The Coordinate Plane and Points
3. Quadrants and Axes
Learning outcomes
- I can identify the four quadrants of the coordinate plane.
- I can determine the signs of coordinates in each quadrant.
- I can identify whether a point lies in a quadrant or on an axis.
- I can classify points based on their coordinates.
- I can use quadrant information to describe the location of points.
Introduction
In the previous lesson, you learned how to plot and name points using ordered pairs. Once points are plotted, we can describe their locations more precisely by identifying the quadrant or axis where they lie.
The x-axis and y-axis divide the coordinate plane into four sections, called quadrants. Each quadrant has its own combination of positive and negative coordinates.
Understanding quadrants helps us quickly identify the location of any point on the coordinate plane.
The Four Quadrants
The x-axis and y-axis divide the coordinate plane into four regions.
The quadrants are numbered using Roman numerals, starting in the upper-right corner and moving counterclockwise.
| Quadrant | Location | Sign of Coordinates |
|---|---|---|
| Quadrant I | Upper right | (+, +) |
| Quadrant II | Upper left | (−, +) |
| Quadrant III | Lower left | (−, −) |
| Quadrant IV | Lower right | (+, −) |
The numbering always follows this order:
- Quadrant I
- Quadrant II
- Quadrant III
- Quadrant IV
Remembering the Signs
Each quadrant has a different combination of positive and negative numbers.
| Quadrant | x-coordinate | y-coordinate |
|---|---|---|
| I | Positive | Positive |
| II | Negative | Positive |
| III | Negative | Negative |
| IV | Positive | Negative |
Many students remember the pattern by starting in Quadrant I and moving counterclockwise.
Identifying the Quadrant of a Point
To determine which quadrant a point belongs to, simply examine the signs of its coordinates.
Example 1
Point:
(4, 3)
- x is positive
- y is positive
Therefore, the point is in Quadrant I.
Example 2
Point:
(-5, 2)
- x is negative
- y is positive
This point is in Quadrant II.
Example 3
Point:
(-4, -6)
Both coordinates are negative.
The point is in Quadrant III.
Example 4
Point:
(7, -3)
- x is positive
- y is negative
This point lies in Quadrant IV.
Points on the Axes
Not every point belongs to a quadrant.
Some points lie directly on an axis.
Points on the x-axis
If the y-coordinate is zero, the point lies on the x-axis.
Examples:
- (4,0)
- (-7,0)
- (0,0)
Points on the y-axis
If the x-coordinate is zero, the point lies on the y-axis.
Examples:
- (0,5)
- (0,-3)
- (0,8)
The Origin
The point
(0,0)
is called the origin.
The origin belongs to both axes, but not to any quadrant.
Classifying Points
Every point can be classified by its location.
| Point | Classification |
|---|---|
| (3,5) | Quadrant I |
| (-2,6) | Quadrant II |
| (-4,-1) | Quadrant III |
| (8,-2) | Quadrant IV |
| (5,0) | x-axis |
| (0,-6) | y-axis |
| (0,0) | Origin |
Real-World Applications
Coordinates and quadrants are used in many fields where precise locations are important.
Examples include:
- GPS navigation
- Robotics
- Engineering
- Architecture
- Computer graphics
- Video game design
- Geographic Information Systems (GIS)
- Scientific data mapping
Worked Examples
Example 1
Determine the quadrant of:
(-8,4)
- x is negative
- y is positive
Answer: Quadrant II
Example 2
Determine the quadrant of:
(6,-5)
- x is positive
- y is negative
Answer: Quadrant IV
Example 3
Where is the point (0,7)?
The x-coordinate is zero.
Answer: On the y-axis
Example 4
Where is the point (-9,0)?
The y-coordinate is zero.
Answer: On the x-axis
Example 5
Where is the point (0,0)?
Both coordinates are zero.
Answer: The origin
Did You Know?
Pilots and air traffic controllers use coordinate systems to monitor the exact positions of aircraft. Similar coordinate systems are also used by ships, satellites, and space missions to navigate accurately over long distances.
Key Terms
| Term | Definition |
|---|---|
| Quadrant | One of the four regions formed by the x-axis and y-axis. |
| Quadrant I | The upper-right region where both coordinates are positive. |
| Quadrant II | The upper-left region where x is negative and y is positive. |
| Quadrant III | The lower-left region where both coordinates are negative. |
| Quadrant IV | The lower-right region where x is positive and y is negative. |
| x-axis | The horizontal axis where y = 0. |
| y-axis | The vertical axis where x = 0. |
| Origin | The point (0,0), where the two axes intersect. |
Key Takeaways
- The coordinate plane is divided into four quadrants.
- Each quadrant has a unique combination of positive and negative coordinates.
- Quadrants are numbered I, II, III, and IV in a counterclockwise direction.
- Points with y = 0 lie on the x-axis.
- Points with x = 0 lie on the y-axis.
- The point (0,0) is called the origin and does not belong to any quadrant.