What is a rectangular coordinate system?

A rectangular coordinate system (also called a Cartesian coordinate system) consists of a horizontal number line (x-axis) and a vertical number line (y-axis) intersecting at a right angle at the origin, used to plot points with x and y coordinates.

How do you determine the quadrant of a point from the signs of its coordinates?

You can determine the quadrant based on coordinate signs: Quadrant I has (+, +), Quadrant II has (-, +), Quadrant III has (-, -), and Quadrant IV has (+, -).

Where are points with an x-coordinate or y-coordinate of zero located?

Any point with an x-coordinate of zero is located on the y-axis, and any point with a y-coordinate of zero is located on the x-axis.

Rectangular Coordinate System in a Plane | Cartesian Plane & Point Plotting

The use of a rectangular coordinate system is presented along with examples, practice questions, and detailed solutions.

Rectangular Coordinate System in a Plane

A rectangular coordinate system in a plane is used to plot points having an \( x \) coordinate and a \( y \) coordinate. A vertical number line, also called the y-axis, and a horizontal number line, also called the x-axis, intersecting at a right angle form a system of coordinates in a plane as shown in Figure 1 below. The point of intersection of the x and y axes is called the origin of the system of coordinates.

The x and y axes split the plane into four quadrants as noted in the figure above.

Cartesian Plane — Fig.1 - Rectangular Coordinate System in a Plane

Note: The rectangular coordinate system is also called the Cartesian coordinate system.

Point Plotting in a Rectangular Coordinate System

Each point in the plane corresponds to an ordered pair \( (x,y) \), where \( x \) and \( y \) are real numbers. \( x \) and \( y \) are called the coordinates of the point, where the \( x \)-coordinate represents the directed distance from the y-axis to the point, and the \( y \)-coordinate represents the directed distance from the x-axis to the point.

Example: Understanding Directed Distances and Point Locations

In the example below:

The \( x \)-coordinate of point \( A \) is \( 4 \) (positive), so point \( A \) is located to the right of the y-axis.
The \( x \)-coordinate of point \( B \) is \( -5 \) (negative), so point \( B \) is located to the left of the y-axis.
The \( y \)-coordinate of point \( A \) is \( 2 \) (positive), so point \( A \) is located above the x-axis.
The \( y \)-coordinate of point \( B \) is \( -2 \) (negative), so point \( B \) is located below the x-axis.

Point Plotting in a Rectangular Coordinate System in a Plane — Fig.2 - Example of Point Plotting in a Rectangular Coordinate System

More practice on plotting points in a rectangular coordinate system is included.

Signs of Coordinates and Quadrants

The signs of the \( x \) and \( y \) coordinates of a given point provide enough information to find the quadrant of that point without plotting it.

Example: Determining Quadrants from Coordinate Signs

Points \( A = (4,2) \) and \( C = (-4,3) \) are both located above the x-axis because their \( y \)-coordinates (\( 2 \) and \( 3 \)) are both positive. But point \( A \) is to the right of the y-axis and hence in Quadrant I because its \( x \)-coordinate (\( 4 \)) is positive.

The \( x \)-coordinate of \( C \), which is \( -4 \), is negative, so point \( C \) is to the left of the y-axis, located in Quadrant II.

Similar remarks apply to points \( E = (-4,-2) \) in Quadrant III and \( G = (4,-2) \) in Quadrant IV.

Quadrant Rules Summary

If \( x > 0 \) and \( y > 0 \), the point is in Quadrant I
If \( x < 0 \) and \( y > 0 \), the point is in Quadrant II
If \( x < 0 \) and \( y < 0 \), the point is in Quadrant III
If \( x > 0 \) and \( y < 0 \), the point is in Quadrant IV

Signs of coordinates in the different quadrant — Fig.3 - Signs of the Coordinates in the Four Different Quadrants of a Rectangular System

Points on the x and y Axes

Any point whose \( x \)-coordinate is equal to zero is located on the y-axis because its distance from the y-axis is equal to zero.

Example: Points \( A = (0,3) \), \( D = (0,-2) \), and \( E = (0,-4) \) all have an \( x \)-coordinate equal to zero and are therefore located on the y-axis (see Figure 4).

Any point whose \( y \)-coordinate is equal to zero is located on the x-axis because its distance from the x-axis is equal to zero.

Example: Points \( G = (6,0) \), \( C = (-2,0) \), and \( B = (-8,0) \) all have a \( y \)-coordinate equal to zero and are therefore located on the x-axis (see Figure 4).

Point on x and y axes of a Rectangular Coordinate System — Fig.4 - Point on the x and y axes

Practice Questions and Detailed Solutions

Part A: Plotting Points

Plot the following points: \( A = (0,0) \; , \; B = (-4,3) \; , \; C = (0,-4) \; , \; D = (5,-5) \; , \; E = (-3,0) \; , \; F = (-2,-3) \; , \; G = (4,0) \; , \; H = (2,5) \)

Point on a Rectangular Coordinate System for part A

Part B: Identifying Coordinates from a Graph

Give the coordinates of all points plotted in the graph below.

Point on a Rectangular Coordinate System

\( A = (0,1) \), \( B = (2,0) \), \( C = (1,3) \), \( D = (-1,-1) \), \( E = (1,-3) \), \( F = (-3,1) \), \( G = (0,-4) \), \( H = (-3,0) \)

Part C: Determining Quadrants and Axes Without Plotting

Without plotting the points, determine in which quadrant or axis each point is located:

\( A = (-32,-89) \): Quadrant III (\( x < 0, y < 0 \))
\( B = (0,45) \): On the y-axis, above the x-axis (\( x = 0, y > 0 \))
\( C = (-88,0) \): On the x-axis, to the left of the y-axis (\( x < 0, y = 0 \))
\( D = (57,89) \): Quadrant I (\( x > 0, y > 0 \))
\( E = (0,-77) \): On the y-axis, below the x-axis (\( x = 0, y < 0 \))
\( F = (45,-38) \): Quadrant IV (\( x > 0, y < 0 \))
\( G = (49,0) \): On the x-axis, to the right of the y-axis (\( x > 0, y = 0 \))
\( H = (-90,-56) \): Quadrant III (\( x < 0, y < 0 \))

Part D: Graphing and Describing Quadrilaterals

Graph each group of points, link them in order, and describe each quadrilateral:

Group 1: \( A = (2,2) \; , \; B = (-4,2) \; , \; C = (-4,-1) \; , \; D = (2,-1) \)

The four given points form a rectangle as shown below.

Group 2: \( A = (1,2) \; , \; B = (-2,2) \; , \; C = (-5,-2) \; , \; D = (5,-2) \)

The four given points form a trapezoid as shown below.

Group 3: \( A = (0,4) \; , \; B = (-2,2) \; , \; C = (0,-4) \; , \; D = (2,2) \)

The four given points form a kite as shown below.