## Transformation of Regions in the PlaneIn this educational example, we explore a transformation from a region $ S $ in the $ uv $-plane to a region $ R $ in the $ xy $-plane. The aim is to find equations for a transformation $ T $ that maps the rectangular region $ S $, aligned with the $ u $- and $ v $-axes, onto the region $ R $.### Description of Regions- **Region $ R $**: - Bounded by the lines $ y = 2x - 4 $, $ y = 2x + 4 $, $ y = 4 - x $, and $ y = 6 - x $. - Graphically represented in the $ xy $-plane as a shaded quadrilateral oriented at an angle.- **Region $ S $**: - Shown as a rectangle in the $ uv $-plane. - The sides of $ S $ are parallel to the $ u $- and $ v $-axes, making it easier to visualize its rectangular shape and orientation.### Visual Explanation- **Graphs**: - The left graph represents the $ uv $-plane with region $ S $ shaded as a rectangle. The axes are labeled $ u $ and $ v $, with the $ u $-axis ranging approximately from -6 to 6 and the $ v $-axis from 1 to 6. - The right graph illustrates the $ xy $-plane, where region $ R $ appears as a rotated rectangle bounded by the provided line equations. The axes are labeled $ x $ and $ y $, both ranging from about 0 to 6.### ObjectiveThe goal is to determine a set of equations that define the transformation $ T $, mapping $ S $ to $ R $. These equations need to account for the orientation and alignment differences between $ S $ and $ R $.### Constraints- The mathematical relationship must ensure that the sides of $ S $ remain parallel to the $ u $ and $ v $ axes during transformation.- The equations provided will be a comma-separated list, aligning the coordinates from the $ uv $-plane to the $ xy $-plane correctly.By understanding and applying transformations, we can map and manipulate geometrical regions effectively across coordinate systems.

A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes as shown in the figure below. (Enter your answers as a comma-separated list of equations.) R is bounded by y = 2x - 4, y = 2x + 4, y = 4 - x, y = 6 - x y WebAssign Plot 2 4 -6

A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes as shown in the figure below. (Enter your answers as a comma-separated list of equations.) R is bounded by y = 2x - 4, y = 2x + 4, y = 4 - x, y = 6 - x y WebAssign Plot 2 4 -6

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

## Transformation of Regions in the Plane

In this educational example, we explore a transformation from a region $ S $ in the $ uv $-plane to a region $ R $ in the $ xy $-plane. The aim is to find equations for a transformation $ T $ that maps the rectangular region $ S $, aligned with the $ u $- and $ v $-axes, onto the region $ R $.

### Description of Regions

- **Region $ R $**:
- Bounded by the lines $ y = 2x - 4 $, $ y = 2x + 4 $, $ y = 4 - x $, and $ y = 6 - x $.
- Graphically represented in the $ xy $-plane as a shaded quadrilateral oriented at an angle.

- **Region $ S $**:
- Shown as a rectangle in the $ uv $-plane.
- The sides of $ S $ are parallel to the $ u $- and $ v $-axes, making it easier to visualize its rectangular shape and orientation.

### Visual Explanation

- **Graphs**:
- The left graph represents the $ uv $-plane with region $ S $ shaded as a rectangle. The axes are labeled $ u $ and $ v $, with the $ u $-axis ranging approximately from -6 to 6 and the $ v $-axis from 1 to 6.
- The right graph illustrates the $ xy $-plane, where region $ R $ appears as a rotated rectangle bounded by the provided line equations. The axes are labeled $ x $ and $ y $, both ranging from about 0 to 6.

### Objective

The goal is to determine a set of equations that define the transformation $ T $, mapping $ S $ to $ R $. These equations need to account for the orientation and alignment differences between $ S $ and $ R $.

### Constraints

- The mathematical relationship must ensure that the sides of $ S $ remain parallel to the $ u $ and $ v $ axes during transformation.
- The equations provided will be a comma-separated list, aligning the coordinates from the $ uv $-plane to the $ xy $-plane correctly.

By understanding and applying transformations, we can map and manipulate geometrical regions effectively across coordinate systems.

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