1. S ECTION 9.2 Translations

2. In Lesson 4.7, you learned that a translation or slide is a transformation that moves all points of a figure the same distance in the same direction. Since vectors can be used to describe both distance and direction, vectors can be used to define translations.

3. Example 1: Draw the translation of the figure along the translation vector. Step 1Draw a line through each vertex parallel to vector. Step 2Measure the length of vector. Locate point G' by marking off this distance along the line through vertex G, starting at G and in the same direction as the vector. Step 3Repeat Step 2 to locate points H', I', and J' to form the translated image.

4. Recall that a vector in the coordinate plane can be written as, where a represents the horizontal change and b is the vertical change from the vector’s tip to its tail. is represented by the ordered pair. Written in this form, called the component form, a vector can be used to translate a figure in the coordinate plane.

5. Example 2: a) Graph ΔTUV with vertices T(–1, –4), U(6, 2), and V(5, –5) along the vector  –3, 2 . The vector indicates a translation 3 units left and 2 units up. (x, y)→(x – 3, y + 2) T(–1, –4)→(–4, –2) U(6, 2)→(3, 4) V(5, –5)→(2, –3)

6. Example 2: b) Graph pentagon PENTA with vertices P(1, 0), E(2, 2), N(4, 1), T(4, –1), and A(2, –2) along the vector  –5, –1 . The vector indicates a translation 5 units left and 1 unit down. (x, y)→(x – 5, y – 1) P(1, 0)→(–4, –1) E(2, 2)→(–3, 1) N(4, 1)→(–1, 0) T(4, –1)→(–1, –2) A(2, –2)→(–3, –3)

7. Example 3: The graph shows repeated translations that result in the animation of the raindrop. a) Describe the translation of the raindrop from position 2 to position 3 in function notation and in words. The raindrop in position 2 is (1, 2). In position 3, this point moves to (–1, –1). Use the translation function (x, y) → (x + a, y + b) to write and solve equations to find a and b. (1 + a, 2 + b) or (–1, –1) 1 + a = –1 2 + b = –1 a = –2 b = –3 Answer: function notation: (x, y) → (x – 2, y – 3). So, the raindrop is translated 2 units left and 3 units down from position 2 to 3.

8. Example 3: The graph shows repeated translations that result in the animation of the raindrop. b) Describe the translation of the raindrop from position 3 to position 4 using a translation vector. (–1 + a, –1 + b) or (–1, –4) –1 + a=–1–1 + b=–4 a=0b=–3 Answer: translation vector: