The Coordinate Plane
The coordinate plane is a grid formed by two number lines crossing at right angles. Every point has an address called an ordered pair.
The Axes and Origin
The x-axis is horizontal; the y-axis is vertical. They cross at the origin (0, 0). The plane is divided into 4 quadrants (I–IV, labelled counter-clockwise from top-right).
Plotting Points
An ordered pair (x, y) gives the location: move x units horizontally, then y units vertically from the origin.
Plot (3, −2): go 3 right, 2 down.
Plot (−4, 1): go 4 left, 1 up.
Reading Graphs
To read a point, trace horizontally to the x-axis to get x, then vertically to the y-axis to get y. Always write (x, y) — x first, y second.
Distance Between Points
For two points on the same horizontal or vertical line, distance = |difference of the changing coordinate|.
Distance from (2, 3) to (7, 3) = |7−2| = 5.
The Four Quadrants at a Glance
Knowing the sign pattern lets you check any answer instantly, before you plot anything.
| Quadrant | Location | Signs (x, y) | Example |
|---|---|---|---|
| I | Top right | (+, +) | (3, 5) |
| II | Top left | (−, +) | (−3, 5) |
| III | Bottom left | (−, −) | (−3, −5) |
| IV | Bottom right | (+, −) | (3, −5) |
Points sitting exactly on an axis belong to no quadrant. (0, 4) is on the y-axis and (−6, 0) is on the x-axis.
The Distance Formula
When two points do not share a row or column, the distance between them is the hypotenuse of a right triangle — which makes this the Pythagorean theorem in disguise.
Distance from (1, 2) to (4, 6)
- Horizontal gap:
4 − 1 = 3 - Vertical gap:
6 − 2 = 4 - Apply
d = √(3² + 4²) = √25 = 5
The order of subtraction does not matter, because squaring removes any negative sign.
The Midpoint Formula
The midpoint is simply the average of the x-values paired with the average of the y-values.
Midpoint of (2, 3) and (8, 7)
x = (2 + 8) / 2 = 5 and y = (3 + 7) / 2 = 5
Midpoint: (5, 5)
Common Mistakes
- Reversing the coordinates.
(3, 5)and(5, 3)are different points. X always comes first, alphabetically and on the page. - Moving vertically before horizontally. Go across first, then up or down.
- Numbering quadrants clockwise. They run counter-clockwise, starting from the top right.
- Forgetting the square root at the end of the distance formula.
Where You Will Use This
- Map apps and GPS, which locate you with a pair of coordinates.
- Every graph in science class, from motion to population growth.
- Video games and animation, where objects are positioned on a coordinate grid.
- Spreadsheets and design software, which address cells and pixels the same way.
Summary
- Two perpendicular number lines meet at the origin,
(0, 0). - An ordered pair is always
(x, y)— across, then up or down. - The four quadrants are numbered counter-clockwise from the top right.
- Distance comes from the Pythagorean theorem; midpoint comes from averaging.
FAQ
Which quadrant is (−3, −5) in? Quadrant III — both negative.
Is (x, y) or (y, x) correct? Always (x, y) — x first.
What is the origin? The point (0, 0), where the two axes cross. Every other point is described by how far it sits from there.
How do I plot a point with a zero in it? A zero means no movement in that direction. (0, 5) stays on the y-axis, five units up.
Why is it called the Cartesian plane? After René Descartes, the French mathematician who connected algebra to geometry by giving every point an address.
Does the distance formula work in three dimensions? Yes, with one more term: d = √((Δx)² + (Δy)² + (Δz)²).
Quick Quiz
Test what you just learned. Choose the best answer for each question.