DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A Name Class 1.3 Date Representing and Describing Transformations Essential Question: How can you describe transformations in the coordinate plane using algebraic representations and using words? Resource Locker Performing Transformations Using Coordinate Notation Explore A transformation is a function that changes the position, shape, and/or size of a figure. The inputs of the function are points in the plane; the outputs are other points in the plane. A figure that is used as the input of a transformation is the preimage. The output is the image. Translations, reflections, and rotations are three types of transformations. Thedecorative tiles shown illustrate all three types of transformations. You can use prime notation to name the image of a point. In the diagram, the transformation T moves point A to point A′ (read “A prime”). You can use function notation to writeT( A ) = A′. Note that a transformation is sometimes called a mapping. Transformation T maps A to A′. A' © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Antony McAulay/Shutterstock A Image T Preimage Coordinate notation is one way to write a rule for a transformation on a coordinate plane. The notation uses an arrow to show how the transformation changes the coordinates of a general point,( x, y ). Find the unknown coordinates for each transformation and draw the image. Then complete the description of the transformation and compare the image to its preimage. (x, y) → (x - 4, y - 3) Rule Preimage (x, y) A(0, 4) → (x, y) → (x - 4, y - 3) A′(0 − 4, 4 − 3) 5 Image (x - 4, y - 3) = B(3, 0) → B′(3 − 4, 0 − 3) = C( 0, 0 ) → C′( 0 − 4, 0 − 3 ) = A′( −4, 1 ) ⎛ ⎞ B′ ⎜ -1 , -3 ⎟ ⎝ ⎠ ⎛ ⎞ C′ ⎜ -4 , -3 ⎟ ⎝ ⎠ y A A' -5 C' The transformation is a translation 4 units (left/right) x 0 C B 5 B' -5 and 3 units (up/down). A comparison of the image to its preimage shows that Possible answer: the image is the same size and shape as the preimage Module1 GE_MNLESE385795_U1M01L3.indd 31 31 . Lesson3 3/20/14 3:18 PM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A B DO NOT Correcti (x, y) → (-x, y) Rule Preimage (x, y) Image (-x, y) (x, y) → (-x, y) R′(-(−4), 3) = → S′(-(−1), 3) = → T′( -(−4), 1 ) = R(-4, 3) → S(-1, 3) T(-4, 1) ⎛ ⎞ R′⎜ 4 , 3 ⎟ ⎝ ⎠ ⎛ ⎞ S′⎜ 1 , 3 ⎟ ⎝ ⎠ ⎛ ⎞ T′⎜ 4 , 1 ⎟ ⎝ ⎠ 5 R -5 y S S' R' 0 T' x 5 T -5 The transformation is a reflection across the (x-axis/y-axis). A comparison of the image to its preimage shows that Possible answer: the image is the same size and shape as the preimage, but it is flipped over the y-axis C . (x, y) → (2x, y) Preimage (x, y) ⎛ ⎞ J ⎜ -1 , 2 ⎟ → ⎝ ⎠ ⎛ ⎞ K⎜ 2 , 2 ⎟ → ⎝ ⎠ ⎛ ⎞ L ⎜ 2 , -4 ⎟ → ⎝ ⎠ Rule (x, y) → (2x, y) ⎛ ⎞ J ′ ⎜2 ⋅ -1 , 2 ⎟ ⎝ ⎠ ⎛ ⎞ K′ ⎜2 ⋅ 2 , 2 ⎟ ⎝ ⎠ ⎛ ⎞ L′ ⎜2 ⋅ 2 , -4 ⎟ ⎝ ⎠ Image (2x, y) ⎛ ⎞ = J ′ ⎜ -2 , 2 ⎟ ⎝ ⎠ ⎛ ⎞ = K′ ⎜ 4 , 2 ⎟ ⎝ ⎠ ⎛ ⎞ = L′ ⎜ 4 , -4 ⎟ ⎝ ⎠ 5 J' J y K K' x -5 The transformation is a (horizontal/vertical) stretch by a factor of 2 . 0 -5 5 L L' Possible answer: the image and the preimage are both right triangles, but they do not have the same size or shape . Reflect 1. Discussion How are the transformations in Steps A and B different from the transformation in Step C? The transformations in Steps A and B preserve the size and shape of the right triangle. The transformation in Step C changes the shape of the right triangle. 