Jugami Pre-Algebra

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The Coordinate Plane

Two number lines, one grid, and an exact address for every point.

A city map, a video game screen and a seat in a theater (row G, seat 12) all use the same trick: two numbers pin down one spot. In math, those two numbers are an ordered pair such as (3, 2), and the grid they live on is the coordinate plane.

This page covers the axes and the origin, the four quadrants, plotting points, distances, reflections, shapes on the grid, and graphing a rule like y = 2x. Then try the practice problems and the Grid Quest game.

Jump to practicePlay Grid Quest
A coordinate plane from −5 to 5 on both axes. The axes cross at the origin and split the plane into four quadrants: I, upper right, with signs plus, plus; II, upper left, minus, plus; III, lower left, minus, minus; IV, lower right, plus, minus xy−55−55I(+, +)II(−, +)III(−, −)IV(+, −)0origin

Axes, origin and ordered pairs

Take a horizontal number line, the x-axis, and a vertical number line, the y-axis, and make them cross at 0. The crossing point is the origin, written (0, 0). Every point now has an address (x, y): the x-coordinate says how far to move left or right from the origin, and the y-coordinate says how far to move up or down.

The order matters, which is why it is called an ordered pair. (2, 5) and (5, 2) are different points. A handy way to remember: x comes before y in the alphabet, so walk across first, then up or down.

Example 1: Read the address

What are the coordinates of a point 6 units left of the origin and 1 unit up? What about a point 4 units right, on the x-axis?

  1. Left means a negative x, so x = −6. Up means a positive y, so y = 1. The point is (−6, 1).
  2. On the x-axis you don’t move up or down at all, so y = 0. The point is (4, 0).

The four quadrants

The axes cut the plane into four regions called quadrants. They are numbered with Roman numerals I, II, III and IV, starting at the upper right and going counterclockwise. The signs of x and y tell you the quadrant right away:

  • Quadrant I: x positive, y positive (+, +)
  • Quadrant II: x negative, y positive (−, +)
  • Quadrant III: x negative, y negative (−, −)
  • Quadrant IV: x positive, y negative (+, −)

A point with a 0 coordinate, such as (0, −3) or (5, 0), sits on an axis and is not in any quadrant.

Example 2: Name the quadrant

Where are (−2, 7), (3, −8) and (0, 4)?

  1. (−2, 7): x is negative and y is positive, so Quadrant II.
  2. (3, −8): x is positive and y is negative, so Quadrant IV.
  3. (0, 4): x is 0, so the point is on the y-axis, not in a quadrant.

Plotting points

To plot (x, y), start at the origin. Move along the x-axis: right if x is positive, left if it is negative. Then move parallel to the y-axis: up if y is positive, down if it is negative. Put a dot there and label it.

Four points plotted on a grid from −6 to 6 across and −5 to 5 up. A at (4, 3) is reached by walking 4 right from the origin and then 3 up. B is at (−3, 2), C at (−5, −4) and D at (2, −3) xy−6−4−2246−4−2244 right3 upA(4, 3)B(−3, 2)C(−5, −4)D(2, −3)
To plot A(4, 3), walk 4 units right from the origin, then 3 units up. The same walk works for every point; negative numbers just send you left or down.

Example 3: Plot and check

Plot C(−5, −4). Which quadrant is it in?

  1. Start at (0, 0). x = −5, so move 5 units left.
  2. y = −4, so move 4 units down, and put the dot there.
  3. Both coordinates are negative, so C is in Quadrant III, as the drawing shows.

Distance on the grid

When two points share a y-coordinate, they lie on the same horizontal line, and the distance between them is the difference of their x-coordinates. When they share an x-coordinate, they lie on a vertical line, and you use the y-coordinates. Distance is never negative, so use the absolute value, a number’s distance from 0: distance = |x₂ − x₁| or |y₂ − y₁|.

If the two points are on opposite sides of an axis, you can also add their distances from that axis. From (−3, 3) to (4, 3) is |−3| + |4| = 3 + 4 = 7 units. If they are on the same side, subtract instead: from (2, 3) to (6, 3) is 6 − 2 = 4 units.

