Translations Draw translated images using repeated reflections. Draw translated images using coordinates. Draw translated images using repeated reflections. The movement of each band member as they progress through the show are examples of translations.

TRANSLATIONS USING COORDINATES A translation is a transformation that moves all points of a figure the same distance in the same direction. One way to symbolize the translation is to write (x, y) → (x + a, y + b).

TRANSLATIONS USING COORDINATES A translation is a transformation that moves all points of a figure the same distance in the same direction. One way to symbolize the translation is to write (x, y) → (x + a, y + b). 2 1.5 1 Example: (x, y) → (x - 5, y - 3) 0.5 -5 -4 -3 -2 -1 1 2 3 4 5 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4

Example 1 Translations in the coordinate plane Rectangle PQRS has vertices P(-3, 5), Q(-4, 2), R(3, 0), S(4, 3). Translate the image (x, y) → (x + 8, y - 5). The translation moved every point of the preimage 8 units right and five units down. 6 5 4 3 2 1 -4 -2 2 4 6 8 10 12 -1 -2 -3 -4 -5 -6

Example 2 Repeated Translations 5 4 Repeated translation results in animation of the star. 3 2 1 -6 -4 -2 2 4 6 8 -1 -2 -3 -4 -5 -6

Example 3 Repeated Translations (-5, 5) (-3, 5) (-1, 5) (1, 5) 5 1 2 3 4 4 (3, 3) Repeated translation results in animation of the star. 3 5 2 (5, 1) 1 6 (7, -1) -6 -4 -2 2 4 6 8 -1 7 -2 -3 -4 -5 -6

TRANSLATIONS BY REPEATED REFLECTIONS Another way to find a translation is to perform a reflection in the first of two parallel lines and then reflect the image in the other parallel line.

Example 4 Find a Translation Using Reflections Since translations are compositions of two reflections, all translations are isometries.