The Coordinate Plane

If you've ever used a map to find a location, you've used something similar to the coordinate plane. Just like a map uses coordinates to pinpoint a spot, the coordinate plane uses two numbers to identify any point in two-dimensional space.

The coordinate plane (also called the Cartesian plane, named after mathematician René Descartes) is formed by two perpendicular number lines. The horizontal line is the \(x\)-axis, and the vertical line is the \(y\)-axis. Where they cross is called the origin, and it's labeled with the coordinates \((0, 0)\).

Ordered Pairs

Every point on the coordinate plane can be described by an ordered pair \((x, y)\). The first number is the \(x\)-coordinate, which tells you how far to move left or right from the origin. The second number is the \(y\)-coordinate, which tells you how far to move up or down.

The word "ordered" is important. The point \((3, 2)\) is not the same as the point \((2, 3)\). The order matters.

Example: Plot the point \((3, 2)\).

Start at the origin \((0, 0)\). Move 3 units to the right (because the \(x\)-coordinate is positive 3). Then move 2 units up (because the \(y\)-coordinate is positive 2). Put a dot there. That's the point \((3, 2)\).

Example: Plot the point \((-3, 5)\).

Start at the origin. Move 3 units to the left (because the \(x\)-coordinate is -3). Then move 5 units up (because the \(y\)-coordinate is positive 5). That's your point.

Example: Plot the point \((2, -4)\).

From the origin, move 2 units right, then 4 units down (because the \(y\)-coordinate is -4).

The Four Quadrants

The coordinate plane is divided into four sections called quadrants. They're numbered using Roman numerals, starting in the upper-right and going counterclockwise.

Quadrant I (top right): Both \(x\) and \(y\) are positive. Example: \((3, 5)\)

Quadrant II (top left): \(x\) is negative, \(y\) is positive. Example: \((-4, 2)\)

Quadrant III (bottom left): Both \(x\) and \(y\) are negative. Example: \((-6, -3)\)

Quadrant IV (bottom right): \(x\) is positive, \(y\) is negative. Example: \((5, -7)\)

Points that lie exactly on the \(x\)-axis or \(y\)-axis are not in any quadrant.

Special Points on the Axes

If a point has an \(x\)-coordinate of 0, it lies on the \(y\)-axis. For example, \((0, 5)\) is on the \(y\)-axis, 5 units above the origin.

If a point has a \(y\)-coordinate of 0, it lies on the \(x\)-axis. For example, \((-3, 0)\) is on the \(x\)-axis, 3 units to the left of the origin.

The origin itself, \((0, 0)\), is where the axes intersect.

Reading Coordinates from a Graph

If you're given a graph with a point plotted on it, you can find its coordinates by seeing how far it is from the origin in each direction.

Look at how far the point is from the \(y\)-axis to find the \(x\)-coordinate. If it's to the right of the \(y\)-axis, the \(x\)-coordinate is positive. If it's to the left, it's negative.

Look at how far the point is from the \(x\)-axis to find the \(y\)-coordinate. If it's above the \(x\)-axis, the \(y\)-coordinate is positive. If it's below, it's negative.

Example: A point is 3 units to the left of the origin and 2 units up. What are its coordinates?

Left means negative \(x\), and up means positive \(y\). So the point is \((-3, 2)\).

Why the Coordinate Plane Matters

The coordinate plane is the foundation for graphing equations, which you'll do constantly in algebra and beyond. When you graph a linear equation like \(y = 2x + 1\), you're plotting points on the coordinate plane and connecting them to show the relationship between \(x\) and \(y\).

The coordinate plane also shows up in real life. Maps use coordinate systems (latitude and longitude). Video games use coordinates to track where characters are on the screen. Architects use coordinates to design buildings. GPS uses a three-dimensional version to pinpoint locations on Earth.

Symmetry and Reflections

The coordinate plane has some interesting symmetry properties.

If you take a point \((x, y)\) and reflect it across the \(x\)-axis, you get the point \((x, -y)\). The \(x\)-coordinate stays the same, but the \(y\)-coordinate changes sign.

Example: The point \((2, 2)\) reflected across the \(x\)-axis becomes \((2, -2)\).

If you reflect a point across the \(y\)-axis, you get \((-x, y)\). The \(y\)-coordinate stays the same, but the \(x\)-coordinate changes sign.

Example: The point \((5, -2)\) reflected across the \(y\)-axis becomes \((-5, -2)\).

If you reflect a point through the origin (rotate it 180 degrees around the origin), both coordinates change sign: \((x, y)\) becomes \((-x, -y)\).

Example: The point \((2, 3)\) reflected through the origin becomes \((-2, -3)\).

Practice Problems

1. Plot the following points on a coordinate plane: \((2, 3)\), \((-4, 1)\), \((0, -3)\), \((3, -2)\)

2. In which quadrant is each point located?

   • \((-5, 6)\)
   • \((4, -7)\)
   • \((-2, -3)\)
   • \((8, 1)\)

3. What is the reflection of \((6, -4)\) across the \(x\)-axis?

4. What is the reflection of \((-3, 5)\) across the \(y\)-axis?

5. A point is 7 units to the right of the origin and 2 units down. What are its coordinates?

6. If a point is in Quadrant III, what can you say about its coordinates?

Answers:

1. \((2, 3)\) is in Quadrant I. \((-4, 1)\) is in Quadrant II. \((0, -3)\) is on the \(y\)-axis. \((3, -2)\) is in Quadrant IV.

2. \((-5, 6)\) is in Quadrant II. \((4, -7)\) is in Quadrant IV. \((-2, -3)\) is in Quadrant III. \((8, 1)\) is in Quadrant I.

3. \((6, 4)\) — the \(x\)-coordinate stays the same, the \(y\)-coordinate changes sign.

4. \((3, 5)\) — the \(y\)-coordinate stays the same, the \(x\)-coordinate changes sign.

5. \((7, -2)\) — right means positive \(x\), down means negative \(y\).

6. Both coordinates are negative.

Things to Watch For

Switching the order of coordinates is the most frequent error. Remember that \(x\) always comes first—think "\(x\) comes before \(y\) in the alphabet" if that helps you remember.

Mixing up left/right and up/down happens when you're working quickly. Just remember: the \(x\)-coordinate controls horizontal movement (left or right), while the \(y\)-coordinate controls vertical movement (up or down).

Don't forget that negative \(x\) means left and negative \(y\) means down. The signs tell you direction, not quality.

When reflecting points, be careful with your signs. Reflection across the \(x\)-axis flips the \(y\)-coordinate (and only the \(y\)-coordinate), not both coordinates.

Points sitting exactly on the \(x\)-axis or \(y\)-axis aren't in any quadrant. If either coordinate is 0, the point lives on an axis.