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Page 1 of 5 Transformations in the Coordinate Plane BEFORE Now WHY? You identified translations, You’ll graph transformations So you can describe movement reflections, and rotations. in a coordinate plane. in a cartoon, as in Ex. 17. In the Real World Word Watch Computer Graphics Computer Review Words and video game programmers use y transformations to create patterns 6 coordinate plane, p. 293 5 transformation, p. 511 and animations. How can you use 4 image, p. 511 coordinates to describe the 3 translation, p. 511 transformation shown? 2 reflection, p. 511 1 rotation, p. 511 The transformation is a translation. You can use coordinate notation O 1 2 3 4 5 6 7 8 x to describe a translation. An arrow is used in coordinate notation to signify “goes to.” Coordinates of Coordinates original point of image (x, y) → (x a, y b) The number a tells how many units The number b tells how many units to shift the figure to the left or right. to shift the figure up or down. EXAMPLE 1 Describing a Translation You can use coordinate notation to describe the translation shown above. Each point on the original figure is moved 4 units to the left and 3 units down. ANSWER In coordinate notation you write this translation as: (x, y) → (x ( 4), y ( 3)) or (x, y) → (x 4, y 3) Your turn now Describe the translation using coordinate notation. 1. A figure is moved 5 units to the right and 4 units up. 2. A figure is moved 7 units to the left and 2 units up. 518 Chapter 10 Geometric Figures Page 2 of 5 EXAMPLE 2 Translating a Figure with Reading Draw quadrilateral JKLM with vertices J( 5, 3), K( 4, 5), Each point on an image is L( 3, 3), and M( 4, 1). Then find the coordinates of labeled with a prime. The the vertices of the image after the translation notation J is read “J prime.” (x, y) → (x 6, y 2), and draw the image. For each vertex of the original figure, add 6 to the x-coordinate and subtract 2 from the y-coordinate. Original Image y 6 J( 5, 3) → J (1, 1) K 5 K( 4, 5) → K (2, 3) J L 4 K 3 L( 3, 3) → L (3, 1) 2 1 J L M( 4, 1) → M (2, 1) M 6 5 4 3 2 O 1 3 4 x Each point on the original M figure is translated 6 units 2 to the right and 2 units down. The graph shows both figures. Your turn now Complete the following exercise. 3. Draw ABC with vertices A( 4, 0), B(0, 4), and C(0, 0). Then find the coordinates of the vertices of the image after the translation (x, y) → (x 4, y 6), and draw the image. Reflections You can also use coordinate notation to describe reflections. y y 3 3 W( 3, 2) E ( 3, 2) E(3, 2) 2 2 X(2, 1) 1 1 5 4 3 2 O 1 2 3 x 5 4 3 O 1 3 x 2 X (2, 1) W ( 3, 2) F ( 1, 2) F(1, 2) 3 3 Reflection in the x-axis: Reflection in the y-axis: Multiply y-coordinate by 1. Multiply x-coordinate by 1. (x, y) → (x, 1 p y) or (x, y) → ( 1 p x, y) or (x, y) → (x, y) (x, y) → ( x, y) Lesson 10.8 Transformations in the Coordinate Plane 519 Page 3 of 5 EXAMPLE 3 Reflecting a Figure Draw parallelogram ABCD with vertices A( 3, 3), B(2, 3), C(4, 1), and D( 1, 1). Then find the coordinates of the vertices of the image after a reflection in the x-axis, and draw the image. For each vertex of the original figure, multiply the y-coordinate by 1. Original Image A y B A( 3, 3) → A ( 3, 3) 2 B(2, 3) → B (2, 3) D C 4 3 O 1 2 3 x C(4, 1) → C (4, 1) D C 2 D( 1, 1) → D ( 1, 1) A B The graph shows both figures. INTERNET Exercises eWorkbook Plus CLASSZONE.COM More Practice, p. 714 Getting Ready to Practice 1. Vocabulary Copy and complete: The figure that results from ? performing a transformation on an original figure is called the __. Describe the translation using coordinate notation. 2. y 3. y M 4 B B 3 2 L M A 1 N A 1 K L N 3 O 1 4 5 x 3 2 O 2 4 5 x C K C Draw GHJ with vertices G ( 4, 1), H( 2, 4), and J (0, 1). Then find the coordinates of the vertices of the image after the specified transformation, and draw the image. 