The transformation of the figure in the Solve It is a translation—moving a figure to a new location without changing its size, shape, or orientation. You can use matrix addition to translate all the vertices of a figure in one step.

Problem 1 Translating a Figure

Kite ABCD has vertices (5, 5), (3, 7), (1, 5), and (3, 1). If you translate it 8 units to the right and 5 units down, what are the coordinates of the vertices of its image A ' B ' C ' D '? Use matrix addition. Draw ABCD and its image.

Think

To translate 8 units to the right, what must you do to each x-coordinate?

Add 8 to each x-coordinate to translate 8 units to the right.

Preimage Vertices	Translation Matrix	Image Vertices
A B C D [ 5 3 1 3 5 7 5 1 ] table with 2 rows and 1 column , row1 column 1 , eh b c d , row2 column 1 , . matrix with 2 rows and 4 columns , row1 column 1 , 5 , column 2 3 , column 3 1 , column 4 3 , row2 column 1 , 5 , column 2 7 , column 3 5 , column 4 1 , end matrix , end table	Image Long Description	A ′ B ′ C ′ D ′ [ 13 11 9 11 0 2 0 − 4 ] table with 2 rows and 1 column , row1 column 1 , eh prime , b prime , c prime , d prime , row2 column 1 , . matrix with 2 rows and 4 columns , row1 column 1 , 13 , column 2 11 , column 3 9 , column 4 11 , row2 column 1 , 0 , column 2 2 , column 3 0 , column 4 negative 4 , end matrix , end table

The vertices of the preimage, A(5, 5), B(3, 7), C(1, 5), and D(3, 1) translate to the vertices A'(13, 0), B'(11, 2), C'(9, 0), and D' ( 11 , − 4 ) open 11 comma negative 4 close  of the image.

✓ Got It?

Image Long Description

1. 1. Reasoning How would you translate the kite image A'B'C'D' to the kite preimage ABCD?
   2. A pentagon has vertices ( 0 , − 5 ) , ( − 1 , − 1 ) , ( − 5 , 0 ) , open 0 comma negative 5 close comma open negative 1 comma negative 1 close comma open negative 5 comma 0 close comma  (1, 3), and (4, 0). Use matrix addition to translate the pentagon 3 units left and 2 units up. What are the vertices of the image? Graph the preimage and the image.

You enlarge or reduce a figure with a dilation. You use scalar multiplication to dilate a figure with center of dilation at the origin. In this book, dilations have their centers at the origin.

A B C 2 [ − 1 3 2 0 2 − 1 ] = A ′ B ′ C ′ [ − 2 6 4 0 4 − 2 ] table with 2 rows and 1 column , row1 column 1 , table with 1 row and 3 columns , row1 column 1 , eh , column 2 b , column 3 c , end table , row2 column 1 , 2 . matrix with 2 rows and 3 columns , row1 column 1 , negative 1 , column 2 3 , column 3 2 , row2 column 1 , 0 , column 2 2 , column 3 negative 1 , end matrix , end table . equals . table with 2 rows and 1 column , row1 column 1 , table with 1 row and 3 columns , row1 column 1 , eh prime , column 2 b prime , column 3 c prime , end table , row2 column 1 , . matrix with 2 rows and 3 columns , row1 column 1 , negative 2 , column 2 6 , column 3 4 , row2 column 1 , 0 , column 2 4 , column 3 negative 2 , end matrix , end table

Image Long Description

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