Understanding The Translation Rule (x, Y) To (x-2, Y+7)

In the realm of geometry, transformations play a crucial role in understanding how shapes and figures can be moved and altered in space. One of the most fundamental transformations is translation, which involves sliding a figure without rotating or reflecting it. In this article, we will delve into the concept of translation, focusing on how it can be represented using mapping rules and how to interpret these rules to understand the direction and magnitude of the translation. The rule as a mapping for the translation of a rectangle is $(x, y) \rightarrow (x-2, y+7)$. Our primary focus will be on deciphering the given mapping rule $(x, y) \rightarrow (x-2, y+7)$ and determining the correct description of the translation it represents. This involves understanding how the coordinates of a point change under the translation and relating these changes to movements in the coordinate plane. Understanding translations is essential not only in mathematics but also in various real-world applications, such as computer graphics, engineering, and physics. For instance, in computer graphics, translations are used to move objects around the screen, while in engineering, they are used to analyze the movement of structures and mechanisms. This comprehensive guide aims to provide a clear and detailed explanation of translations, equipping you with the knowledge and skills to understand and apply this concept effectively. We will explore the underlying principles of translations, discuss how they are represented mathematically, and work through examples to solidify your understanding. By the end of this article, you will be able to confidently interpret mapping rules and describe the corresponding translations, as well as appreciate the broader significance of translations in various fields.

Decoding the Mapping Rule (x, y) → (x-2, y+7)

The mapping rule $(x, y) \rightarrow (x-2, y+7)$ is a concise way to represent a translation in the coordinate plane. To fully understand this rule, we need to break it down and analyze how it affects the coordinates of a point. The notation $(x, y)$ represents the original coordinates of a point, while $(x-2, y+7)$ represents the coordinates of the image of that point after the translation. The arrow, denoted by $\rightarrow$, indicates the transformation or mapping that occurs. Let's examine the changes in the x and y coordinates separately. The x-coordinate changes from $x$ to $x-2$. This means that the x-coordinate of the image is 2 units less than the x-coordinate of the original point. In the coordinate plane, decreasing the x-coordinate corresponds to a movement to the left. Therefore, the translation involves a shift of 2 units to the left. The y-coordinate changes from $y$ to $y+7$. This means that the y-coordinate of the image is 7 units more than the y-coordinate of the original point. In the coordinate plane, increasing the y-coordinate corresponds to a movement upwards. Therefore, the translation involves a shift of 7 units upwards. Combining these two movements, we can conclude that the mapping rule $(x, y) \rightarrow (x-2, y+7)$ represents a translation of 2 units to the left and 7 units upwards. It's important to note that the order in which we analyze the x and y coordinates does not affect the final result. We can first consider the change in the y-coordinate and then the change in the x-coordinate, or vice versa. The key is to correctly interpret the signs and magnitudes of the changes in each coordinate. A negative change in the x-coordinate indicates a movement to the left, while a positive change indicates a movement to the right. Similarly, a negative change in the y-coordinate indicates a movement downwards, while a positive change indicates a movement upwards. Understanding these conventions is crucial for accurately interpreting mapping rules and describing translations.

Analyzing the Options

Now that we have decoded the mapping rule $(x, y) \rightarrow (x-2, y+7)$, we can evaluate the given options and determine which one correctly describes the translation. We have established that the translation involves a shift of 2 units to the left and 7 units upwards. Let's consider each option:

• A. a translation of 2 units down and 7 units to the right This option states that the translation is 2 units down and 7 units to the right. However, we know that the translation is 2 units to the left (not down) and 7 units upwards (not to the right). Therefore, this option is incorrect.

• B. a translation of 2 units down and 7 units to the left This option states that the translation is 2 units down and 7 units to the left. While the 7 units to the left part is consistent with our analysis, the 2 units down part is incorrect. The translation is 2 units to the left, but it is 7 units upwards, not down. Therefore, this option is also incorrect.

• C. a translation of 2 units to the left and 7 units up This option accurately describes the translation. We have determined that the mapping rule $(x, y) \rightarrow (x-2, y+7)$ represents a shift of 2 units to the left (due to the $x-2$ term) and 7 units upwards (due to the $y+7$ term). Therefore, this option is the correct answer.

Why Option C is the Correct Answer

Option C,