1. Understanding Coordinates
Coordinates tell us the exact position of a point on the Cartesian plane. Every point is represented in the form \((x, y)\), which is called an ordered pair. The two values show how far the point is from the origin along the x-axis and the y-axis.
The order is important: the first value is always x-coordinate, the second is y-coordinate.
1.1. Definition
Coordinate of a point is an ordered pair of the form \((x, y)\), where:
- x: horizontal distance from the origin
- y: vertical distance from the origin
1.2. Interpreting a Coordinate
To locate a point:
- Move along the x-axis to the value of x.
- From there, move parallel to the y-axis to the value of y.
1.2.1. Example
Point \((3, -4)\):
- Move 3 units right (positive x)
- Move 4 units down (negative y)
2. Signs of Coordinates
The sign of the coordinates tells us which direction the point lies from the origin. Positive and negative signs indicate right/left and up/down movements.
2.1. Sign Rules
- Positive x → right side of y-axis
- Negative x → left side of y-axis
- Positive y → above x-axis
- Negative y → below x-axis
2.2. Visual Sense of Signs
If both coordinates are positive, the point lies in the upper right region. If both are negative, the point lies in the lower left region.
3. Understanding Quadrants
The Cartesian plane is divided into four regions called quadrants. These quadrants help in identifying where a point lies based on the signs of its coordinates.
The quadrants are numbered in the counterclockwise direction starting from the top-right region.
3.1. Quadrant Details
| Quadrant | Coordinate Signs |
|---|---|
| Quadrant I | \((+, +)\) |
| Quadrant II | \((-, +)\) |
| Quadrant III | \((-, -)\) |
| Quadrant IV | \((+, -)\) |
3.2. How to Identify the Quadrant
Look at the signs:
- If x and y are positive → Quadrant I
- If x is negative and y is positive → Quadrant II
- If both are negative → Quadrant III
- If x is positive and y is negative → Quadrant IV
3.3. Axis Cases
If any coordinate is zero, the point does not lie in any quadrant. It lies on one of the axes.
3.3.1. Examples
- \((0, 5)\): lies on y-axis
- \((4, 0)\): lies on x-axis
- \((0, 0)\): origin
4. Examples to Understand Coordinates and Quadrants
These examples help in getting a complete idea of how coordinates determine the quadrant.
4.1. Example Set
- \((2, 3)\) → Quadrant I
- \((-5, 4)\) → Quadrant II
- \((-3, -7)\) → Quadrant III
- \((6, -2)\) → Quadrant IV
- \((0, -3)\) → on y-axis
- \((4, 0)\) → on x-axis