Math Core

A number line lets you give every point on a line its own address. The coordinate plane does the same thing for a flat surface, using two numbers instead of one. It's the map you'll use for every graph in algebra.

Two number lines

Take a horizontal number line and a vertical number line, and cross them at 0. The horizontal one is the $x$ -axis. The vertical one is the $y$ -axis. The point where they cross is the origin.

On the $x$ -axis, positive numbers are to the right of the origin and negative numbers are to the left.
On the $y$ -axis, positive numbers are above the origin and negative numbers are below.

Every point in the plane gets an address called an ordered pair, written $(x, y)$ .

Definition

Ordered pair

An ordered pair $(x, y)$ gives the location of a point. The $x$ -coordinate tells how far to move left or right from the origin. The $y$ -coordinate tells how far to move up or down. The origin is $(0, 0)$ .

Plotting points

To plot a point, always start at the origin. Move along the $x$ -axis first, then move up or down.

Right is positive x, up is positive y.Open in grapher →

$A(4, 3)$ : right 4, up 3.
$B(-2, 5)$ : left 2, up 5.
$C(-5, -1)$ : left 5, down 1.
$D(3, -4)$ : right 3, down 4.

A point with a 0 in it sits on an axis. $(0, 6)$ is on the $y$ -axis, because you don't move left or right at all. $(-3, 0)$ is on the $x$ -axis.

Common mistake

Order matters. $(2, 5)$ and $(5, 2)$ are different points. The $x$ -coordinate always comes first, the same way you walk to a building before you take the elevator up. A common slip is to go up first and then over.

The four quadrants

The axes split the plane into four regions called quadrants. They are numbered with Roman numerals, starting in the upper right and going counterclockwise.

The signs of x and y tell you the quadrant.Open in grapher →

Signs in each quadrant

Quadrant	$x$	$y$	Example
I	positive	positive	$(2, 7)$
II	negative	positive	$(-2, 7)$
III	negative	negative	$(-2, -7)$
IV	positive	negative	$(2, -7)$

Points on an axis are not in any quadrant.

Worked example: Naming quadrants

Which quadrant is each point in?

$(-8, 1)$
$(6, -6)$
$(0, -4)$

Solutions.

$x$ is negative and $y$ is positive: Quadrant II.
$x$ is positive and $y$ is negative: Quadrant IV.
$x$ is 0, so the point is on the $y$ -axis. It isn't in any quadrant.

Distance along a grid line

When two points share the same $y$ -coordinate, they lie on a horizontal line. The distance between them is the difference of their $x$ -coordinates. When they share the same $x$ -coordinate, use the $y$ -coordinates instead.

Worked example: Horizontal and vertical distance

Find the distance between $(-3, 2)$ and $(5, 2)$ .

Both points have $y = 2$ , so they are on a horizontal line. Subtract the $x$ -coordinates: $5 - (-3) = 8$ . The distance is 8 units.

You can also count: from $-3$ to 0 is 3 units, and from 0 to 5 is 5 more, for $3 + 5 = 8$ .

Worked example: A rectangle on the grid

A rectangle has corners at $(-1, 4)$ , $(5, 4)$ , $(5, -2)$ and $(-1, -2)$ . Find its perimeter and area.

(-1, 4)(5, 4)(5, -2)(-1, -2)(-1 + 6t/(2pi), 4)(5, 4 - 6t/(2pi))(5 - 6t/(2pi), -2)(-1, -2 + 6t/(2pi))Open in grapher →

The top side goes from $x = -1$ to $x = 5$ , so its length is $5 - (-1) = 6$ . The right side goes from $y = -2$ to $y = 4$ , so its length is $4 - (-2) = 6$ .

It's a square with side 6. The perimeter is $4 \times 6 = 24$ units and the area is $6 \times 6 = 36$ square units.

Reflecting a point

Flip a point over the $x$ -axis and it lands the same distance on the other side, so its $y$ -coordinate changes sign. Flip it over the $y$ -axis and its $x$ -coordinate changes sign.

$(4, -3)$ reflected over the $x$ -axis is $(4, 3)$ .
$(4, -3)$ reflected over the $y$ -axis is $(-4, -3)$ .

Tip

Before you plot, say the signs out loud: "negative, positive" means left, then up, which is Quadrant II. If your dot lands somewhere else, you mixed up an axis.

Practice

Practice 1

Start at the origin. Move 4 units left and 3 units up. What are the coordinates of the point where you end?

Enter a point like (2, -3)

Practice 2

Which quadrant contains the point $(5, -2)$ ?

Practice 3

What are the coordinates of point $P$ ?

(-3, -2)Open in grapher →

Enter a point like (2, -3)

Practice 4

Which point lies on the $y$ -axis?

Practice 5

How far apart are the points $(-4, 2)$ and $(5, 2)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Reflect the point $(3, -5)$ over the $x$ -axis. What are the new coordinates?

Enter a point like (2, -3)

Practice 7

A rectangle has corners at $(-2, 1)$ , $(4, 1)$ , $(4, -3)$ and $(-2, -3)$ . What is its area in square units?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Three corners of a rectangle are $(-1, 4)$ , $(3, 4)$ and $(3, -2)$ . What is the fourth corner?

Enter a point like (2, -3)