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The coordinate plane

The point (3, −2) is 3 steps right of the origin and 2 steps down. Swap the numbers and you land somewhere else entirely.

Two number lines at right angles

A number line places every number at a spot along one direction. Put a second number line at right angles to the first, crossing at zero, and you can place a point anywhere on a flat surface. The horizontal line is the x-axis and the vertical line is the y-axis. Where they cross is the origin. This arrangement is called the rectangular coordinate system, and the flat surface is the coordinate plane. Map grids, spreadsheet cells and the pixels on a screen all use the same idea of finding a place with two numbers.

Every point is named by an ordered pair (x, y). The first number, the x-coordinate, tells you how far to move left or right from the origin. The second number, the y-coordinate, tells you how far to move up or down. To plot (3, −2), start at the origin, move 3 units right, then 2 units down, and mark the point. The word ordered matters. The point (−2, 3) is 2 units left and 3 units up, in a different part of the plane. The origin itself is (0, 0).

The four quadrants, numbered counterclockwise from the upper right. (3, −2) is in quadrant IV and (−2, 3) is in quadrant II.

Four quadrants

The two axes cut the plane into four regions called quadrants. OpenStax numbers them with Roman numerals, starting at the upper right and going counterclockwise. In quadrant I both coordinates are positive. In quadrant II x is negative and y is positive. In quadrant III both are negative. In quadrant IV x is positive and y is negative. A point with a zero coordinate sits on an axis rather than in a quadrant: (4, 0) is on the x-axis and (0, −5) is on the y-axis. The signs alone tell you the quadrant, before you draw anything.

BackgroundThe plane is often called Cartesian, after René Descartes. In 1637 he published, at Leiden, a book on method with three appendices, and the last of them, La Géométrie, applied algebra to geometry. That idea, that a curve can be described by an equation between two quantities, is what this whole course rests on. Descartes himself measured lengths from a single reference line, often at a slant, and his book has no grid of two perpendicular axes. The pair of axes was added later by his commentators, starting with Frans van Schooten, who printed the first Latin edition in 1649. The grid in today's textbooks is their tidier version, but the name has stuck.

The graph lab below asks you to plot points. Read the ordered pair, move along the x-axis first, then up or down, and tap the grid. If you place the point with its coordinates swapped, the lab tells you so. The score counts only first tries. Notice how quickly you start reading the signs as directions: negative x means left, negative y means down.

Reading a point works the same way in reverse. Drop a line from the point straight down or up to the x-axis to read x, then across to the y-axis to read y, and write them in that order. On a printed graph, check the scale first: one square is not always one unit.
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Drag to turn. Scroll or pinch to zoom.

Plot the ordered pair shown in the panel, then press Next point. Ten points cover all four quadrants and both axes.
  1. Where is the point (3, −2)?
    • Quadrant I
    • Quadrant II
    • Quadrant III
    • Quadrant IV
  2. Where is the point (−5, 4)?
    • Quadrant I
    • Quadrant II
    • Quadrant III
    • Quadrant IV
  3. Where is the point (−1, −7)?
    • Quadrant I
    • Quadrant II
    • Quadrant III
    • Quadrant IV
  4. Where is the point (6, 2)?
    • Quadrant I
    • Quadrant II
    • Quadrant III
    • Quadrant IV
  5. Where is the point (0, −5)?
    • Quadrant I
    • Quadrant II
    • On the x-axis
    • On the y-axis
  6. Where is the point (4, 0)?
    • Quadrant I
    • Quadrant II
    • On the x-axis
    • On the y-axis
  7. From the origin, which point is 2 left and 5 up?
    • (−2, 5)
    • (5, −2)
    • (2, −5)
    • (−5, 2)
  8. What is the y-coordinate of the point (−8, 3)?
  9. In which quadrant are both coordinates negative?
    • Quadrant III
    • Quadrant I
    • Quadrant II
    • Quadrant IV
  10. Which point is the origin?
    • (0, 0)
    • (1, 1)
    • (0, 1)
    • (1, 0)
Key points
  • An ordered pair (x, y): x first for left and right, y second for up and down.
  • The axes cross at the origin (0, 0).
  • Quadrants I to IV go counterclockwise from the upper right.
  • A point with a zero coordinate lies on an axis.