Coordinate Transformation Find The Range Of Transformed Points

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In mathematics, coordinate transformations play a crucial role in simplifying problems and gaining new perspectives on geometric objects. A coordinate transformation is a function that maps points from one coordinate system to another. This article delves into the concept of coordinate transformations, specifically focusing on a linear transformation in two dimensions. We will explore how to determine the range of a given domain under a specific transformation. The problem at hand involves a transformation defined by f(x, y) = (-2x, y) and a domain consisting of three points: (0, 5), (8, -1), and (-6, 4). Our objective is to find the range of this transformation, which means determining the images of these points under the given function. This exercise will not only solidify our understanding of coordinate transformations but also enhance our ability to apply these concepts in various mathematical contexts.

Coordinate transformations are fundamental tools in various branches of mathematics, including geometry, calculus, and linear algebra. A coordinate transformation is a mapping that changes the coordinates of a point or a set of points from one system to another. These transformations can involve translations, rotations, scaling, shearing, and reflections, or a combination of these. Understanding coordinate transformations is essential for simplifying complex problems, as they allow us to view objects and equations from different perspectives. For instance, a rotation can simplify the analysis of a figure with rotational symmetry, or a translation can center an object at the origin, making calculations easier. In higher dimensions, coordinate transformations are used extensively in fields such as computer graphics, robotics, and image processing. The key idea is to maintain the underlying geometric relationships while changing the numerical representation of the points. This flexibility is what makes coordinate transformations so powerful and versatile in mathematical and scientific applications.

Let's consider the specific problem we are tasked with. We are given a coordinate transformation defined by the function f(x, y) = (-2x, y). This transformation takes a point (x, y) in the Cartesian plane and maps it to a new point (-2x, y). Notice that this transformation involves scaling the x-coordinate by a factor of -2 while leaving the y-coordinate unchanged. This means that the transformation effectively stretches the plane horizontally by a factor of 2 and reflects it across the y-axis. The domain of this transformation is given by the set of three points: (0, 5), (8, -1), and (-6, 4). The domain is the set of input points, and we want to find the range, which is the set of output points that result from applying the transformation to the domain. In other words, we need to find the images of the points (0, 5), (8, -1), and (-6, 4) under the transformation f(x, y) = (-2x, y). This involves substituting the coordinates of each point into the transformation function and calculating the resulting coordinates. The range will then consist of these transformed points. This problem is a classic example of how coordinate transformations alter the positions of points in a plane and how we can mathematically determine these new positions.

Applying the Transformation

To find the range of the given coordinate transformation, we need to apply the function f(x, y) = (-2x, y) to each point in the domain. The domain consists of the points (0, 5), (8, -1), and (-6, 4). Let's apply the transformation to each point individually:

  1. For the point (0, 5):

    • f(0, 5) = (-2 * 0, 5) = (0, 5). So, the image of the point (0, 5) under the transformation is (0, 5). This means that the point (0, 5) remains unchanged by this transformation, as the x-coordinate is multiplied by -2, but since it's 0, the result is still 0.

  2. For the point (8, -1):

    • f(8, -1) = (-2 * 8, -1) = (-16, -1). Thus, the image of the point (8, -1) under the transformation is (-16, -1). The x-coordinate is transformed from 8 to -16, which means it is stretched horizontally by a factor of 2 and reflected across the y-axis, while the y-coordinate remains the same.

  3. For the point (-6, 4):

    • f(-6, 4) = (-2 * -6, 4) = (12, 4). Hence, the image of the point (-6, 4) under the transformation is (12, 4). Here, the x-coordinate is transformed from -6 to 12, which also involves a horizontal stretch by a factor of 2 and a reflection across the y-axis, with the y-coordinate staying constant.

By applying the transformation to each point in the domain, we have found their corresponding images. These images form the range of the coordinate transformation for the given domain.

Determining the Range

Now that we have applied the coordinate transformation f(x, y) = (-2x, y) to each point in the domain, we can determine the range. The range is the set of output points obtained after applying the transformation. From our calculations, we found the following images:

  • The image of (0, 5) is (0, 5).
  • The image of (8, -1) is (-16, -1).
  • The image of (-6, 4) is (12, 4).

Therefore, the range of the transformation for the given domain is the set of points (0, 5), (-16, -1), and (12, 4). This means that the original points in the domain have been transformed to these new locations in the Cartesian plane. The transformation has stretched the x-coordinates by a factor of 2 and reflected them across the y-axis, while the y-coordinates have remained unchanged. Understanding how transformations affect points in space is crucial for many applications, including computer graphics, physics simulations, and engineering designs. By finding the range, we have successfully mapped the original domain to its transformed counterpart, providing a clear picture of the effect of the coordinate transformation.

In conclusion, we have successfully determined the range of the coordinate transformation f(x, y) = (-2x, y) for the given domain (0, 5), (8, -1), and (-6, 4). By applying the transformation to each point in the domain, we found the corresponding points in the range to be (0, 5), (-16, -1), and (12, 4). This exercise demonstrates the fundamental principles of coordinate transformations and their impact on geometric objects. Coordinate transformations are a cornerstone of many mathematical and scientific disciplines, allowing us to manipulate and analyze spatial data effectively. Understanding how transformations work is essential for solving a wide range of problems, from simple geometric mappings to complex simulations in physics and engineering. The ability to find the range of a transformation given a domain is a key skill in these fields, and this article has provided a clear and concise example of how to accomplish this. The process involves applying the transformation function to each point in the domain and collecting the resulting points, which then form the range. This approach can be extended to more complex transformations and domains, making it a valuable tool in mathematical analysis.