If The Point G ( − 4 , 6 G (-4, 6 G ( − 4 , 6 ] Was Moved Horizontally 4 Units To The Left, Which Of The Following Is The Correct Mapping To Define The Translation?A. ( X , Y ) → ( X , Y − 4 (x, Y) \rightarrow (x, Y-4 ( X , Y ) → ( X , Y − 4 ] B. ( X , Y ) → ( X + 4 , Y (x, Y) \rightarrow (x+4, Y ( X , Y ) → ( X + 4 , Y ] C. $(x, Y)
Introduction
In geometry, a translation is a fundamental concept that involves moving a point or an object from one location to another without changing its orientation or size. When a point is moved horizontally, it means that its x-coordinate changes, while its y-coordinate remains the same. In this article, we will explore the concept of translation mapping and determine the correct mapping to define the translation of point G (-4, 6) when it is moved horizontally 4 units to the left.
Understanding Translation Mapping
A translation mapping is a function that takes a point (x, y) and maps it to a new point (x', y') based on a specific transformation. In the case of a horizontal translation, the x-coordinate of the point changes, while the y-coordinate remains the same. Mathematically, this can be represented as:
(x, y) → (x' + d, y)
where (x, y) is the original point, (x' + d, y) is the new point, and d is the distance of the translation.
Analyzing the Options
Let's analyze the given options to determine the correct mapping to define the translation of point G (-4, 6) when it is moved horizontally 4 units to the left.
Option A: (x, y) → (x, y - 4)
This option suggests that the y-coordinate of the point changes by -4 units, while the x-coordinate remains the same. However, this is not a horizontal translation, as the x-coordinate does not change. Therefore, this option is incorrect.
Option B: (x, y) → (x + 4, y)
This option suggests that the x-coordinate of the point changes by +4 units, while the y-coordinate remains the same. This is a horizontal translation, as the x-coordinate changes, and the y-coordinate remains the same. Therefore, this option is a strong candidate for the correct mapping.
Option C: (x, y) → (x - 4, y)
This option suggests that the x-coordinate of the point changes by -4 units, while the y-coordinate remains the same. However, this is not a horizontal translation of 4 units to the left, as the x-coordinate changes in the opposite direction. Therefore, this option is incorrect.
Conclusion
Based on the analysis of the options, the correct mapping to define the translation of point G (-4, 6) when it is moved horizontally 4 units to the left is:
(x, y) → (x + 4, y)
This mapping represents a horizontal translation of 4 units to the left, as the x-coordinate changes by +4 units, while the y-coordinate remains the same.
Final Answer
The final answer is:
B. (x, y) → (x + 4, y)
Introduction
In our previous article, we explored the concept of translation mapping and determined the correct mapping to define the translation of point G (-4, 6) when it is moved horizontally 4 units to the left. In this article, we will answer some frequently asked questions related to translation mapping to help you better understand this concept.
Q: What is translation mapping?
A: Translation mapping is a function that takes a point (x, y) and maps it to a new point (x', y') based on a specific transformation. In the case of a horizontal translation, the x-coordinate of the point changes, while the y-coordinate remains the same.
Q: What is the difference between a horizontal and vertical translation?
A: A horizontal translation involves changing the x-coordinate of a point, while a vertical translation involves changing the y-coordinate of a point. In a horizontal translation, the y-coordinate remains the same, while in a vertical translation, the x-coordinate remains the same.
Q: How do I determine the correct mapping for a translation?
A: To determine the correct mapping for a translation, you need to identify the type of translation (horizontal or vertical) and the distance of the translation. For a horizontal translation, the x-coordinate changes by the distance of the translation, while the y-coordinate remains the same. For a vertical translation, the y-coordinate changes by the distance of the translation, while the x-coordinate remains the same.
Q: What is the formula for a translation mapping?
A: The formula for a translation mapping is:
(x, y) → (x' + d, y)
where (x, y) is the original point, (x' + d, y) is the new point, and d is the distance of the translation.
Q: Can I use a translation mapping to move a point in a 3D space?
A: Yes, you can use a translation mapping to move a point in a 3D space. However, you need to consider the x, y, and z coordinates of the point and apply the translation mapping accordingly. For example, if you want to move a point (x, y, z) horizontally 4 units to the left, you would use the following mapping:
(x, y, z) → (x + 4, y, z)
Q: Are there any limitations to translation mapping?
A: Yes, there are some limitations to translation mapping. For example, you cannot use a translation mapping to rotate or scale a point or object. Translation mapping only allows you to move a point or object from one location to another without changing its orientation or size.
Q: Can I use translation mapping to solve real-world problems?
A: Yes, you can use translation mapping to solve real-world problems. For example, you can use translation mapping to determine the new location of a building or a monument after a natural disaster or a construction project. You can also use translation mapping to calculate the distance and direction of a point or object in a 2D or 3D space.
Conclusion
In this article, we answered some frequently asked questions related to translation mapping to help you better understand this concept. We hope that this article has provided you with a deeper understanding of translation mapping and its applications in mathematics and real-world problems.