2. © Houghton Mifflin Harcourt Publishing Company A comparison of the image to its preimage shows that For each transformation, what rule could you use to map the image back to the preimage? A. (x, y) → (x + 4, y + 3); B. (x, y) → (-x, y); C. (x, y) → (0.5x, y) Module 1 GE_MNLESE385795_U1M01L3.indd 32 32 Lesson 3 3/20/14 3:17 PM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A Describing Rigid Motions Using Coordinate Notation Explain 1 Some transformations preserve length and angle measure, and some do not. A rigid motion (or isometry) is a transformation that changes the position of a figure without changing the size or shape of the figure. Translations, reflections, and rotations are rigid motions. Properties of Rigid Motions • Rigid motions preserve distance. • Rigid motions preserve collinearity. • Rigid motions preserve angle measure. • Rigid motions preserve parallelism. • Rigid motions preserve betweenness. If a figure is determined by certain points, then its image after a rigid motion is determined by the images of those points. This is true because of the betweenness and collinearity properties of rigid motions. For example, suppose △ABC is determined by its vertices, points A, B, and C. You can find the image of △ABC by finding the images of points A, B, and C and connecting them with segments. Example 1 Use coordinate notation to write the rule that maps each preimage to its image. Then identify the transformation and confirm that it preserves length and angle measure. B' Preimage A(1, 2) B(4, 2) C(3, −2) → → → Image A′(−2, 1) B′(−2, 4) C′(2, 3) -5 © Houghton Mifflin Harcourt Publishing Company The x-coordinate of each image point is the opposite of the y-coordinate of its preimage. The y-coordinate of each image point equals the x-coordinate of its preimage. √(3 − 4) 2 + (-2 − 2) 2 2 Since A' B 0 x 5 C -5 △ ABC and △A′B′C′. Use the Distance Formula as needed. ―――――――― = √― 17 ―――――――― AC = √(3 − 1) + (-2 − 2) = √― 20 BC = C' ° counterclockwise around the origin given by The transformation is a rotation of 90 the rule (x, y) → (−y, x). AB = 3 y A Look for a pattern in the coordinates. Find the length of each side of 5 2 A′ B′ = 3 B′ C′ = ―――――――― + (3 − 4) √(2 − (-2)) 2 2 _ = √17 ____ A′ C′ = √ (2 − (-2)) + (3 − 1) 2 2 _ = √20 AB = A′ B′ , BC = B′ C′ , and AC = A′ C′ , the transformation preserves length. Find the measure of each angle of △ABC and △A′B′C′. Use a protractor. m ∠A = 63°, m∠B = 76°, m∠C = 41° m ∠A′ = 63°, m∠B′ = 76°, m∠C′ = 41° Since m ∠A = m∠A′ , m∠B = m∠B′ , and m∠C = m∠C′ , the transformation preserves angle measure. Module1 GE_MNLESE385795_U1M01L3.indd 33 33 Lesson3 3/20/14 3:17 PM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A B Preimage P(-3, -1) → Q(3, -1) → R(1, −4) → DO NOT Correcti Image P′(−3, 1) Q′(3, 1) R′(1, 4) 5 Look for a pattern in the coordinates. -5 The x-coordinate of each image point equals the x-coordinate of its preimage. P' P y R' Q' Q 0 The y-coordinate of each image point is the opposite of the y-coordinate of its preimage. -5 x 5 R reflection across the x-axis The transformation is a x, y) → (x, -y) . given by the rule ( Find the length of each side of △PQR and △P′Q′R′. P′ Q′ = 6 PQ = 6 QR = ―――――――――― √( 1− 3 ) ( 2 + −4 − -1 ) 2 Q′ R′ = ―― 13 =√ ―――――――――― √(1 − -3 ) + (−4 − -1 ) ―― = √ 25 = 5 PR = 2 ―――――――――― √(1 − 3 ) + (4 − 1 ) ―― 2 2 √ 13 ―――――――――― P′ R′ = √( 1 − -3 ) + ( 4 − 1 ) ―― = √ 25 = 5 = 2 2 2 Since PQ = P' Q' , QR = Q' R' , and