Answered: Suppose S is the unit square in the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 19E

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Suppose S is the unit square in the first quadrant of the uv-plane.
Describe the image of the transformation T: x = 2u, y = 2v.

Expert Solution

Step 1

Given that $S$ is the unit square in the first quadrant of the $u v$ -plane.

We have to find the image of the transformation $T : x = 2 u, y = 2 v$ .

Step 2

Let us consider the unit square on $u v$ -plane,

$S = \{(u, v) | 0 \leq u \leq 1, 0 \leq v \leq 1\}$ .

The transformation $T : x = 2 u, y = 2 v$ .

$x = 2 u u = \frac{x}{2}$

The limit of $u$ transformed into $0 \leq \frac{x}{2} \leq 1$

$0 \leq x \leq 2$

And,

$y = 2 v v = \frac{y}{2}$

The limit of $v$ transformed into $0 \leq \frac{y}{2} \leq 1$

$0 \leq y \leq 2$

The image of the transformation $T$ becomes $i m a g e (T) = \{(x, y) | 0 \leq x \leq 2, 0 \leq y \leq 2\}$ .

The image of $T$ in the $x y$ -plane:

Therefore, the image of the transformation $T : x = 2 u, y = 2 v$ is the square with vertices

$(0, 0), (0, 2), (2, 0),$ and $(2, 2)$ .

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Suppose S is the unit square in the first quadrant of the uv-plane.Describe the image of the transformation T: x = 2u, y = 2v.