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# X Coordinate|Definition & Meaning

## Definition

On a **Cartesian** graph, the x-coordinate (or the **abscissa**) of a point represents how far **along** it is on the x-axis. The **x-axis** normally points to the left and right, so the x-coordinate tells us about the **horizontal** position of the point. For coordinates in the ordered pair form **(x, y)**, the x-coordinate is always the **first** coordinate.

The x-coordinate **4** in (**4**, 3) is the signed **distance** from the origin to the point along the **x-axis**, as shown in **figure 1**.

## Rectangular Coordinates

The **rectangular** coordinates, also known as the **Cartesian** coordinates, are the **x** and** y** coordinates on a graph in a two-dimensional plane. They are the signed **distances** from the origin to the point in the **horizontal** and **vertical** directions.

The signs with the **x** and **y** coordinate specify the **direction** of the distance from the origin. If a point’s coordinates are **(a, b)**, then their distance from the origin is the **absolute** value of **a** and** b**.

Both the **x** and** y** coordinates form the Cartesian or **orthogonal** coordinate **system**. Without the y-coordinate, the x-coordinate alone cannot trace a point in a **2D** plane.

These **numerical** coordinates **specify** the position of a point by using two coordinate **axes**, the x and y-axis. For point **A** in a 2D plane, two lines, which are **perpendicular** to both axes, passing through point **A**, meet each axis at a number. These two **numbers** are the rectangular coordinates of point **A**.

The **x** and **y** coordinates are called **rectangular** coordinates, as locating the point through the coordinates forms a rectangle in the x-y plane. For example, point **(7, 4)** is located in the x-y plane, as shown in **figure 2**.

## Ordered Pair

The rectangular coordinates are written in **parentheses**, separated by a **comma** as **(x, y)**; the first coordinate is always the **x-coordinate**, and the second, is the y-coordinate. This notation (x, y) is known as an **ordered pair**. The x-coordinate is known as the **abscissa**, and the y-coordinate as **ordinate**.

For example,** (2, 3)** is an ordered pair showing the **horizontal** distance as **2**(x-coordinate) units and the **vertical** distance as** 3**(y-coordinate) units from the origin.

Each **point** in the Cartesian coordinate system is specified by a **unique** ordered pair.

## X-Axis and Y-axis

An axis is a reference **line** used to measure the distance from a certain **reference** point. The two axes in a two-dimensional **plane** are the x and y-axis. The two axes are placed in such a way that they are **perpendicular** to each other.

The horizontal axis is known as the **x-axis**, while the vertical axis is the **y-axis**. The **left-right** direction is the horizontal direction, and the **up-down** direction is the vertical direction.

### Origin

The **x** and **y** axes can be considered as two number lines placed at **right** angles. The point at which both axes **intersect** is known as the **origin**. It is denoted by **O** with the ordered pair as **(0, 0)**. It is the **reference** point from where any **point** can be traced by moving some units left, right, up, or down.

## Directions of Positive and Negative X and Y Axes

The **direction** of the positive and negative **x** and **y** axes are discussed below.

### Positive X-axis

The **positive** x-axis moves from the origin **O** from left to right. The x-coordinate increases as we move further from **left** to **right**.

For example, point **(5, 3)** has the **x-coordinate** as positive five. It means that from the origin, going **five** units from left to right tells how much the point is along the **positive** x-axis. But to completely trace the point in a **2D** plane, the y-coordinate** 3** should also be considered.

### Negative X-axis

The **negative** x-axis has a direction **opposite** to the positive x-axis. From origin **O**, the negative x-axis increases as we move from **right** to **left**. For example, point **(-3, 4)** has the x-coordinate as negative three. It means that from the origin, going **three** units from right to **left** shows the point along the negative x-axis.

### Positive Y-axis

The **positive** y-axis increases in the upper direction from the origin **O**. This means that the y-coordinate increases as going more and more in the **upper** direction. For example, point **(1, 6)** has a positive **six** as a y-coordinate. It means that going six units **up** from the **origin** shows how much the point is far up on the positive y-axis.

### Negative Y-axis

The direction of the **negative** y-axis is the opposite of the positive y-axis. The negative y-axis **increases** from the origin point in the **downward** direction. For example, point **(2, -4)** has a negative four as the y-coordinate. By moving **four** units down from the origin, the y-coordinate -4 is located.

**Figure 3** shows the **origin** and the positive and negative **x** and **y**-axis.

## Four Quadrants

“**Quad**” means four. The **x** and** y** axes divide the Cartesian plane into **four** parts, known as the quadrants. The four **quadrants** are numbered in ascending order in a **counter-clockwise** direction. The four quadrants are discussed below.

### First Quadrant

In the first quadrant, both the **x** and **y** coordinates are **positive**. This is the **top-right** quadrant which includes the positive **x** and **y**-axis. For example, the ordered pair **(1, 3)** belongs in the first quadrant as both the x-coordinate **1** and the y-coordinate **3** are positive.