4. (x, y) → (x 4, y 4) 5. (x, y) → (x 2, y 3) 6. Reflect GHJ in the x-axis. 7. Reflect GHJ in the y-axis. 520 Chapter 10 Geometric Figures Page 4 of 5 Practice and Problem Solving with Homework Describe the transformation using coordinate notation. Example Exercises 8. 9. y M y M B 1 8–9, 17 4 5 2 10–16, 18 3 B 2 C 3 13–16 A 2 N N Online Resources 4 O 1 4 x A C CLASSZONE.COM 4 D 1 2 3 4 x • More Examples 2 • eTutorial Plus 3 D 2 L L Draw the triangle with the given vertices. Then find the coordinates of the vertices of the image after the specified translation, and draw the image. 10. P(1, 1), Q(3, 5), R(5, 4); (x, y) → (x 2, y 4) Art 11. F( 2, 3), G(3, 3), H(3, 1); (x, y) → (x 3, y 6) 12. L( 6, 0), M( 6, 4), N( 3, 4); (x, y) → (x, y 5) Draw rectangle FGHJ with vertices F( 2, 3), G(3, 3), H(3, 1), and J ( 2, 1). Then find the coordinates of the vertices of the image after the specified transformation, and draw the image. 13. (x, y) → (x 3, y 6) 14. (x, y) → (x 7, y) 15. Reflect FGHJ in the x-axis. 16. Reflect FGHJ in the y-axis. 17. Motion Transformations can be used to show motion. Use coordinate notation to describe the transformation of the drummer from one picture to the next picture. y y y y 3 3 2 2 1 1 O 1 2 3 x O 1 2 3 x O 1 2 3 x O 1 2 3 x s Motion Photography can show 18. Explain Draw a rectangle with vertices K(0, 0), L( 3, 0), M( 3, 4), and motion by using repeated N(0, 4). Find the length, width, and area of the rectangle. Then translate flashes to record the rectangle 2 units to the right. Find the length, width, and area of the overlapping images. Capturing a drum stroke image. Explain your results. may require 200 flashes in 4 seconds. What is this 19. Critical Thinking Draw one triangle with vertices (2, 1), (3, 4), and rate as a unit rate? (4, 1) and a second triangle with vertices ( 4, 1), ( 3, 4), and ( 2, 1). Describe two different transformations that could move the first triangle onto the second. Lesson 10.8 Transformations in the Coordinate Plane 521 Page 5 of 5 Rotations In Exercises 20–23, the vertices of a triangle are given. Find the coordinates of the vertices of the image after a rotation of 90 clockwise about the origin. The coordinate notation (x, y) → (y, x) describes such a rotation. 20. F(0, 2), G( 3, 1), H( 1, 1) 21. L(2, 2), M(4, 1), N(2, 2) 22. R( 3, 3), S( 3, 0),T( 1, 0) 23. W(0, 0), X(5, 3), Y(3, 4) 24. Critical Thinking The vertices of quadrilateral ABCD are A( 3, 4), B(2, 4), C(3, 2), and D( 4, 1). After a translation, the coordinates of A are ( 5, 1). Describe the translation using coordinate notation. Then find the coordinates of B , C , and D . 25. Challenge Line segment EF has endpoints E(4, 3) and F(4, 3). Its image after a 180 rotation has endpoints E ( 4, 3) and F ( 4, 3). Describe a 180 rotation using coordinate notation. Mixed Review Solve the equation. Check your solution. (Lesson 7.5) 26. 3x 5 23 27. 8 4x 20 28. 35 3 16x 29. Choose a Strategy Use a strategy from Problem Solving Strategies the list to solve the following problem. Explain your choice of strategy. s Guess, Check, and Revise s Write an Equation You paid three times as much for lunch s Look for a Pattern as you paid for a snack. Altogether, you spent $5. How much did you pay for the snack? for lunch? Basic Skills Copy and complete the statement. 30. 0.53 kg ? __ g 31. 8.9 m __ cm ? 32. 62 mL ? __ L Test-Taking Practice INTERNET 33. Extended Response Describe a y State Test Practice 4 combination of two transformations CLASSZONE.COM that would move figure A to figure B. 3 A 2 Then describe a combination of two 1 transformations that would move figure B to figure A. 4 3 2 O 1 2 3 4 x B 522 Chapter 10 Geometric Figures

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