Math Core

In grade 5 you plotted points using only positive numbers, so everything lived in one corner of the grid. Now that you have negative numbers, you can extend both axes past zero. The result is a full coordinate plane that stretches in every direction, like a map with a center point.

Extending the axes

The coordinate plane is made from two number lines that cross at $0$ :

The x-axis is horizontal. Positive numbers go right, negative numbers go left.
The y-axis is vertical. Positive numbers go up, negative numbers go down.

They cross at the origin, $(0, 0)$ .

A point is named by an ordered pair $(x, y)$ . The first number, the x-coordinate, tells how far to move left or right from the origin. The second number, the y-coordinate, tells how far to move up or down.

Worked example: Plotting points

Plot $A(3, 2)$ , $B(-4, 1)$ , $C(-2, -3)$ and $D(1, -4)$ .

$A(3, 2)$ : from the origin, go right 3, then up 2.
$B(-4, 1)$ : go left 4, then up 1.
$C(-2, -3)$ : go left 2, then down 3.
$D(1, -4)$ : go right 1, then down 4.

A(3, 2), B(-4, 1), C(-2, -3) and D(1, -4).Open in grapher →

Common mistake

Always move along the x-axis first. $(-4, 1)$ and $(1, -4)$ are different points. Swapping the numbers puts the point in a completely different place.

The four quadrants

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

The quadrants are numbered counterclockwise, starting at the upper right.Open in grapher →

Signs in each quadrant

quadrant	x-coordinate	y-coordinate	example
I	positive	positive	$(3, 2)$
II	negative	positive	$(-4, 1)$
III	negative	negative	$(-2, -3)$
IV	positive	negative	$(1, -4)$

A point on an axis is not in any quadrant. For example, $(0, 5)$ is on the y-axis and $(-3, 0)$ is on the x-axis.

Reflections

Remember that opposites are mirror images across $0$ . On the coordinate plane, changing a coordinate to its opposite flips the point across an axis.

Change the sign of the y-coordinate: the point reflects across the x-axis. $(4, 3)$ becomes $(4, -3)$ .
Change the sign of the x-coordinate: the point reflects across the y-axis. $(4, 3)$ becomes $(-4, 3)$ .
Change both signs: the point lands in the opposite quadrant. $(4, 3)$ becomes $(-4, -3)$ .

Q is the reflection of P across the x-axis. R is the reflection of P across the y-axis.Open in grapher →

Distance between points

When two points share an x-coordinate or a y-coordinate, they sit on the same vertical or horizontal line. You can find the distance between them using absolute value.

Worked example: Points on opposite sides of an axis

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

Both points have $y = 2$ , so they are on the same horizontal line. Compare the x-coordinates, $-3$ and $5$ .

$-3$ is $|-3| = 3$ units from the y-axis.
$5$ is $|5| = 5$ units from the y-axis.

They are on opposite sides of the y-axis, so add: $3 + 5 = 8$ units.

Worked example: Points on the same side of an axis

Find the distance between $(1, -6)$ and $(1, -2)$ .

Both points have $x = 1$ , so they are on the same vertical line. Compare the y-coordinates: $|-6| = 6$ and $|-2| = 2$ .

They are on the same side of the x-axis, so subtract: $6 - 2 = 4$ units.

Tip

Check a distance by counting grid squares from one point to the other. Distances are never negative.

Practice

Practice 1

In which quadrant is the point $(-5, -2)$ ?

Points E, F, G and H.Open in grapher →

Practice 2

Use the graph above. What are the coordinates of point $E$ ?

Enter a point like (2, -3)

Practice 3

Use the graph above. Which point is not in any quadrant?

Practice 4

Reflect the point $(6, -7)$ across the x-axis. What are the coordinates of the new point?

Enter a point like (2, -3)

Practice 5

Reflect the point $(-2, 5)$ across the y-axis. What are the coordinates of the new point?

Enter a point like (2, -3)

Practice 6

Find the distance between $(-4, 3)$ and $(7, 3)$ .

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

Practice 7

Find the distance between $(-2, -8)$ and $(-2, -3)$ .

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

Practice 8

On a map, the town square is at the origin. The library is at $(-1, 4)$ . The pool is 9 blocks straight south of the library. What are the coordinates of the pool? (North is the positive y-direction.)

Enter a point like (2, -3)