Chapter 12.4, Problem 12E

a.

To determine

To describe: The translation from figure 1 to figure 2 using coordinate notation.

Answer to Problem 12E

The coordination notation for the translation from figure 1 to figure 2 is:

$(x,y)\to (x+4,y)$

Explanation of Solution

Given information:

The following figures:

To describe the translation from figure 1 to figure 2, obtain the coordinates of figure 1 and figure 2 from the given graph. Let the sides of figure 1 be called $A,B,C,D,E,F$ . Therefore, coordinates of each vertex will be:

  $A(0,1),B(3,1),C(2,2),D(3,3),E(0,3),F(1,2)$

Let the sides of figure 2 be called $A\text{'},B\text{'},C\text{'},D\text{'},E\text{'},F\text{'}$ . Therefore, coordinates of each vertex will be:

  $A\text{'}(4,1),B\text{'}(7,1),C\text{'}(6,2),D\text{'}(7,3),E\text{'}(4,3),F\text{'}(5,2)$

It is observed that the $x$ coordinates increase by $4$ units while the $y$ coordinates remain the same. Therefore, the coordinate notation will be as follows:

$(x,y)\to (x+4,y)$

b.

To determine

To describe: The translation from figure 2 to figure 3 using coordinate notation.

Answer to Problem 12E

The coordination notation for the translation from figure 2 to figure 3 is:

$(x,y)\to (x+4,y)$

Explanation of Solution

Given information:

The following figures:

To describe the translation from figure 2 to figure 3, obtain the coordinates of figure 2 and figure 3 from the given graph. Let the sides of figure 2 be called $A\text{'},B\text{'},C\text{'},D\text{'},E\text{'},F\text{'}$ . Therefore, coordinates of each vertex will be:

  $A\text{'}(4,1),B\text{'}(7,1),C\text{'}(6,2),D\text{'}(7,3),E\text{'}(4,3),F\text{'}(5,2)$

Let the sides of figure 3 be called $A\text{'}\text{'},B\text{'}\text{'},C\text{'}\text{'},D\text{'}\text{'},E\text{'}\text{'},F\text{'}\text{'}$ . Therefore, coordinates of each vertex will be:

  $A\text{'}\text{'}(8,1),B\text{'}\text{'}(11,1),C\text{'}\text{'}(10,2),D\text{'}\text{'}(11,3),E\text{'}\text{'}(8,3),F\text{'}\text{'}(9,2)$

It is observed that the $x$ coordinates increase by $4$ units while the $y$ coordinates remain the same. Therefore, the coordinate notation will be as follows:

$(x,y)\to (x+4,y)$

c.

To determine

To draw: Figure 4 using translations from part a and b

Answer to Problem 12E

Figure 4 will be obtained as:

Explanation of Solution

Given information:

The following figures:

To draw figure 4, a coordinate notation is required. As a coordinate notation describing the translations in the given figures is obtained in subpart a and b, the same coordinate notation will be used here to make translation from figure 3 to figure 4. The coordinates of figure 3 are as follows:

$A\text{'}\text{'}(8,1),B\text{'}\text{'}(11,1),C\text{'}\text{'}(10,2),D\text{'}\text{'}(11,3),E\text{'}\text{'}(8,3),F\text{'}\text{'}(9,2)$

Let the coordinates of figure 4 be $A\text{'}\text{'}\text{'},B\text{'}\text{'}\text{'},C\text{'}\text{'}\text{'},D\text{'}\text{'}\text{'},E\text{'}\text{'}\text{'},F\text{'}\text{'}\text{'}$ .

On using the coordinate notation:

$(x,y)\to (x+4,y)$

The coordinates of figure 4 will be:

$A\text{'}\text{'}\text{'}(12,1),B\text{'}\text{'}\text{'}(15,1),C\text{'}\text{'}\text{'}(14,2),D\text{'}\text{'}\text{'}(15,3),E\text{'}\text{'}\text{'}(12,3),F\text{'}\text{'}\text{'}(13,2)$

The graph drawn using the above coordinates will be as follows: