Example 1

1. Find the image of D ( 1 , − 2 ) d open 1 comma negative 2 close  under the translation described by the vector 〈 1 , 4 〉 . left pointing angle bracket , 1 comma 4 , right pointing angle bracket . .

   

   〈 1 , 4 〉 left pointing angle bracket , 1 comma 4 , right pointing angle bracket  is a translation 1 unit right and 4 units up, the same as ( x , y ) → ( x + 1 , y + 4 ) . open x comma y close rightwards arrow open x plus 1 comma y plus 4 close .

   D ( 1 , − 2 ) d open 1 comma negative 2 close  translates to ( 1 + 1 , − 2 + 4 ) , open 1 plus 1 comma negative 2 plus 4 close comma  or (2, 2).

2. Find the image of D ( 1 , − 2 ) d open 1 comma negative 2 close  under the composition of translations 〈 1 , 4 〉 left pointing angle bracket , 1 comma 4 , right pointing angle bracket  followed by 〈 − 4 , − 1 〉 . left pointing angle bracket . negative 4 comma negative 1 . right pointing angle bracket . .

   Use vector addition: 〈 1 , 4 〉 + 〈 − 4 , − 1 〉 = 〈 1 + ( − 4 ) , 4 + ( − 1 ) 〉 = 〈 − 3 , 3 〉 . left pointing angle bracket , 1 comma 4 , right pointing angle bracket . plus . left pointing angle bracket . negative 4 comma negative 1 . right pointing angle bracket . equals . left pointing angle bracket . 1 plus open negative 4 close comma 4 plus open negative 1 close . right pointing angle bracket . equals . left pointing angle bracket , negative 3 comma 3 , right pointing angle bracket . .

   D ( 1 , − 2 ) d open 1 comma negative 2 close  translates to ( 1 − 3 , − 2 + 3 ) , or  ( − 2 , 1 ) . open 1 minus 3 comma negative 2 plus 3 close comma or open negative 2 comma 1 close .

Example 2

Use matrices to find the image of Δ M F H cap delta m f h  under the translation 〈 4 , − 5 〉 . left pointing angle bracket , 4 comma negative 5 , right pointing angle bracket . .

x - coordinate y - coordinate M F H [ − 1 2 3 − 1 − 5 2 ] Write a matrix for  Δ M F H made up of the three vertices .   [ 4 4 4 − 5 − 5 − 5 ] Write the translation matrix with the translation vector three times . [ − 1 2 3 − 1 − 5 2 ] + [ 4 4 4 − 5 − 5 − 5 ] = [ 3 6 7 − 6 − 10 − 3 ]  Add the matrices . table with 2 rows and 1 column , row1 column 1 , table with 2 rows and 3 columns , row1 column 1 , table with 2 rows and 1 column , row1 column 1 , x minus . coordinate , row2 column 1 , y minus . coordinate , end table , column 2 table with 2 rows and 1 column , row1 column 1 , m f h , row2 column 1 , . matrix with 2 rows and 3 columns , row1 column 1 , negative 1 , column 2 2 , column 3 3 , row2 column 1 , negative 1 , column 2 negative 5 , column 3 2 , end matrix , end table , column 3 table with 2 rows and 1 column , row1 column 1 , cap writeamatrixfor . cap delta m f h , row2 column 1 , madeupofthethreevertices . . , end table , row2 column 1 , , column 2 . matrix with 2 rows and 3 columns , row1 column 1 , 4 , column 2 4 , column 3 4 , row2 column 1 , negative 5 , column 2 negative 5 , column 3 negative 5 , end matrix , column 3 table with 2 rows and 1 column , row1 column 1 , cap writethetranslationmatrixwith , row2 column 1 , thetranslationvectorthreetimes . . , end table , end table , row2 column 1 , . matrix with 2 rows and 3 columns , row1 column 1 , negative 1 , column 2 2 , column 3 3 , row2 column 1 , negative 1 , column 2 negative 5 , column 3 2 , end matrix . plus . matrix with 2 rows and 3 columns , row1 column 1 , 4 , column 2 4 , column 3 4 , row2 column 1 , negative 5 , column 2 negative 5 , column 3 negative 5 , end matrix . equals . matrix with 2 rows and 3 columns , row1 column 1 , 3 , column 2 6 , column 3 7 , row2 column 1 , negative 6 , column 2 negative 10 , column 3 negative 3 , end matrix . cap addthematrices . . , end table

The image of Δ M F H cap delta m f h  is Δ M ′ F ′ H ′ cap delta , m prime , f prime , h prime  with M ′ ( 3 , − 6 ) , F ′ ( 6 , − 10 ) , and  H ′ ( 7 , − 3 ) . m prime , open 3 comma negative 6 close comma , f prime , open 6 comma negative 10 close comma , and , h prime , open 7 comma negative 3 close .