The Coordinate Plane

Two numbers pin down any point, once you agree which one comes first.

Give directions from the town hall without naming a single street.

"Four blocks east, three blocks south." That is enough for anyone to find the spot.

four blocks east, three blocks south town hall (4, −3) (−3, 2) the minus is a direction, not a smaller distance

Write it as (4, -3). The minus does not mean a shorter walk — it means the other way.

Go the opposite way on both and you get (-3, 2): three blocks west, two north.

Two numbers and one agreement — across before up — locate any point on the plane. The four quadrants are just the four combinations of signs.

Distance between two points as a right triangle the distance formula is the Pythagorean theorem in disguise A(1, 1) B(5, 4) 4 across 3 up d^2 = 4^2 + 3^2 d^2 = 16 + 9 = 25 d = 5 subtract the x values for the run, the y values for the rise, then use a^2 + b^2 = c^2

Across before up

The first number always tells you how far across, the second how far up.

(4, -3) means four right and three down; (-3, 4) is a different point entirely.

Alphabetical order helps: x before y, across before up.

Order is everything

(2, 5) and (5, 2) are different places. Swapping them reflects the point across the diagonal, which is rarely what anybody meant.

The four quadrants

The axes cut the plane into four regions, numbered anticlockwise from the top right.

Quadrant I has both coordinates positive; II has x negative; III has both negative; IV has y negative.

Knowing the signs tells you the quadrant without plotting anything.

The four quadrants, numbered anticlockwise from the top right, with the sign pattern of the coordinates in each four quadrants, numbered anticlockwise from the top right xy I(+, +)II(−, +)III(−, −)IV(+, −) (−3, 4) (4, −3) the signs name the quadrant (−3, 4): x negative, y positive → up and left: quadrant II (4, −3): x positive, y negative → right and down: quadrant IV a point on an axis is inno quadrant at all

Signs name the region

A point with a negative x and a positive y is up and to the left — quadrant II, every time.

Points on the axes

A point on the x-axis has y = 0, because it is neither up nor down.

A point on the y-axis has x = 0. The origin has both zero.

Points on an axis belong to no quadrant — the quadrants are the regions between the axes.

Zero is a real coordinate

(0, -4) is a genuine point, four below the origin on the y-axis. It is not "nowhere" and it is not in quadrant IV.

Reflections

Flipping across the x-axis changes the sign of y and leaves x alone.

Flipping across the y-axis changes the sign of x. Only the coordinate that had to move changes.

So (3, 5) reflected in the x-axis is (3, -5).

The point (3, 5) reflected in the x-axis lands on (3, −5) and reflected in the y-axis lands on (−3, 5) reflecting changes the sign of one coordinate — the one that crosses the mirror xy 55 33 (3, 5) (3, −5) (−3, 5) in the x-axis (3, 5) → (3, −5) x stays put, y changes sign in the y-axis (3, 5) → (−3, 5) y stays put, x changes sign only the coordinate that hadto move changes

Which axis, which sign

Reflecting in the x-axis changes y. It feels backwards, but the x-axis is the mirror, so the across-distance stays put.

Distance between points

If two points share an x-coordinate, the distance between them is the gap in the y values.

Count the steps on the grid, or subtract — but subtract the smaller from the larger, since distance is never negative.

From (2, -3) to (2, 5) is 5 - (-3) = 8 units.

Crossing zero

Going from -3 to 5 covers three steps up to zero and five more beyond it. That is eight, not two, which subtracting carefully also gives.

Worked examples

Naming a quadrant

In which quadrant does (-6, -2) lie?

  1. The x is negative, so the point is left of the y-axis. First number, across.
  2. The y is negative, so it is below the x-axis. Second number, up or down.
  3. Left and down is quadrant III. Numbering runs anticlockwise from the top right.

Quadrant III

Reflecting

Reflect (3, 5) across the x-axis.

  1. The x-axis is the mirror, so the across-distance is unchanged. x stays 3.
  2. The point flips from above to below, so y changes sign. 5 becomes -5.
  3. The image is (3, -5). Same distance from the axis, other side.

(3, -5)

Distance

How far is it from (2, -3) to (2, 5)?

  1. Both have x = 2, so this is a vertical distance. Only the y values differ.
  2. Subtract the smaller from the larger: 5 - (-3). Two minuses make a plus.
  3. = 8 units. Count on the grid to check: three up to zero, then five more.

8

Words to know

Origin
The point (0,0) where the two axes cross.
x-coordinate
The first number — how far across.
y-coordinate
The second number — how far up or down.
Quadrant
One of the four regions the axes divide the plane into.
Reflection
A flip across an axis, which changes the sign of one coordinate.

Mistakes to watch for

Plotting the coordinates in the wrong order. (2,5) is two across and five up; (5,2) is a different point. The first number is always the across one.

Reading a negative coordinate as a shorter distance. (-3, 2) is three blocks west, not a smaller number of blocks east.

Changing the wrong sign when reflecting. Reflecting in the x-axis changes y, because the x-axis is the mirror and across-distances do not move.

Putting points on an axis into a quadrant. (0,-4) lies on the y-axis and belongs to no quadrant at all.

Subtracting the wrong way for distance. From -3 to 5 is 8 units; writing -3 - 5 = -8 gives a negative, and a distance is never negative.

Descartes is said to have thought of this while watching a fly on his ceiling, and wondering how to describe exactly where it was.

Every digital map, every game world and every spreadsheet cell reference uses the same idea.

Latitude and longitude are coordinates on a sphere, and they follow the same convention of naming a direction with a sign.

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