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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1 PRECALCULUS AND ADVANCED TOPICS Lesson 24: Matrix Notation Encompasses New Transformations! Classwork Example 1 Determine the following: a. ( 1 0 3 )( ) 0 1 −2 b. ( 1 0 −7 )( ) 0 1 12 c. ( 1 0 3 )( 0 1 −2 5 ) 1 d. ( 1 0 −1 )( 0 1 −7 −3 ) 6 e. ( 9 12 1 )( −3 −1 0 f. ( 1 0 )( 0 1 ) g. ( 0 ) 1 1 )( 0 0 ) 1 Lesson 24: Matrix Notation Encompasses New Transformations! This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M1-TE-1.3.0-07.2015 S.121 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1 PRECALCULUS AND ADVANCED TOPICS Example 2 Can the reflection about the real axis () = ̅ be expressed in matrix notation? Exercises 1. Express a reflection about the vertical axis in matrix notation. Prove that it produces the desired reflection by using matrix multiplication. 2. Express a reflection about both the horizontal and vertical axes in matrix notation. Prove that it produces the desired reflection by using matrix multiplication. 3. Express a reflection about the vertical axis and a dilation with a scale factor of 6 in matrix notation. Prove that it produces the desired reflection by using matrix multiplication. Lesson 24: Matrix Notation Encompasses New Transformations! This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M1-TE-1.3.0-07.2015 S.122 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 24 M1 PRECALCULUS AND ADVANCED TOPICS Explore the transformation given by each matrix below. Use the graph of the rectangle provided to assist in the exploration. Describe the effect on the graph of the rectangle, and then show the general effect of the transformation by using matrix multiplication. Matrix 4. ( 1 0 0 ) 1 5. ( 0 1 1 ) 0 6. ( 0 0 0 ) 0 7. ( 1 0 1 ) 1 8. ( 1 1 0 ) 1 Transformation of the Rectangle Lesson 24: Matrix Notation Encompasses New Transformations! This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M1-TE-1.3.0-07.2015 General Effect of the Matrix S.123 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 PRECALCULUS AND ADVANCED TOPICS Lesson Summary 11 All matrices in the form ( 21 12 22 ) correspond to a transformation of some kind. 1 0 ) reflects all coordinates about the horizontal axis. 0 −1 −1 0 The matrix ( ) reflects all coordinates about the vertical axis. 0 1 1 0 The matrix ( ) is the identity matrix and corresponds to a transformation that leaves points alone. 0 1 0 0 The matrix ( ) is the zero matrix and corresponds to a dilation of scale factor 0. 0 0 The matrix ( Problem Set 1. What matrix do you need to use to reflect the following points about the -axis? What is the resulting matrix when this is done? Show all work and sketch it. 3 3 b. ( ) a. ( ) 2 0 −4 −3 c. ( ) d. ( ) 3 −2 3 5 e. ( ) f. ( ) −3 4 2. What matrix do you need to use to reflect the following points about the -axis? What is the resulting matrix when this is done? Show all work and sketch it. 0 2 a. ( ) b. ( ) 3 3 −2 −3 c. ( ) d. ( ) 3 −3 −3 3 f. ( ) e. ( ) 4 −3 3. What matrix do you need to use to dilate the following points by a given factor? What is the resulting matrix when this is done? Show all work and sketch it. 3 1 b. ( ), a factor of 2 a. ( ), a factor of 3 2 0 1 1 −4 c. ( ), a factor of 1 d. ( ), a factor of 2 −2 −6 e. 1 9 ( ), a factor of 3 3 Lesson 24: f. ( √3 ), a factor of √2 √11 Matrix Notation Encompasses New Transformations! This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M1-TE-1.3.0-07.2015 S.124 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 PRECALCULUS AND ADVANCED TOPICS 4. What matrix will rotate the given point by the angle? What is the resulting matrix when this is done? Show all work and sketch it. 1 1 a. ( ), radians b. ( ), radians 0 2 0 3 1 1 c. ( ), radians d. ( ), radians 0 6 0 4 √3 e. √2 6 ( 2 ), radians 1 2 f. √2 2 √3 g. 4 ( 2 ), radians ( 2 ), radians h. 1 2 1 ( ), − radians 6 0 5. For the transformation shown below, find the matrix that will transform point to ’, and verify your answer. 6. In this lesson, we learned [ 7. Describe the transformation and the translations in the diagram below. Write the matrices that will perform the tasks. What is the area that these transformations and translations have enclosed? 0 1 ] will produce a reflection about the line = . What matrix will produce a 1 0 3 reflection about the line = −? Verify your answers by testing the given point ( ) and graphing them on the 1 coordinate plane. Lesson 24: Matrix Notation Encompasses New Transformations! This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M1-TE-1.3.0-07.2015 S.125 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 24 NYS COMMON CORE MATHEMATICS CURRICULUM M1 PRECALCULUS AND ADVANCED TOPICS 8. 9. Given the kite figure below, answer the following questions. a. Explain how you would create the star figure above using only rotations. b. Explain how to create the star figure above using reflections and rotation. c. Explain how to create the star figure above using only reflections. Explain your answer. Given the rectangle below, answer the following questions. a. Can you transform the rectangle ′′′′ above using only rotations? Explain your answer. b. Describe a way to create the rectangle ′′′′. c. Can you make the rectangle ′′′′ above using only reflections? Explain your answer. Lesson 24: Matrix Notation Encompasses New Transformations! This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M1-TE-1.3.0-07.2015 S.126 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.