3.1 Sequences of Transformations

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1 Name lass Date 3.1 Sequences of Transformations Essential Question: What happens when ou appl more than one transformation to a figure? Eplore ombining Rotations or Reflections transformation is a function that takes points on the plane and maps them to other points on the plane. Transformations can be applied one after the other in a sequence where ou use the image of the first transformation as the preimage for the net transformation. Resource Locker Find the image for each sequence of transformations. Using geometr software, draw a triangle and label the vertices,, and. Then draw a point outside the triangle and label it P. Rotate 3 around point P and label the image as. Then rotate 5 around point P and label the image as. Sketch our result. Make a conjecture regarding a single rotation that will map to. heck our conjecture, and describe what ou did. Houghton Mifflin Harcourt Publishing ompan Using geometr software, draw a triangle and label the vertices D, E, and F. Then draw two intersecting lines and label them j and k. Reflect DEF across line j and label the image as D E F. Then reflect D E F across line k and label the image as D E F. Sketch our result. onsider the relationship between DEF and D E F. Describe the single transformation that maps DEF to D E F. How can ou check that ou are correct? Module Lesson 1

2 Reflect 1. Repeat Step using other angle measures. Make a conjecture about what single transformation will describe a sequence of two rotations about the same center.. Make a conjecture about what single transformation will describe a sequence of three rotations about the same center. 3. Discussion Repeat Step, but make lines j and k parallel instead of intersecting. Make a conjecture about what single transformation will now map DEF to D E F. heck our conjecture and describe what ou did. Eplain 1 ombining Rigid Transformations In the Eplore, ou saw that sometimes ou can use a single transformation to describe the result of appling a sequence of two transformations. Now ou will appl sequences of rigid transformations that cannot be described b a single transformation. Eample 1 Draw the image of after the given combination of transformations. Reflection over line l then translation along v ν Step 1 Draw the image of after a reflection across line l. Label the image. ν l Step Translate along v. Label this image. ν l l Houghton Mifflin Harcourt Publishing ompan Module 3 11 Lesson 1

3 18 rotation around point P, then translation along v, then reflection across line l ppl the rotation. Label the image. ppl the translation to. Label the image. P l ppl the reflection to. Label the image. ν Reflect. re the images ou drew for each eample the same size and shape as the given preimage? In what was do rigid transformations change the preimage? 5. Does the order in which ou appl the transformations make a difference? Test our conjecture b performing the transformations in Part in a different order.. For Part, describe a sequence of transformations that will take back to the preimage. Houghton Mifflin Harcourt Publishing ompan Your Turn Draw the image of the triangle after the given combination of transformations. 7. Reflection across l then 9 rotation around 8. Translation along v then 18 rotation around point P point P then translation along u P l G E F P u v Module Lesson 1

4 Eplain ombining Nonrigid Transformations Eample Draw the image of the figure in the plane after the given combination of transformations. (, ) ( 3_, 3_ ) (-, ) ( + 1, - ) 1. The first transformation is a dilation b a factor of 3. ppl the dilation. Label the image D.. ppl the reflection of D across the -ais. Label this image D. 3. ppl the translation of D. Label this image ' ' 'D '. 8 D D D D 8 (, ) (3, ) ( 1_, - 1_ ) 1. The first transformation is a [horizontal/vertical] stretch b a factor of. ppl the stretch. Label the image.. The second transformation is a dilation b a factor of combined with a reflection. ppl the transformation to image.. Label the Reflect 9. If ou dilated a figure b a factor of, what transformation could ou use to return the figure back to its preimage? If ou dilated a figure b a factor of and then translated it right units, write a sequence of transformations to return the figure back to its preimage. 1. student is asked to reflect a figure across the -ais and then verticall stretch the figure b a factor of. Describe the effect on the coordinates. Then write one transformation using coordinate notation that combines these two transformations into one. - Houghton Mifflin Harcourt Publishing ompan Module Lesson 1

5 Your Turn Draw the image of the figure in the plane after the given combination of transformations. 11. (, ) ( - 1, - 1) (3, ) (-, -) 1. (, ) ( 3_, - ) ( - 5, + ) Eplain 3 Predicting the Effect of Transformations Eample 3 Predict the result of appling the sequence of transformations to the given figure. LMN is translated along the vector -, 3, reflected across the -ais, and then reflected across the -ais. L Houghton Mifflin Harcourt Publishing ompan Predict the effect of the first transformation: translation along the vector -, 3 will move the figure left units and up 3 units. Since the given triangle is in Quadrant II, the translation will move it further from the - and -aes. It will remain in Quadrant II. - Predict the effect of the second transformation: Since the triangle is in Quadrant II, a reflection across the -ais will change the orientation and move the triangle into Quadrant I. - M N - - Predict the effect of the third transformation: reflection across the -ais will again change the orientation and move the triangle into Quadrant IV. The two reflections are the equivalent of rotating the figure 18 about the origin. The final result will be a triangle the same shape and size as LMN in Quadrant IV. It has been rotated 18 about the origin and is farther from the aes than the preimage. Module Lesson 1

