# x and y coordinates

## What are x and y coordinates?

X and y coordinates are, respectively, the horizontal and vertical addresses of a point in any two-dimensional (2D) space, such as a sheet of paper or a computer display screen. Together, these coordinates help identify the exact location of a point.

In the Cartesian coordinate system, the x and y coordinates are part of the x-axis and y-axis in a 2D space. For a point in space, the x and y coordinates are written as an ordered pair, (x, y). The first number represents the point's position on the x-axis, and the second number represents its position on the y-axis. The coordinates can also be written as (x,y), without a space after the comma.

The order of the x and y coordinates in the ordered pair is important. The x coordinate always comes first, followed by the y coordinate. That is why (3, 4) is not the same as (4, 3).

(3, 4) refers to a point three units to the right of zero and four units above zero.

(4, 3) refers to a point four units to the right of zero and three units above zero.

The two axes intersect perpendicularly at the origin or zero location. The x and y coordinates of this location are written as (0, 0) or (0,0).

## Important x and y coordinate terms

The x and y axes on which the x and y coordinates are plotted form a coordinate plane. The system was invented by French mathematician René Descartes and is known as the Cartesian coordinate system.

The coordinate plane is required to represent any point in a given 2D space. The plane, formed by the intersection of the two axes, is two-dimensional because the location of any point on this plane requires two data points:

- its distance on the x-axis
- its distance on the y-axis

These distances are represented by the x-coordinate and y-coordinate, respectively.

The x value of the point (x, y) is known as the *abscissa*. It represents the distance of the point from the origin or along the horizontal x-axis. The y value of the point (x, y) is known as the *ordinate*. It represents the vertical or perpendicular distance of the point from the origin or from the x-axis.

The point at which a line intercepts the x-axis is called the *x-**intercept*, and the point at which it intercepts the y-axis is called the *y-intercept*. The y coordinate of an x-intercept is 0, and the x coordinate of a y-intercept is 0. If the equation of a line is available (y = mx + b), plugging in x = 0 into the equation yields the y-intercept. Similarly, plugging in y = 0 gives the x-intercept.

The coordinate plane is divided into four quadrants:

- Quadrant 1 is in the top right.
- Quadrant 2 is in the top left.
- Quadrant 3 is in the bottom left.
- Quadrant 4 is in the bottom right.

## Representing x and y coordinates with examples

Any point in a 2D space is represented by x and y coordinates as an ordered pair, either of which can be zero, positive or negative.

If either value is zero, the point is represented as the following:

- (0, y): The x coordinate is zero, so the point lies on the y-axis.
- (0, 10): The point is on the y axis and 10 units
*above*. - (0, -10): The point is on the y axis and 10 units
*below*.

- (0, 10): The point is on the y axis and 10 units
- (x, 0): The y coordinate is zero, so the point lies on the x-axis.
- (10, 0): The point is on the x axis and 10 units to the
*right*of zero. - (-10, 0): The point is on the x axis and 10 units to the
*left*of zero.

- (10, 0): The point is on the x axis and 10 units to the

If both the x and y coordinates are zero (0, 0), the point is on the origin, which is where the x-axis and y-axis intersect.

If both x and y coordinates are non-zero, the point lies somewhere on the 2D coordinate plane in one of its four quadrants.

### Example 1

Consider point M in the coordinate plane here.

M lies one unit to the right of zero and two units above zero. So, its x coordinate is (1), and its y coordinate is (2). Together, its (x, y) coordinates are represented on the 2D coordinate plane as the following:

M = (1, 2)

Point M is in Quadrant 1.

### Example 2

Consider point N in the coordinate plane here.

N lies three units to the left of zero and four units below zero. So, its x coordinate is (-3), and its y coordinate is (-4). Together, its (x, y) coordinates are represented on the 2D coordinate plane as the following:

N = (-3, -4)

Point N is in Quadrant 3.

## Positive and negative values in the 4 quadrants

Depending on the location of the point in one of the four quadrants on the coordinate plane, the x and y coordinates will have positive or negative values. If the x coordinate is in the left part of the plane, it has a negative value, and if it is in the right, its value is positive.

Similarly, if the y coordinate is on the top part of the plane, its value is positive. If it is in the bottom plane, it has a negative value. The left, right, top and bottom parts of the plane are determined by the location of the point from the origin or zero value.

Quadrant |
Point location |
X value(positive/negative) |
Y value(positive/negative) |
(x, y) |

Quadrant 1 |
Top right |
Positive |
Positive |
(+, +) |

Quadrant 2 |
Top left |
Negative |
Positive |
(-, +) |

Quadrant 3 |
Bottom left |
Negative |
Negative |
(-, -) |

Quadrant 4 |
Bottom right |
Positive |
Negative |
(+, -) |

### Example 1

(2, 5): The point is in Quadrant 1, two units to the *right* of zero and five units *above* zero.

### Example 2

(-2, 5): The point is in Quadrant 2, two units to the *left* of zero and five units *above* zero.

### Example 3

(-2, -5): The point is in Quadrant 3, two units to the *left* of zero and five units *below* zero.

### Example 4

(2, -5): The point is in Quadrant 4, two units to the *right* of zero and five units *below* zero.

## Uses of x and y coordinates

The x and y coordinates of a point are required to find the distance of that point from the declared origin of a 2D space. The coordinates are also used to find the midpoint and slope of a line and to determine its linear equation.

The linear equation of a line is represented as y = mx + b:

- m = slope = change in y / change in x
- x = x coordinate, "how far along"
- y = y coordinate, "how far up"
- b = value of y when x = 0

Here's what the (x, y) coordinate pair looks like if the value of x is known and if the equation is expressed as y = 2x + 2:

X coordinate |
Y coordinate |
Slope (m) |
y = 2x + 2 |
(x, y) |

0 |
2 |
2 |
2 |
(0, 2) |

1 |
4 |
2 |
4 |
(1, 4) |

2 |
6 |
2 |
6 |
(2, 6) |

3 |
8 |
2 |
8 |
(3, 8) |

4 |
10 |
2 |
10 |
(4, 10) |

To graph the equation y = 2x + 2, each coordinate in each ordered pair is located on a coordinate grid. Then, the x and y coordinates are connected to form a straight line.

*See also: mathematical symbols*.