Chapter 7.8, Problem 15E

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 15E

Explanation of Solution

Given information: The given vertices are (0,0),(0,3),(3,0),(3,3) and horizontal stretch ,

k =2.

Calculation:

A square hasvertices at (0,0),(0,3),(3,0),(3,3). To find the image of the square after

horizontal stretch k =2, first write the vertices as column matrices . Then multiply each column matrix by the horizontal stretch transformation matrices.

The horizontal transformation matrices is,

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

  $\begin{array}{l}\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}0\\ 0\end{array}\right]=\left[\begin{array}{c}2\times 0+0\times 0\\ 0\times 0+1\times 0\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}0\\ 3\end{array}\right]=\left[\begin{array}{c}2\times 0+0\times 3\\ 0\times 0+1\times 3\end{array}\right]=\left[\begin{array}{c}0\\ 3\end{array}\right]\\ \left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}3\\ 0\end{array}\right]=\left[\begin{array}{c}2\times 3+0\times 0\\ 0\times 3+1\times 0\end{array}\right]=\left[\begin{array}{c}6\\ 0\end{array}\right],\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}3\\ 3\end{array}\right]=\left[\begin{array}{c}2\times 3+0\times 3\\ 0\times 3+1\times 3\end{array}\right]=\left[\begin{array}{c}6\\ 3\end{array}\right].\end{array}$

So, the vertices of the image are (0,0),(0,3),(6,0),(6,3).

Blow figure shows the square and its image.