# Locating Points in Quadrants and on Axes

An *ordered pair* $\,(x,y)\,$
is a pair of numbers,
separated by a comma, and enclosed in parentheses.
The order that the numbers are listed
makes a difference:
$\,(5,3)\,$ is different from $\,(3,5)\,.$
Thus, the name *ordered pair* is appropriate.

The number that is listed first is called the
*first coordinate* or the
$x$-*value*.
The number that is listed second is called the
*second coordinate* or the
$y$-*value*.

For example, $\,(5,3)\,$ is an ordered pair: the first coordinate is $\,5\,$ and the second coordinate is $\,3\,.$ Alternatively, the $x$-value is $\,5\,$ and the $y$-value is $\,3\,.$

Partial translation: For two ordered pairs to be equal, the first coordinates must be equal, and the second coordinates must be equal.

The *coordinate plane* (also called the
$\,xy$-*plane*) is a device to
‘picture’ ordered pairs.
Each ordered pair corresponds to a
point in the coordinate plane,
and each point in the coordinate plane
corresponds to an ordered pair.
For this reason, ordered pairs are often called
*points*.

The process of showing where a point ‘lives’
in a coordinate plane is called
*plotting the point*.
To plot the point
$\,(1,-2)\,$:

- Start at the point $\,(0,0)\,$ (look at the diagram below).
- Move $\,1\,$ to the right.
- Move down $\,2\,.$

To plot the point $\,(-2,1)\,$:

- Start at the point $\,(0,0)\,.$
- Move $\,2\,$ to the left.
- Move up $\,1\,.$

Notice that the $x$-value tells you how to move left/right: if the $x$-value is positive, move right; if the $x$-value is negative, move left.

Notice that the $y$-value tells you how to move up/down: if the $y$-value is positive, move up; if the $y$-value is negative, move down.

The *quadrants* (see below)
divide the coordinate plane into four regions.

Quadrant I is the set of all points $\,(x,y)\,$ with $\,x\gt 0\,$ and $\,y\gt 0\,.$

Quadrant II is the set of all points $\,(x,y)\,$ with $\,x\lt 0\,$ and $\,y\gt 0\,.$

Quadrant III is the set of all points $\,(x,y)\,$ with $\,x\lt 0\,$ and $\,y\lt 0\,.$

Quadrant IV is the set of all points $\,(x,y)\,$ with $\,x\gt 0\,$ and $\,y\lt 0\,.$

Roman numerals (I, II, III, IV) are conventionally used to talk about the four quadrants. You start numbering the quadrants in the upper right, and then proceed counter-clockwise.

The $x$-axis is the set of all points
$\,(x,0)\,,$ for all real numbers $\,x\,.$
The $x$-axis is the *horizontal* axis
(think of the horizon).
The
$x$-axis separates the upper
two quadrants (I and II) from the bottom
two quadrants (III and IV).
Points on the $x$-axis do not belong to
*any* quadrant.

The $y$-axis is the set of all
points $\,(0,y)\,,$
for all real numbers $\,y\,.$
The $y$-axis is the
*vertical* axis.
The
$y$-axis
separates the right
two quadrants (I and IV) from the left two
quadrants (II and III).
Points on the $y$-axis do not
belong to *any* quadrant.

The *origin* is the point
$\,(0,0)\,.$
The origin is the *only* point that
lies on *both* the
$x$-axis
and the
$y$-axis.

Points with positive
$x$-values
lie to the *right* of the
$y$-axis.

Points with negative
$x$-values
lie to the
*left* of the
$y$-axis.

Points with positive
$y$-values
lie
*above* the
$x$-axis.

Points with negative
$y$-values
lie
*below* the
$x$-axis.