A horizontal segment from (−3, 3) to (4, 3): 3 units left of the y-axis plus 4 units right, 7 units long. A vertical segment from (5, 1) to (5, −4), 5 units long xy−6−4−2246−4−224(−3, 3)(4, 3)34(5, 1)(5, −4)5|−3| + |4| = 7|1 − (−4)| = 5
Counting squares agrees with the absolute value: the horizontal segment is 3 + 4 = 7 units long, and the vertical one is |1 − (−4)| = 5 units long.

Example 4: Find a distance

How far is (−2, −6) from (−2, 3)?

  1. The x-coordinates match (both −2), so the points are on a vertical line. Use the y-coordinates.
  2. |3 − (−6)| = |3 + 6| = 9.
  3. Or: −6 is 6 below the x-axis and 3 is 3 above it, so 6 + 3 = 9 units.

Reflections across the axes

A reflection flips a point over a line as if the line were a mirror. The image is the same distance from the mirror, on the other side.

  • Across the x-axis: x stays the same and y changes sign. (x, y) → (x, −y).
  • Across the y-axis: y stays the same and x changes sign. (x, y) → (−x, y).

The mark ′ (read “prime”) is used to name the new point, so the image of P is P′.

Point P at (4, 3). Reflected across the x-axis it lands at P prime, (4, −3). Reflected across the y-axis it lands at Q, (−4, 3). Both axes are drawn as mirror lines xy−6−4−2246−4−224P(4, 3)P′(4, −3)Q(−4, 3)Across the x-axis: (x, y) → (x, −y)Across the y-axis: (x, y) → (−x, y)
P(4, 3) is 3 units above the x-axis, so its mirror image P′ is 3 units below: (4, −3). It is 4 units right of the y-axis, so its image Q is 4 units left: (−4, 3).

Example 5: Reflect a point

Reflect (−5, 2) across the x-axis, and then reflect the result across the y-axis.

  1. Across the x-axis, x stays and y changes sign: (−5, 2) → (−5, −2).
  2. Across the y-axis, y stays and x changes sign: (−5, −2) → (5, −2).
  3. Both signs flipped, and the point moved from Quadrant II to Quadrant IV.

Polygons on the grid: perimeter and area

Plot the corners (the vertices) of a shape and join them in order to get a polygon. When the sides run along grid lines, use the distance rule to find each side length. Then the perimeter is the sum of the side lengths, and for a rectangle the area is length × width.

A rectangle with corners (−4, 3), (3, 3), (3, −2) and (−4, −2). Its width is 7 and its height is 5, so its area is 35 square units and its perimeter is 24 units xy−6−4−2246−4−224(−4, 3)(3, 3)(3, −2)(−4, −2)width 7height 5Area = 7 × 5 = 35 square unitsPerimeter = 2 × (7 + 5) = 24 units
Width: |3 − (−4)| = 7. Height: |3 − (−2)| = 5. So the area is 35 square units, and the perimeter is 7 + 5 + 7 + 5 = 24 units.

Example 6: A garden plan

A garden is drawn with corners at (1, 1), (9, 1), (9, 5) and (1, 5). Each grid unit is 1 meter. Find its perimeter and area.

  1. Length: |9 − 1| = 8 m. Width: |5 − 1| = 4 m.
  2. Perimeter: 8 + 4 + 8 + 4 = 24 meters.
  3. Area: 8 × 4 = 32 square meters.

Graphing a table: y = 2x

A rule like y = 2x makes a table of pairs: x = 1 gives y = 2, x = 2 gives y = 4, and so on. Each row of the table is an ordered pair, so you can plot it. For y = 2x the points line up on a straight line that passes through the origin.

This is the same idea as an equivalent-ratio table on the Ratios and Rates page. In every pair, y : x = 2 : 1, so the rule describes a proportional relationship, and 2 is the unit rate: y goes up by 2 each time x goes up by 1.

A table for y = 2x with x from 0 to 5 and y from 0 to 10, and the same six points plotted on a grid with equal units on both axes. The points lie on a straight line through the origin, because y : x is always 2 : 1 xy24246810y = 2xy = 2xxy0012243648510y : x is always 2 : 1
Each row of the table becomes a point. Because y is always 2 times x, the points line up on a straight line through (0, 0), rising 2 units for every 1 unit across.

Example 7: From a story to a graph

Pencils cost $3 each. Make a table for 0 to 4 pencils and say what the graph looks like.

