Chapter 2.4, Problem 34E

a.

To determine

To find: the vertex matrix for the given graph and make conjecture about the resulting figure if it is multiplied by the given vertex matrix.

Answer to Problem 34E

Vertex matrix = $\left[\begin{array}{cccc}\begin{array}{l}-1\\ -1\end{array}& \begin{array}{l}1\\ 2\end{array}& \begin{array}{l}4\\ 0\end{array}& \begin{array}{l}2\\ -3\end{array}\end{array}\right]$

Conjecture: The figure would be enlarged but not proportional because of 3 and 2 in the given matrix.

Explanation of Solution

Given information:

The given figure is as follows:

  

Multiply the vertex matrix by $\left[\begin{array}{cc}\begin{array}{l}3\\ 0\end{array}& \begin{array}{l}0\\ 2\end{array}\end{array}\right]$ .

Calculation:

The vertex matrix is $\left[\begin{array}{cccc}\begin{array}{l}-1\\ -1\end{array}& \begin{array}{l}1\\ 2\end{array}& \begin{array}{l}4\\ 0\end{array}& \begin{array}{l}2\\ -3\end{array}\end{array}\right]$ . If the vertex matrix will be multiplied by the given matrix, then the figure would be enlarged but not proportional.

b.

To determine

To graph: the resulting vertex matrix at the given situation.

Answer to Problem 34E

  

Explanation of Solution

Given information:

The given figure is as follows:

  

The given situation is that multiply the vertex matrix by $\left[\begin{array}{cc}\begin{array}{l}3\\ 0\end{array}& \begin{array}{l}0\\ 2\end{array}\end{array}\right]$ and then graphs the resulting vertex matrix on the grid paper.

Vertex matrix = $\left[\begin{array}{cccc}\begin{array}{l}-1\\ -1\end{array}& \begin{array}{l}1\\ 2\end{array}& \begin{array}{l}4\\ 0\end{array}& \begin{array}{l}2\\ -3\end{array}\end{array}\right]$

Calculation:

Multiplication of the vertex matrix by $\left[\begin{array}{cc}\begin{array}{l}3\\ 0\end{array}& \begin{array}{l}0\\ 2\end{array}\end{array}\right]$ as follows:

  $\left[\begin{array}{cc}\begin{array}{l}3\\ 0\end{array}& \begin{array}{l}0\\ 2\end{array}\end{array}\right]\cdot \left[\begin{array}{cccc}\begin{array}{l}-1\\ -1\end{array}& \begin{array}{l}1\\ 2\end{array}& \begin{array}{l}4\\ 0\end{array}& \begin{array}{l}2\\ -3\end{array}\end{array}\right]=\left[\begin{array}{cccc}\begin{array}{l}-3\\ -2\end{array}& \begin{array}{l}3\\ 4\end{array}& \begin{array}{l}12\\ 0\end{array}& \begin{array}{l}6\\ -6\end{array}\end{array}\right]$

Now, the graph for vertex matrix (black color) and resulting vertex matrix after multiplication (red color) is as follows:

  

c.

To determine

To compare: the result with your conjecture, and find why this is often called a shear.

Answer to Problem 34E

It looks as if it's growing, but no proportionally.

Explanation of Solution

Given information:

The given figure is as follows:

  

Calculation:

There is no proportionally, but it appears as if it's growing.