PR = P' R' , the transformation preserves length. Find the measure of each angle of △PQR and △P′Q′R′. Use a protractor. m∠P ' = 37° , m∠Q' = 56° , m∠R' = 87° Since m∠P = m∠P' , m∠Q = m∠Q' , and m∠R = m∠R' , the transformation preserves angle measure. Reflect 3. How could you use a compass to test whether corresponding lengths in a preimage and image are the same? Place the point of the compass on one endpoint of the segment in the preimage and open it to the length of the segment. Without adjusting the compass, move the point of the compass to an endpoint of the corresponding segment in the image and make an arc. If the lengths © Houghton Mifflin Harcourt Publishing Company m∠P = 37° , m∠Q = 56° , m∠R = 87° are the same, the arc will pass through the other endpoint of the segment in the image. 4. Look back at the transformations in the Explore. Classify each transformation as a rigid motion or not a rigid motion. A. rigid motion; B. rigid motion; C. not a rigid motion Module 1 GE_MNLESE385795_U1M01L3.indd 34 34 Lesson 3 3/20/14 3:17 PM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A Your Turn Use coordinate notation to write the rule that maps each preimage to its image. Then identify the transformation and confirm that it preserves length and angle measure. 5. Preimage D(-4, 4) E(2, 4) F(−4, 1) Image D′(4, -4) E′(-2, -4) F′(4, -1) → → → Each coordinate maps to its opposite. The transformation is a rotation of 180° around -5 the origin given by the rule (x, y) → (−x, -y). m∠D = m∠D′ = 90° DE = D′ E′ = 6 ― EF = E′ F′ = √45 DF = D'F' = 3 5 D y E x F 0 m∠E = m∠E′ = 27° 5 F' E' -5 m∠F = m∠F′ = 63° D' The transformation preserves length and angle measure. 6. Preimage S(-3, 4) T(2, 4) U(−2, 0) Image S′(-2, 2) T′(3, 2) U′(-1, -2) → → → 5 S S' y T T' x-coordinates: image is 1 more than preimage y-coordinates: image is 2 less than preimage The transformation is a translation given by the rule (x, y) → (x + 1, y - 2). © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Mary Hockenbery/Flickr/Getty Images ST = S′ T′ = 5 TU = T′U′ = √32 ― SU = S′ U′ = √― 17 x -5 m∠S = m∠S′ = 76° m∠T = m∠T′ = 45° U 0 5 U' -5 m∠U = m∠U′ = 59° The transformation preserves length and angle measure. Explain 2 Describing Nonrigid Motions Using Coordinate Notation Transformations that stretch or compress figures are not rigid motions because they do not preserve distance. The view in the fun house mirror is an example of a vertical stretch. Module 1 GE_MNLESE385795_U1M01L3.indd 35 35 Lesson 3 3/20/14 3:17 PM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A DO NOT Correcti Example 2 Use coordinate notation to write the rule that maps each preimage to its image. Then confirm that the transformation is not a rigid motion. △JKL maps to triangle △J′K′L′. Preimage Image J(4, 1) → L(0, -3) → K(-2, -1) → J′(4, 3) K′(-2, -3) L′(0, -9) Look for a pattern in the coordinates. x-coordinate of each image point equals the x-coordinate of its preimage. The y-coordinate of each image point is 3 times the y-coordinate of its preimage. The transformation is given by the rule (x, y) → (x, 3y). The Compare the length of a segment of the preimage to the length of the corresponding segment of the image. ____ ――――――― 2 2 2 2 JK = √(-2 − 4) + (-1 − 1) J′ K′ = (-2 − 4) + (-3 − 3) ― = √40 = _ √ 72 Since JK ≠ J′K′ , the transformation is not a rigid motion. △MNP maps to triangle △M′ N′ P′. Preimage