6 Square HIJK is rotated 9 clockwise about the origin and then dilated b a factor of, which maps (, ) (, ). Predict the effect of the first transformation: H I - K - J - Predict the effect of the second transformation: - The final result will be Your Turn Predict the result of appling the sequence of transformations to the given figure. 13. Rectangle GHJK is reflected across the -ais and translated along the vector 5,. K J TUV is horizontall stretched b a factor of 3, which maps (, ) ( 3, ), and then translated along the vector, U G H Houghton Mifflin Harcourt Publishing ompan T V - Module 3 1 Lesson 1

7 Elaborate 15. Discussion How man different sequences of rigid transformations do ou think ou can find to take a preimage back onto itself? Eplain our reasoning. 1. Is there a sequence of a rotation and a dilation that will result in an image that is the same size and position as the preimage? Eplain our reasoning. 17. Essential Question heck-in In a sequence of transformations, the order of the transformations can affect the final image. Describe a sequence of transformations where the order does not matter. Describe a sequence of transformations where the order does matter. Houghton Mifflin Harcourt Publishing ompan Module 3 11 Lesson 1

8 Evaluate: Homework and Practice Draw and label the final image of after the given sequence of transformations. 1. Reflect over the -ais and then translate b, -3. Online Homework Hints and Help Etra Practice. Rotate 9 degrees clockwise about the origin and then reflect over the -ais Translate b,, rotate 9 degrees counterclockwise around, and reflect over the -ais. 8. Reflect over the -ais, translate b -3, -1, and rotate 18 degrees around the origin Houghton Mifflin Harcourt Publishing ompan - Module 3 1 Lesson 1

9 Draw and label the final image of after the given sequence of transformations. 5. (, ) (, 1_ 3 ) (-, -). (, ) ( - 3_, _ 3 ) ( +, - ) ( _ 3, - 3_ ) Predict the result of appling the sequence of transformations to the given figure. Houghton Mifflin Harcourt Publishing ompan 7. is translated along the vector -3, -1, reflected across the -ais, and then reflected across the -ais is translated along the vector -1, -3, rotated 18 about the origin, and then dilated b a factor of Module 3 13 Lesson 1

10 In Eercises 9 1, use the diagram. Fill in the blank with the letter of the correct image described. 9. is the result of the sequence: G reflected over a vertical line and then a horizontal line. G 1. is the result of the sequence: D rotated 9 clockwise around one of its vertices and then reflected over a horizontal line. 11. is the result of the sequence: E translated and then rotated 9 counterclockwise. F 1. is the result of the sequence: D rotated 9 counterclockwise and then translated. E D hoose the correct word to complete a true statement. 13. combination of two rigid transformations on a preimage will alwas/sometimes/never produce the same image when taken in a different order. 1. double rotation can alwas/sometimes/never be written as a single rotation. 15. sequence of a translation and a reflection alwas/sometimes/never has a point that does not change position. 17. sequence of rigid transformations will alwas/sometimes/never result in an image that is the same size and orientation as the preimage. 1. sequence of a reflection across the -ais and then a reflection across the -ais alwas/sometimes/never results in a 18 rotation of the preimage. 18. sequence of a rotation and a dilation will alwas/sometimes/never result in an image that is the same size and orientation as the preimage. 19. QRS is the image of LMN under a sequence of transformations. an each of the following sequences be used to create the image, QRS, from the preimage, LMN? Select es or no. a. Reflect across the -ais and then Yes No translate along the vector,. b. Translate along the vector, Yes No and then reflect across the -ais. c. Rotate 9 clockwise about the Yes No origin, reflect across the -ais, and then rotate 9 counterclockwise about the origin. d. Rotate 18 about the origin, reflect Yes No across the -ais, and then translate along the vector,. S M L R - - Q N Houghton Mifflin Harcourt Publishing ompan Module 3 1 Lesson 1

11 . teacher gave students this puzzle: I had a triangle with verte at (1, ) and verte at (3, ). fter two rigid transformations, I had the image shown. Describe and show a sequence of transformations that will give this image from the preimage. H.O.T. Focus on Higher Order Thinking 1. nalze Relationships What two transformations would ou appl to to get DEF? How could ou epress these transformations with a single mapping rule in the form of (, ) (?,?)? - D - F E Houghton Mifflin Harcourt Publishing ompan Image redits: JP/ Shutterstock. Multi-Step Muralists will often make a scale drawing of an art piece before creating the large finished version. muralist has sketched an art piece on a sheet of paper that is 3 feet b feet. a. If the final mural will be 39 feet b 5 feet, what is the scale factor for this dilation? b. The owner of the wall has decided to onl give permission to paint on the lower half of the wall. an the muralist simpl use the transformation (, ) 1 (, ) in addition to the scale factor to alter the sketch for use in the allowed space? Eplain. 3. ommunicate Mathematical Ideas s a graded class activit, our teacher asks our class to reflect a triangle across the -ais and then across the -ais. Your classmate gets upset because he reversed the order of these reflections and thinks he will have to start over. What can ou sa to our classmate to help him? Module 3 15 Lesson 1

12 Lesson Performance Task The photograph shows an actual snowflake. Draw a detailed sketch of the arm of the snowflake located at the top left of the photo (1: on a clock face). Describe in as much detail as ou can an translations, reflections, or rotations that ou see. Then describe how the entire snowflake is constructed, based on what ou found in the design of one arm. Houghton Mifflin Harcourt Publishing ompan Image redits: Mark assino/superstock Module 3 1 Lesson 1