### Second Quadrant

In the second quadrant, the **x-coordinate** is **negative**, and the **y-coordinate** is **positive**. This is the **top-left** part of the Cartesian plane, including the negative x-axis and the positive y-axis. For example, point **(-1, 3)** lies in the second quadrant, moving left **one** unit and moving up **three** units from the origin.

### Third Quadrant

In the third quadrant, both the **x** and **y** coordinates are **negative**. This is the **bottom-left** quadrant in the **x-y** plane with both the x and y axis negative. For example, the ordered pair **(-1, -3)** belongs in the third quadrant, moving **one** unit left and **three** units down from the origin.

### Fourth Quadrant

In the fourth quadrant, the x-coordinate is **positive**, and the y-coordinate is **negative**. This is the **bottom-right** quadrant with a positive x-axis and a negative y-axis. For example, point **(1, -3)** belongs in the fourth quadrant, moving **one** unit right and **three** units down from the origin O.

**Figure 4** shows the points (1, 3), (-1, 3), (-1, -3), and (1, -3) in the first, second, third, and fourth **quadrants**, respectively.

## X-coordinate in Different Dimensions

Different dimensions require a different number of **coordinates** to define a **point** or object in space. The **x-coordinate** in a different number of dimensions is discussed below.

### One Dimension

One axis can be considered as a **number line**. A line is a one-dimensional figure, so it requires only **one** number to specify a point on a number line. The number line contains two **half-lines**, one positive half-line from to **positive** infinity and one **negative** half-line from **zero** to negative infinity.

### Two Dimensions

Two dimensions consist of two axes, the **x** and **y**-axis. The x and y-coordinate specify a point in a **2D** plane. **Two** numbers are required to identify a **point** in a 2D plane which determines the **horizontal** and **vertical** distance from the origin **O**.

### Three Dimensions

A three-dimensional space consists of an **x**, **y**, and **z** coordinate. Three lines drawn at **right** angles to each other are the x, y, and z axis in **space**. The point where the three axes intersect is the **origin** O.

The x-coordinate shows how much to move **left** or **right** from the origin. The y-coordinate shows how much to move **up** and **down** from the origin, and the z-coordinate shows how much to move **backward** or **forward** to get from the origin to the point in space.

## Distance Between Two Points in an X-Y Plane

The **distance** between two points in an** x-y** plane can be found by using the x and y-coordinates of the points. If the **coordinates** of point **C** are (x_{1}, y_{1}) and those of point **D** are (x_{2}, y_{2}), the distance between point **C** and **D** can be calculated by using the **formula**:

\[ \text{Distance} = \sqrt{ {( y_2 \ – \ y_1 )}^2 + {( x_2 \ – \ x_1 )}^2 } \]

## Examples Involving the X Coordinate

### Example 1 – Identifying X and Y Coordinates of an Ordered Pair

Point **Q** is shown in an **x-y** plane in **figure 5**. Identify the x and y **coordinates** and write the ordered pair of the point **Q**.

### Solution

At first, two **lines** that pass through point **Q** and are **perpendicular** to the two axes are drawn such that they meet the **x** and **y** axis at the x and y **coordinates** of point Q, respectively. Hence, the x-coordinate is **4**, and the y-coordinate is **2**. It can be written as an ordered pair as **(4, 2)**.

### Example 2 – Identifying the Quadrants of Different Ordered Pairs

In which quadrants do the ordered pairs (-9, -3), (-1.4, 12), (3, 7), and (8, -5) lie?

### Solution

The point **(-9, -3)** lies in the **third** quadrant as both the **x** and **y**-coordinates are negative.

The point** (-1.4, 12)** lies in the **second** quadrant as the abscissa is negative, but the ordinate is positive.

Point** (3, 7)** lies in the **first** quadrant as both the x and y coordinates are positive.

The ordered pair **(8, -5)** lies in the **fourth** quadrant as the abscissa is positive, but the ordinate is negative.

### Example 3 – Finding the Distance Between Two Points

Find the **distance** between two points, **T** with coordinates **(2, 6)** and **R** with coordinates **(5, -8)**.

### Solution

The **formula** for the distance between two points is given:

\[ \text{Distance} = \sqrt{ {( y_2 \ – \ y_1 )}^2 + {( x_2 \ – \ x_1 )}^2 } \]

Taking the **coordinates** of **T** as (x_{1}, y_{1}) and the coordinates of **R** as (x_{2}, y_{2}).

So:

x_{1} = 2, y_{1} = 6, x_{2} = 5, y_{2} = -8

Putting the **values** in the distance formula gives:

\[ \text{Distance} = \sqrt{ {( – 8 \ – \ 6 )}^2 + {( 5 \ – \ 2 )}^2 } \]

\[ \text{Distance} = \sqrt{ {( -14 )}^2 + {( 3 )}^2 } \]

\[ \text{Distance} = \sqrt{ 196 + 9 } \]

\[ \text{Distance} = \sqrt{ 205 } \]

**Distance = 14.3**

*All the images are created using GeoGebra.*