What Are The Coordinates Of The Pre-image Of A Point, Given That The Image Of The Point Under The Transformation $r_{y=-x}(x, Y) ightarrow(-4,9]$?

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ightarrow(-4,9]$.

In the captivating realm of mathematics, transformations play a pivotal role in altering the position and orientation of geometric figures. Among these transformations, reflections hold a special significance, mirroring figures across a specific line. In this article, we embark on a journey to unravel the concept of reflections, particularly those occurring across the line y = -x, and delve into the intriguing task of determining the pre-image of a point after such a transformation. Let's consider the problem at hand: The image of a point is given by the rule ry=−x(x,y)ightarrow(−4,9]r_{y=-x}(x, y) ightarrow(-4,9]. Our mission is to unearth the coordinates of its pre-image, the original point before the transformation took place. To solve this mathematical puzzle, we must first grasp the fundamental principles governing reflections across the line y = -x. This line, with its equation y = -x, forms a diagonal that elegantly bisects the second and fourth quadrants of the coordinate plane. When a point undergoes reflection across this line, its x and y coordinates undergo a fascinating transformation: they swap places, and their signs are negated. To illustrate, let's take an arbitrary point (x, y). Upon reflection across the line y = -x, this point metamorphoses into the point (-y, -x). This transformation serves as the cornerstone of our quest to find the pre-image. Now, armed with this knowledge, let's tackle the problem head-on. We are given that the image of a point after reflection across the line y = -x is (-4, 9). Our objective is to determine the coordinates of the original point, the pre-image. To achieve this, we must reverse the transformation that occurred during reflection. Remember, reflection across y = -x swaps the x and y coordinates and negates their signs. Therefore, to reverse this process, we must perform the opposite operations: swap the coordinates back and negate their signs once more.

Decoding the Reflection Rule

Understanding the Rule of Reflection over y = -x: At its core, the reflection over the line y = -x involves a systematic exchange and negation of coordinates. This mathematical dance can be elegantly expressed as follows: a point with coordinates (x, y) transforms into a new point with coordinates (-y, -x). This rule, seemingly simple, holds the key to unlocking the secrets of reflections across this specific line. To truly grasp the essence of this transformation, let's delve into the mechanics of how this rule operates. Imagine a point residing in the coordinate plane. When reflected over the line y = -x, this point embarks on a journey to a new location. The path it takes is perpendicular to the line of reflection, and the distance it travels on either side of the line remains equal. This ensures that the image is a true mirror reflection of the original point. Now, let's dissect the coordinate transformation. The x-coordinate of the original point becomes the negated y-coordinate of the reflected point, and conversely, the y-coordinate of the original point becomes the negated x-coordinate of the reflected point. This exchange and negation create a symmetrical relationship between the original point and its image, perfectly mirroring each other across the line y = -x. To solidify our understanding, let's consider a concrete example. Suppose we have a point with coordinates (2, 3). When reflected over the line y = -x, this point will transform into a new point with coordinates (-3, -2). Notice how the x and y coordinates have swapped places, and their signs have been flipped. This transformation exemplifies the rule of reflection over y = -x in action. Now, with a firm grasp of this rule, we can confidently tackle problems involving reflections across the line y = -x. Whether we are tasked with finding the image of a point or, as in our current problem, uncovering the pre-image, the principles we have discussed will serve as our guiding light.

Reverse Transformation

Applying the Reverse Transformation to Find the Pre-Image: Our objective is to find the original point, or the pre-image, given the image (-4, 9) after reflection across the line y = -x. As we've established, the reflection rule swaps the x and y coordinates and negates their signs. To reverse this process, we simply apply the same rule again. The beauty of reflections lies in their reversibility. If reflecting a point across a line transforms it to its image, reflecting the image across the same line will bring you back to the original point. This principle is crucial in our quest to find the pre-image. Starting with the image (-4, 9), we need to swap the coordinates and negate their signs. Swapping the coordinates gives us (9, -4). Now, we negate the signs of both coordinates, resulting in (-9, 4). Therefore, the pre-image of the point (-4, 9) after reflection across the line y = -x is (-9, 4). This elegant solution demonstrates the power of understanding the underlying principles of transformations. By reversing the reflection rule, we were able to effortlessly trace our steps back to the original point. To further solidify our understanding, let's consider another example. Suppose the image of a point after reflection across y = -x is (5, -2). To find the pre-image, we swap the coordinates, giving us (-2, 5), and then negate the signs, resulting in (2, -5). Thus, the pre-image is (2, -5). This consistent application of the reverse transformation allows us to confidently solve a wide range of problems involving reflections across the line y = -x. In essence, finding the pre-image involves a simple yet powerful technique: applying the reflection rule in reverse. This method provides a clear and concise path to uncovering the original point, showcasing the symmetrical nature of reflections.

Solution

Determining the Pre-Image Coordinates: Following the reverse transformation process, we arrive at the pre-image coordinates. Starting with the image (-4, 9), we swapped the coordinates and negated their signs. This yielded the point (-9, 4). Therefore, the pre-image of the point whose image is (-4, 9) under the reflection ry=−xr_{y=-x} is indeed (-9, 4). This meticulous application of the reverse transformation ensures that we have accurately pinpointed the original point. The pre-image, in essence, is the point that, when reflected across the line y = -x, transforms into the given image. By understanding the mechanics of reflection and applying the reverse transformation, we can confidently navigate the world of geometric transformations. To recap, the process involves two key steps: swapping the coordinates and negating their signs. This simple yet powerful technique allows us to unravel the mysteries of reflections and uncover the pre-images of points with precision. In this specific problem, the pre-image (-9, 4) represents the point that, when mirrored across the line y = -x, perfectly aligns with the image (-4, 9). This symmetrical relationship underscores the essence of reflections and their transformative power. Now, armed with this knowledge, we can confidently tackle a variety of problems involving reflections across the line y = -x, whether it be finding images or, as in this case, unearthing the pre-images of points. The key lies in understanding the reflection rule and applying it judiciously, either in its original form or in reverse.

Therefore, based on our calculations and understanding of reflections across the line y = -x, the coordinates of the pre-image are (-9, 4), which corresponds to option A. This article has not only provided the solution to the problem but also delved into the underlying principles of reflections, empowering you to confidently tackle similar mathematical challenges.

In conclusion, understanding the principles of geometric transformations, particularly reflections, is crucial for solving mathematical problems involving geometric figures. By grasping the rule of reflection across the line y = -x and applying the reverse transformation, we can effectively determine the pre-image of a point. This article has provided a comprehensive guide to this process, equipping you with the knowledge and skills to confidently navigate the world of transformations. Remember, mathematics is not merely about memorizing formulas; it's about understanding the underlying concepts and applying them creatively to solve problems. As you continue your mathematical journey, embrace the power of transformations and their ability to reveal the hidden symmetries and relationships within the world of geometry. Just as we successfully unraveled the pre-image in this problem, you too can conquer any mathematical challenge with a combination of knowledge, understanding, and a touch of ingenuity.

  • Reflections
  • Pre-image
  • Geometric Transformations
  • Line y = -x
  • Coordinate Plane
  • Reverse Transformation
  • Mathematics