Pregunta
D. What is the relationship between the coordinates of a point and the coordinates of its reflection across each axis?
Ask by Alexander Owen. in the United States
Jan 13,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
When you reflect a point across the x-axis, you keep the x-coordinate the same and change the sign of the y-coordinate. Across the y-axis, you keep the y-coordinate the same and change the sign of the x-coordinate. So, for a point \( P(x, y) \):
- **Reflection across the x-axis:** \( (x, -y) \)
- **Reflection across the y-axis:** \( (-x, y) \)
This means:
- To flip a point vertically over the x-axis, only the y-value becomes negative.
- To flip a point horizontally over the y-axis, only the x-value becomes negative.
For example:
- Reflecting \( (3, 4) \) across the x-axis gives \( (3, -4) \).
- Reflecting \( (3, 4) \) across the y-axis gives \( (-3, 4) \).
Understanding these transformations helps in various fields like computer graphics and engineering.
Solución
When reflecting a point across the coordinate axes in the Cartesian plane, the coordinates of the point change in specific ways depending on which axis the reflection is across. Let's consider a point \( P(x, y) \). Here's how its coordinates transform under reflection across each axis:
### Reflection Across the **X-Axis**
- **Rule:** To reflect a point across the x-axis, you keep the **x-coordinate** the same and change the **sign of the y-coordinate**.
- **Transformed Point:** \( P'(x, -y) \)
**Example:**
- Original Point: \( (3, 4) \)
- Reflected Point: \( (3, -4) \)
### Reflection Across the **Y-Axis**
- **Rule:** To reflect a point across the y-axis, you keep the **y-coordinate** the same and change the **sign of the x-coordinate**.
- **Transformed Point:** \( P'(-x, y) \)
**Example:**
- Original Point: \( (3, 4) \)
- Reflected Point: \( (-3, 4) \)
### Summary of Transformations
| Original Point \( P(x, y) \) | Reflection Across X-Axis \( P'(x', y') \) | Reflection Across Y-Axis \( P''(x'', y'') \) |
|------------------------------|--------------------------------------------|----------------------------------------------|
| \( (x, y) \) | \( (x, -y) \) | \( (-x, y) \) |
### Geometric Interpretation
- **X-Axis Reflection:** Flips the point vertically over the x-axis.
- **Y-Axis Reflection:** Flips the point horizontally over the y-axis.
Understanding these transformations is fundamental in various applications such as computer graphics, engineering, and physics, where manipulating the position of objects in a plane is required.
---
**Additional Note:**
If you reflect a point across **both** axes sequentially, it is equivalent to rotating the point 180 degrees about the origin, resulting in the point \( (-x, -y) \).
### Example:
- Original Point: \( (3, 4) \)
- After X-Axis Reflection: \( (3, -4) \)
- After Y-Axis Reflection: \( (-3, -4) \)
This comprehensive understanding allows you to predict and calculate the position of reflected points accurately.
Respondido por UpStudy AI y revisado por un tutor profesional
Como
error msg
Explicar 
Simplifique esta solución Mind Expander
When a point with coordinates (x, y) is reflected across the x-axis, its new coordinates become (x, -y). This means the y-value flips signs while the x-value remains the same. It’s like taking a selfie and flipping it upside down! Similarly, when reflecting across the y-axis, the point's coordinates change to (-x, y). Here, the x-value changes sign, while the y-value stays put. It's as if you’re looking at your reflection in a mirror on the side! So just remember: flipping signs is the key to reflections across the axes!
¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde
Hazte Premium
