Chapter 12.4, Problem 21E

To determine

To calculate: Whether the final image is the same after switching the order of translation.

Answer to Problem 21E

The final image will be the same after switching the order of translation.

Explanation of Solution

Given information:

The following coordinate notations:

$(x,y)\to (x-3,y-4)$

$(x,y)\to (x+2,y-6)$

To determine whether the final image is the same after switching the order of translation, assume a figure $ABC$ with the following coordinates:

$A(3,3),B(4,5),C(5,3)$

After translation using the first coordinate notation, $(x,y)\to (x-3,y-4)$ , the coordinates of the image will be:

$A\text{'}(0,-1),B\text{'}(1,1),C\text{'}(2,-1)$

After translating using the second coordinate notation, $(x,y)\to (x+2,y-6)$ , the coordinates of the final image will be:

$A\text{'}\text{'}(2,-7),B\text{'}\text{'}(3,-5),C\text{'}\text{'}(4,-7)$

A graph representing the translations will be as follows:

Now, reverse the order of translation.

After translating using the second coordinate notation, $(x,y)\to (x+2,y-6)$ , the coordinates of the image will be:

$A\text{'}(5,-3),B\text{'}(6,-1),C\text{'}(7,-3)$

After translation using the first coordinate notation, $(x,y)\to (x-3,y-4)$ , the coordinates of the final image will be:

$A\text{'}\text{'}(2,-7),B\text{'}\text{'}(3,-5),C\text{'}\text{'}(4,-7)$

A graph representing the translations will be as follows:

Conclusion: It is observed that the coordinates of the final image is same in both cases, therefore, on switching the order of translation, the final image remains same.