  1. The rule is y = 3x, where x is the number of pencils and y is the cost in dollars.
  2. Pairs: (0, 0), (1, 3), (2, 6), (3, 9), (4, 12).
  3. They lie on a straight line through the origin, rising 3 dollars for each pencil. The ratio y : x is always 3 : 1.

How do we know?

Why does each quadrant have its own signs? Right of the y-axis the x-axis shows positive numbers, and left of it negative ones. Above the x-axis the y-axis shows positive numbers, and below it negative ones. Each quadrant is one choice of left or right and one choice of up or down, so it has one pair of signs.

Why does reflecting across the x-axis change only y? The mirror is horizontal, so the point moves straight up or down, and x does not change. The image must be the same distance from the mirror, so |y| stays the same, but the point is now on the other side, so the sign of y changes.

Why do the points of y = 2x make a straight line? Every step of 1 to the right goes up exactly 2. Steps that are always the same size and direction can’t bend, so the points line up. Starting at (0, 0), the line goes through the origin.

History: Descartes and Fermat

The grid is called the Cartesian plane after the French thinker René Descartes, who published the idea of describing geometry with algebra in 1637, in La Géométrie. Another Frenchman, the lawyer and mathematician Pierre de Fermat, worked out the same idea on his own at about the same time. He described it in a short work, Introduction to Plane and Solid Loci, which he sent to mathematicians in Paris, but it was not published in his lifetime. It was printed in 1679, after his death.

Their pictures did not look quite like ours. Wikipedia notes that both Descartes and Fermat measured lengths from a single axis. The habit of drawing a pair of axes came later, after Frans van Schooten and his students translated La Géométrie into Latin in 1649 and explained it. Long before either of them, in the 1300s, the French scholar Nicole Oresme drew graphs that plotted one changing quantity against another. MacTutor calls this a type of coordinate geometry, but it did not yet link algebra and geometry the way Descartes and Fermat did.

You may hear a story that Descartes got the idea while lying in bed watching a fly crawl on the ceiling. It is a popular legend, and there is no good evidence that it happened, but it is a nice way to remember the idea: two numbers are enough to say exactly where the fly is.

Practice problems

Try each one, then open it to compare.

1. Which quadrant is (−4, −9) in?

Both coordinates are negative, so Quadrant III.

2. Is (0, −7) in a quadrant?

No. Its x-coordinate is 0, so it is on the y-axis, 7 units below the origin.

3. Starting at the origin, you walk 2 units left and 5 units up. Where are you?

Left is negative x and up is positive y: (−2, 5), in Quadrant II.

4. How far apart are (−6, 4) and (3, 4)?

Same y, so use x: |3 − (−6)| = 9 units (or 6 + 3 = 9, since they are on opposite sides of the y-axis).

5. Reflect (7, −1) across the y-axis.

y stays and x changes sign: (−7, −1).

6. Reflect (−3, −8) across the x-axis.

x stays and y changes sign: (−3, 8).

7. A rectangle has corners (−2, −1), (4, −1), (4, 3) and (−2, 3). Find its perimeter and area.

Width |4 − (−2)| = 6, height |3 − (−1)| = 4. Perimeter = 6 + 4 + 6 + 4 = 20 units; area = 6 × 4 = 24 square units.

8. The graph of y = 4x passes through (5, ?) and (?, 36). Fill in the blanks.

y = 4 × 5 = 20, so (5, 20). And 36 ÷ 4 = 9, so (9, 36).

Game: Grid Quest

Ten problems: name the quadrant, reflect a point across an axis, find a distance, find the perimeter or area of a rectangle, and fill in a point on a graph like y = 2x. Type a quadrant as II (or 2), a point as (3, -2), or a whole number such as 7. Press Enter. One point per correct answer. Your best score is saved only on this device. About 4 minutes.

Talk About It

  • A friend plots (3, 5) by going 3 up and 5 right. Where did their point land, and how would you help them remember the order?
  • Why isn’t a point like (0, 6) in any quadrant?
  • If you reflect a point across the x-axis and then across the y-axis, where does it end up compared with where it started?
  • A table has the pairs (1, 3), (2, 5), (3, 7). Do the points line up? Is y : x always the same? What is different from y = 2x?

Sources

Credits: All diagrams on this page are original SVG drawings made for Jugami.