M(-2, 2) → P(-2, -2) → N(4, 0) → Image M′(-4, 1) N′(8, 0) P′(-4, -1) The x-coordinate of each image point is twice the x-coordinate of its preimage. The y-coordinate of each image point is half the y-coordinate of its preimage. ( ) 1 (x, y) → 2x, __2 y . Compare the length of a segment of the preimage to the length of the corresponding segment of the image. ―――――――― ―――――――――― = √( 4 − -2 ) + ( 0 − 2 ) ――――― = √ 6 + -2 ―― MN = √(x 2 − x 1) 2 + (y 2 − y 1) 2 2 2 = Since ―――――――― ―――――――――――― = √( 8 − -4 ) + ( 0 − 1 ) ――――― = √ 12 + -1 ―― M′ N′ = 2 2 2 2 √ 40 = MN ≠ M′ N′ , the transformation is not a rigid motion. Module1 GE_MNLESE385795_U1M01L3.indd 36 √(x 2 − x 1) 2 + (y 2 − y 1) 2 36 2 2 √145 © Houghton Mifflin Harcourt Publishing Company The transformation is given by the rule Lesson3 3/20/14 3:16 PM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A Reflect 7. How could you confirm that a transformation is not a rigid motion by using a protractor? If any angle measure in the preimage is different from the corresponding angle measure in the image, then the transformation is not a rigid motion. Therefore, use the protractor to check corresponding angles. If all angle measures are preserved, then check lengths. Your Turn Use coordinate notation to write the rule that maps each preimage to its image. Then confirm that the transformation is not a rigid motion. 8. △ ABC maps to triangle △ A′ B′ C′. Preimage A(2, 2) → C(2, -4) → B(4, 2) → 9. △ RST maps to triangle △ R′S′ T ′ . Image Preimage B′(6, 3) S(4, 2) A′(3, 3) C′(3, -6) x, y) → (1.5x, 1.5y) (_ Image R(-2, 1) → R′(-1, 3) T(2, -2) → T ′(1, -6) ( → 1 x, 3y (x, y) → _ 2 ) S′(2, 6) ―――――――― 37 √(4 − (-2)) +( 2 − 1) = √― ―――――――― 18 R′ S′ = √(2 − (-1)) + (6 − 3) = √― AB is horizontal and AB = 2. _ A′ B′ is horizontal and A′B′ = 3. RS = Since AB ≠ A′ B′, the transformation is not a rigid motion. 2 2 2 2 Since RS ≠ R′ S′, the transformation is not a rigid motion. Elaborate 10. Critical Thinking To confirm that a transformation is not a rigid motion, do you have to check the length of every segment of the preimage and the length of every segment of the image? Why or why not? No; once you find a segment of the preimage whose length is not equal to the length of © Houghton Mifflin Harcourt Publishing Company the corresponding segment of the image, you can stop checking lengths. You only need to find one pair whose lengths are not equal in order to confirm that the transformation is not a rigid motion. 11. Make a Conjecture A polygon is transformed by a rigid motion. How are the perimeters of the preimage polygon and the image polygon related? Explain. The perimeters are equal. Each side of the preimage polygon is transformed to a side of the image polygon with the same length. The sum of the side lengths of the preimage is equal to the sum of the side lengths of the image. 12. Essential Question Check-In How is coordinate notation for a transformation, such as (x, y) → (x + 1, y - 1), similar to and different from algebraic function notation, such as ƒ(x) = 2x + 1? In both cases, the notation shows how an input is changed by the transformation or function. In coordinate notation, the input is a point of the coordinate plane. In algebraic function notation, the input is a real number. Module 1 GE_MNLESE385795_U1M01L3.indd 37 37 Lesson 3 3/20/14 3:16 PM

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