Question
How do the signs of the coordinates affect the position in the coordinate plane?
Answers
Answered by
GPT-4o mini
In a two-dimensional coordinate plane, the position of a point is determined by its coordinates, usually represented as \((x, y)\). The signs of these coordinates indicate the location of the point in relation to the axes, which define four distinct quadrants. Here's how the signs affect the position:
1. **First Quadrant (I)**:
- Coordinates: \(x > 0\), \(y > 0\)
- Description: Both the x-coordinate and y-coordinate are positive. Points in this quadrant lie to the right of the y-axis and above the x-axis.
2. **Second Quadrant (II)**:
- Coordinates: \(x < 0\), \(y > 0\)
- Description: The x-coordinate is negative, and the y-coordinate is positive. Points in this quadrant are to the left of the y-axis and above the x-axis.
3. **Third Quadrant (III)**:
- Coordinates: \(x < 0\), \(y < 0\)
- Description: Both the x-coordinate and y-coordinate are negative. Points here are to the left of the y-axis and below the x-axis.
4. **Fourth Quadrant (IV)**:
- Coordinates: \(x > 0\), \(y < 0\)
- Description: The x-coordinate is positive, and the y-coordinate is negative. Points in this quadrant are to the right of the y-axis and below the x-axis.
Additionally, points on the axes have unique classifications:
- **On the x-axis**: \(y = 0\) (coordinates like \((x, 0)\), where \(x\) can be positive or negative).
- **On the y-axis**: \(x = 0\) (coordinates like \((0, y)\), where \(y\) can be positive or negative).
- **Origin**: The point \((0, 0)\) is at the intersection of the x-axis and the y-axis.
Understanding the signs of \(x\) and \(y\) helps to quickly determine where a point lies in the coordinate plane and identify its quadrant or its position relative to the axes.
1. **First Quadrant (I)**:
- Coordinates: \(x > 0\), \(y > 0\)
- Description: Both the x-coordinate and y-coordinate are positive. Points in this quadrant lie to the right of the y-axis and above the x-axis.
2. **Second Quadrant (II)**:
- Coordinates: \(x < 0\), \(y > 0\)
- Description: The x-coordinate is negative, and the y-coordinate is positive. Points in this quadrant are to the left of the y-axis and above the x-axis.
3. **Third Quadrant (III)**:
- Coordinates: \(x < 0\), \(y < 0\)
- Description: Both the x-coordinate and y-coordinate are negative. Points here are to the left of the y-axis and below the x-axis.
4. **Fourth Quadrant (IV)**:
- Coordinates: \(x > 0\), \(y < 0\)
- Description: The x-coordinate is positive, and the y-coordinate is negative. Points in this quadrant are to the right of the y-axis and below the x-axis.
Additionally, points on the axes have unique classifications:
- **On the x-axis**: \(y = 0\) (coordinates like \((x, 0)\), where \(x\) can be positive or negative).
- **On the y-axis**: \(x = 0\) (coordinates like \((0, y)\), where \(y\) can be positive or negative).
- **Origin**: The point \((0, 0)\) is at the intersection of the x-axis and the y-axis.
Understanding the signs of \(x\) and \(y\) helps to quickly determine where a point lies in the coordinate plane and identify its quadrant or its position relative to the axes.