The Coordinate Plane: Plotting Points in Four Quadrants

Two number lines at right angles

The coordinate plane is a horizontal number line and a vertical one crossing at zero.

• The horizontal line is the $x$-axis.
• The vertical line is the $y$-axis.
• Where they cross is the origin, at $(0, 0)$.

Because both axes are full number lines, they run into the negatives, and that is what gives the plane its four regions.

Reading an ordered pair

A point is named by an ordered pair:

$$(x, \; y)$$ $$x = \text{how far across} \qquad y = \text{how far up or down}$$

The $x$ always comes first. $(3, 5)$ and $(5, 3)$ are different points, which is what the word “ordered” is recording.

Pair	Meaning
$(4, 3)$	$4$ right, $3$ up
$(-4, 3)$	$4$ left, $3$ up
$(-4, -3)$	$4$ left, $3$ down
$(4, -3)$	$4$ right, $3$ down

Start at the origin every time, move across first, then up or down.

The four quadrants

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

Quadrant	$x$	$y$	Position
I	$+$	$+$	right and up
II	$-$	$+$	left and up
III	$-$	$-$	left and down
IV	$+$	$-$	right and down

The signs alone identify the quadrant, so $(-7, 2)$ is in Quadrant II without anything being plotted.

A point on an axis is in no quadrant. $(0, 4)$ sits on the $y$-axis and $(-3, 0)$ on the $x$-axis, so neither belongs to a quadrant at all.

Reflections across an axis

Changing the sign of one coordinate flips the point across the other axis.

$$(3, 4) \;\xrightarrow{\text{flip } x}\; (-3, 4) \qquad (3, 4) \;\xrightarrow{\text{flip } y}\; (3, -4)$$

Changing the $x$ moves it across the $y$-axis, which is the mirror it crosses. Changing both signs reflects through the origin, landing at $(-3, -4)$ diagonally opposite.

Distance along a grid line

When two points share a coordinate, the distance between them is the difference in the other one — with the sign stripped off:

$$\text{distance} = |a - b|$$

From $(2, -3)$ to $(2, 5)$ the $x$ is the same, so the points lie on a vertical line:

$$|-3 - 5| = 8 \text{ units apart}$$

This is the same distance on a number line as before — the vertical grid line is a number line, and the shared $x$ says which one.

Why the plane needs all four quadrants

Plotting only the first quadrant covers everything positive, and plenty of data is not.

A temperature graph over a year has to go below zero. A profit graph has to show losses. A map with a fixed reference point needs somewhere to put everything west and south of it.

The deeper reason is that one origin can serve a whole picture. Choose any point as $(0, 0)$ — a starting position, a launch time, a break-even level. Every other point then gets coordinates relative to it, in whichever direction it lies.

Staying positive would mean putting the origin below and left of all the data. That works for a chart. It cannot work for anything that moves in every direction.

The plane is also the foundation of graphing equations. A line like $y = 2x - 5$ passes through three quadrants, so cutting the plane down to one would hide most of it.

Worked examples

Common mistakes

Practice problems