant IV} \ (-, -) & (+, -) \end{array}$$
Points that sit exactly on an axis are not considered to be in any quadrant.
Ordered Pairs
Every point on the coordinate plane can be described by two numbers called an ordered pair, written as $(x, y)$.
- The first number (x-coordinate) tells you how far to go left or right from the origin
- The second number (y-coordinate) tells you how far to go up or down
The word “ordered” is crucial here. $(3, 5)$ and $(5, 3)$ are completely different points! The order matters because the first number always represents horizontal position and the second number always represents vertical position.
Think of it like giving directions: “Go 3 blocks east, then 5 blocks north” is different from “Go 5 blocks east, then 3 blocks north.”
Notation and Terminology
| Term | Meaning | Example |
|---|---|---|
| Coordinate plane | A flat surface with perpendicular x and y axes | The grid where we plot points |
| x-axis | The horizontal number line | The line going left-right through the origin |
| y-axis | The vertical number line | The line going up-down through the origin |
| Origin | The point where the axes intersect | $(0, 0)$ |
| Ordered pair | Two numbers $(x, y)$ that locate a point | $(3, -2)$ |
| x-coordinate | The first number; horizontal position | In $(3, -2)$, the x-coordinate is 3 |
| y-coordinate | The second number; vertical position | In $(3, -2)$, the y-coordinate is $-2$ |
| Quadrant | One of four regions created by the axes | Quadrant I, II, III, or IV |
| Plot | To mark a point on the coordinate plane | Plot the point $(4, 1)$ |
Examples
Plot the point $(3, 4)$ on the coordinate plane.
Solution:
Start at the origin $(0, 0)$.
Step 1: Look at the x-coordinate, which is 3. Move 3 units to the right along the x-axis.
Step 2: Look at the y-coordinate, which is 4. From where you are, move 4 units up.
Step 3: Mark the point. You have plotted $(3, 4)$.
This point is in Quadrant I because both coordinates are positive.
Think of it this way: You are in an elevator building. The x-coordinate tells you which floor to get off on horizontally (like walking down a hallway), and the y-coordinate tells you how high up to go in the elevator. Always do horizontal first, then vertical.
A point is located 2 units to the left of the origin and 5 units up. What are its coordinates?
Solution:
Moving left from the origin means the x-coordinate is negative.
Moving up means the y-coordinate is positive.
So the coordinates are $(-2, 5)$.
This point is in Quadrant II (negative x, positive y).
Without plotting, determine which quadrant each point is in (or if it is on an axis):
a) $(-4, -7)$ b) $(5, -3)$ c) $(0, 6)$
Solution:
a) $(-4, -7)$: The x-coordinate is negative (left of origin), and the y-coordinate is negative (below origin). Negative x and negative y means Quadrant III.
b) $(5, -3)$: The x-coordinate is positive (right of origin), and the y-coordinate is negative (below origin). Positive x and negative y means Quadrant IV.
c) $(0, 6)$: The x-coordinate is 0, which means this point is exactly on the y-axis. Points on an axis are not in any quadrant. This point sits on the positive y-axis, 6 units above the origin.
Find the distance between the points $(2, 5)$ and $(7, 5)$.
Solution:
Notice that both points have the same y-coordinate (5). This means they are on the same horizontal line - one is directly to the left or right of the other.
When two points share the same y-coordinate, the distance between them is simply the difference in their x-coordinates:
$$\text{Distance} = |7 - 2| = |5| = 5 \text{ units}$$
We use absolute value because distance is always positive.
Similarly: If two points have the same x-coordinate (they are directly above/below each other), the distance is the absolute difference of their y-coordinates.
For example, the distance between $(3, -2)$ and $(3, 6)$ is: $$|6 - (-2)| = |6 + 2| = |8| = 8 \text{ units}$$
Find the distance between the points $(1, 2)$ and $(4, 6)$.
Solution:
These points do not share an x-coordinate or a y-coordinate, so they are diagonal from each other. We cannot simply subtract one coordinate.
Instead, we use the distance formula, which comes from the Pythagorean theorem. Imagine drawing a right triangle where the diagonal between your two points is the hypotenuse.
The distance formula is: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
Let $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 6)$.
Step 1: Find the horizontal distance: $x_2 - x_1 = 4 - 1 = 3$
Step 2: Find the vertical distance: $y_2 - y_1 = 6 - 2 = 4$
Step 3: Apply the formula: $$d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
The distance between $(1, 2)$ and $(4, 6)$ is 5 units.
Why this works: We created a right triangle with legs of 3 and 4. The Pythagorean theorem tells us that $3^2 + 4^2 = 5^2$, so the hypotenuse (the distance we want) is 5. If you have studied the Pythagorean theorem, you might recognize 3-4-5 as a famous right triangle!
