Chapter 9 - Transformations. Transformation: operation that maps (or moves) a preimage onto an image.
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1 Chapter 9 - Transformations Transformation: operation that maps (or moves) a preimage onto an image. 4 Types of transformations: I. Reflections II. Translations III. Rotations IV. Dilations 3 Parts of a transformation: (1) Pre image (starting point) (2) Rule or operation [reflection(flip), translation(slide), or rotation(turn)] (3) Image (ending point) TRANSFORMATIONS MAPPING ~ MAPPING REFLECTIONS TRANSLATIONS ROTATIONS GLIDE REFLECTIONS DILATIONS ISOMETRY MAPPING : MAPS EVERY SEGMENT TO A SEGMENT PRESERVE: DISTANCE (LENGTHS) ANGLE MEASURE AREA SIMILARITY MAPPING ~ MAPPING: MAPS ANY POLYGON TO A ~ POLYGON ~ MAPPINGS PRESERVE: ANGLE MEASURE ORIENTATION (clockwise/counterclockwise) PARALLELISM FUNCTION NOTATION: One to One Mapping: Function Example 1: g(x) = 3x 5 a. Find g(4) = b. Find the image of -5. c. Find the pre-image of 19. Example 2: h(x, y) (3x, y 2) a) h(3,0) b) (4,-8) 1
2 Reflection Reflection A reflection in line m maps every point P to a point P l such that: If P is not on line m, m is the perpendicular bisector of PP. If P is on line m, then P = P l. Notation: rm means reflection in line m Graph the points to form a figure. Reflect each figure over the x-axis. Draw the image in a different color. Then write the coordinates of the image points. What pattern do you notice? x y x y Write the algebraic rule for a reflection over the x-axis: Graph the points to form a figure. Reflect each figure over the y-axis. Draw the image in a different color. Then write the coordinates of the image points. What pattern do you notice? x y x y Write the algebraic rule for a reflection over the y-axis: 2
3 Graph the points to form a figure. Reflect each figure over the line y = x (drawn for you on the first graph). Draw the image in a different color. Then write the coordinates of the image points. What pattern do you notice? x y x y Write the algebraic rule for a reflection over the line y = x: Graph the points to form a figure. Reflect each figure over the line y = -x (drawn for you on the first graph). Draw the image in a different color. Then write the coordinates of the image points. What pattern do you notice? x y x y Write the algebraic rule for a reflection over the line y = -x: 3
4 Examples: Graph the image of the figure using the indicated reflection. Use the algebraic rule to find the coordinates of the vertices of the image of each figure after the given reflection. 7. D(2, -3), E(1, 4), N(9, -12); over the y-axis 8. T(-4, 19), H(3, 13), A(5, -6), W(-6, -8); over the x-axis 9. W(4, -7), I(0, 3), R(-1, -1), E(-5, 5); across the line x= -4 Write a rule to describe each transformation. 4
5 Given: A(-2,5), B ( -3,4), C( 4,3), D (0, -2), E (-3, 6) find the following: a. rx: A b. ry: B c. rx= 5: C d. ry = 3: D e. rx=y: E Given: rk: A A l, find the standard form of line k. a. A (5,0), A l (9,0) b. A(4,1), A l (4,3) c. A(5,1), A l (1,5) d. A(0,2), A i (4,6) e. A (-1,2), A l (4,5) Given: MAY maps to DAY by reflecting over AY. a) MY maps to. b) AY maps to. c) MAY maps to. d) AYD maps to. e) MA. f) D. g) YAD. M A Y D 6. Find the reflection image of each point over the given reflection line. a) rx=4 (4,1) = f) ry=4 (8,4) = b) rx=2 (-3,3)= g) rx=-1 (5,3) = c) rx=2 (7,2) = h) rx=3 (8,-2) = d) ry = -1 (4,2) = i) rx=1 (-3,4) = e) ry=-2 (3,2) = j) ry=2 (-3,-2) = 5
6 Rotation Rotation Counterclockwise is POSITIVE (think of the quadrants). Clockwise is NEGATIVE Notation for Rotations: R O, 270 means a counterclockwise rotation about point O of 270. R O, -90 means a clockwise rotation about point O of 90. H O means a rotation about point O of 180. (Half-turn) H O is equivalent to and. Find an equivalent rotation for the following: 1. R O, R O, R O, R O, -180 In the diagram, O is the center of equilateral PST. State the images of points P, S, and T for each rotation. P 5. R O, R O, R O, 360 Name each image point: 8. R T, 60(S) = 9. R T, -60(P) = 10. R O, 240(P) = T O S The diagonals of regular hexagon ABCDEF form 6 equilateral s. Complete the following: 11. R O, 60:E 12. R O, -60: D 13. R O, 120: F 14. R D, 60: O A F 15. R B, -60(O) = 16. H O (A) = B O E 17. A reflection in FC maps B to and D to. 18. If k is the perpendicular bisector of FE, then R k(a) =. 19. If a translation maps A to B, then it also maps O to and E to. C D Examples: B A E 1. ABCDE is a regular pentagon. Name the image of point E for a counterclockwise 72 degree rotation about X. X 2. Name the image of point A for a clockwise 216 rotation about X. C D 6
7 The hexagon GIKMPR and FJN are regular. The dashed line segments form 30 angles. 3. Find the angle of rotation about O that maps Q to F. 4. Find the image of point P after a rotation of 240 about point O. 5. Find the angle of rotation about O that maps to. Graph the image of the figure and give its coordinates. 1) 7
8 Translation Translations You can write rules to describe a translation. Example 1 - On the coordinate plane, graph PQRS where P(-1, -2) Q(1, 2) R(-2, 2) S(1, -2). Use the rule (x, y) (x + 3, y -1) to find the translation of the image of PQRS. Example 2 - Write the rule to describe each translation. You can use a vector to describe a translation. Example 3 - <2, 3> means to go <-1, 2> means to go <0, -2> means to go Example 4- Find the image of ACE with vertices A(9,3) C(-3, -2) E(0, -7) under the translation -5,4. