Chapter 7.8, Problem 18E

To determine

To find:the vertices of image of a square with the given vertices after the given transformation by using matrices. Then sketch the square and its image.

Answer to Problem 18E

Explanation of Solution

Given information: The given vertices are (1,1),(3,2),(0,3),(2,4) and vertical shrink

  $k=\frac{1}{2}$ .

Calculation:

A square hasvertices (1,1),(3,2),(0,3),(2,4) .To find the image of the square after

vertical shrink $k=\frac{1}{2}$ , first write the vertices as column matrices . Then multiply each

column matrix by the horizontal stretch transformation matrices

The vertical transformation matrices is,

  $\left[\begin{array}{cc}1& 0\\ 0& \frac{1}{2}\end{array}\right]$ and vertices as column matrices are $\left[\begin{array}{c}1\\ 1\end{array}\right],\left[\begin{array}{c}3\\ 2\end{array}\right],\left[\begin{array}{c}0\\ 3\end{array}\right],\left[\begin{array}{c}2\\ 4\end{array}\right]$ .

  $\begin{array}{l}\left[\begin{array}{cc}1& 0\\ 0& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}1\times 1+0\times 1\\ 0\times 1+1\times \frac{1}{2}\end{array}\right]=\left[\begin{array}{c}1\\ \frac{1}{2}\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}3\\ 2\end{array}\right]=\left[\begin{array}{c}1\times 3+0\times 2\\ 0\times 3+\frac{1}{2}\times 2\end{array}\right]=\left[\begin{array}{c}3\\ 1\end{array}\right]\\ \left[\begin{array}{cc}1& 0\\ 0& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}0\\ 3\end{array}\right]=\left[\begin{array}{c}1\times 0+0\times 3\\ 0\times 0+\frac{1}{2}\times 3\end{array}\right]=\left[\begin{array}{c}0\\ \frac{3}{2}\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}2\\ 4\end{array}\right]=\left[\begin{array}{c}1\times 2+0\times 4\\ 0\times 2+\frac{1}{2}\times 4\end{array}\right]=\left[\begin{array}{c}2\\ 2\end{array}\right].\end{array}$

So, the vertices of the image are (1, $\frac{1}{2}$ ),(3,1),(0, $\frac{3}{2}$ ),(2,2).

Blow figure shows the square and its image.