Key Properties and Rules
Coordinate Signs by Quadrant
| Quadrant | x-coordinate | y-coordinate |
|---|---|---|
| I | Positive (+) | Positive (+) |
| II | Negative (−) | Positive (+) |
| III | Negative (−) | Negative (−) |
| IV | Positive (+) | Negative (−) |
Points on the Axes
- If $x = 0$, the point is on the y-axis
- If $y = 0$, the point is on the x-axis
- If both $x = 0$ and $y = 0$, the point is the origin
Distance Between Points
Horizontal distance (same y-coordinate): $$d = |x_2 - x_1|$$
Vertical distance (same x-coordinate): $$d = |y_2 - y_1|$$
Diagonal distance (distance formula): $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
Plotting Points: The Process
- Start at the origin $(0, 0)$
- Move horizontally according to the x-coordinate (right if positive, left if negative)
- Move vertically according to the y-coordinate (up if positive, down if negative)
- Mark the point
Real-World Applications
Maps and GPS
Every GPS coordinate is an ordered pair showing latitude and longitude. When your phone says you are at $(40.7128°, -74.0060°)$, it is placing you on Earth’s coordinate system - that happens to be New York City!
Video Games
In virtually every video game, character positions are tracked using coordinates. When a game developer writes code to make a character jump, they are changing the y-coordinate. When the character runs left or right, the x-coordinate changes. The entire game world exists on a coordinate plane (or in 3D games, a coordinate space with x, y, and z).
Seating Charts
Finding your seat at a concert, stadium, or airplane uses coordinate thinking. “Row 12, Seat F” is really just another way of writing an ordered pair. The row tells you one direction, and the seat number tells you the other.
Data and Graphs
Every graph you see in the news - stock prices over time, COVID cases by date, poll numbers throughout an election - uses the coordinate plane. Time usually goes on the x-axis, and the measured quantity goes on the y-axis. Each data point is plotted as an ordered pair.
Architecture and Design
Architects and engineers use coordinate systems to specify exact locations for every element of a building. “Place this column at position $(15, 22)$” is much more precise than “put it over there somewhere.”
Screen Pixels
Your computer or phone screen is a coordinate plane. Every pixel has an $(x, y)$ position. When you tap a specific spot on your screen, your device registers the coordinates of your touch.
Self-Test Problems
Problem 1: Plot the following points and identify which quadrant (or axis) each is in: $A(4, 3)$, $B(-2, 5)$, $C(-3, -1)$, $D(6, 0)$.
Show Answer
- $A(4, 3)$: Start at origin, go 4 right, then 3 up. Quadrant I (both positive)
- $B(-2, 5)$: Start at origin, go 2 left, then 5 up. Quadrant II (negative x, positive y)
- $C(-3, -1)$: Start at origin, go 3 left, then 1 down. Quadrant III (both negative)
- $D(6, 0)$: Start at origin, go 6 right, no vertical movement. On the positive x-axis (not in any quadrant because y = 0)
Problem 2: A point is located 5 units to the right of the origin and 3 units below the origin. Write its coordinates and identify the quadrant.
Show Answer
5 units to the right means x = 5 (positive). 3 units below means y = -3 (negative).
The coordinates are $(5, -3)$.
Positive x and negative y means the point is in Quadrant IV.
Problem 3: Without graphing, determine whether the point $(-7, 4)$ is closer to the x-axis or the y-axis.
Show Answer
The distance from a point to the x-axis equals the absolute value of its y-coordinate: $|4| = 4$ units.
The distance from a point to the y-axis equals the absolute value of its x-coordinate: $|-7| = 7$ units.
Since $4 < 7$, the point $(-7, 4)$ is closer to the x-axis.
Problem 4: Find the distance between the points $(-3, 2)$ and $(5, 2)$.
Show Answer
Both points have the same y-coordinate (2), so they lie on the same horizontal line.
The distance is the absolute difference of the x-coordinates: $$|5 - (-3)| = |5 + 3| = |8| = 8 \text{ units}$$
Problem 5: Use the distance formula to find the distance between $(−1, 3)$ and $(3, 6)$.
Show Answer
Using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$:
Let $(x_1, y_1) = (-1, 3)$ and $(x_2, y_2) = (3, 6)$.
$$d = \sqrt{(3 - (-1))^2 + (6 - 3)^2}$$ $$d = \sqrt{(3 + 1)^2 + (3)^2}$$ $$d = \sqrt{4^2 + 3^2}$$ $$d = \sqrt{16 + 9}$$ $$d = \sqrt{25} = 5 \text{ units}$$
Summary
- The coordinate plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis.
- The origin $(0, 0)$ is where the axes intersect - the starting point for locating all other points.
- The axes divide the plane into four quadrants: I (upper right, both positive), II (upper left, x negative), III (lower left, both negative), and IV (lower right, y negative).
- An ordered pair $(x, y)$ gives the exact location of a point - always x first (horizontal), then y (vertical).
- To plot a point, start at the origin, move horizontally according to x, then vertically according to y.
- Distance between points on the same horizontal or vertical line is found by subtracting coordinates.
- The distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ finds the distance between any two points, even diagonal ones.
- Coordinate systems appear everywhere: maps, video games, data graphs, architecture, and your phone screen.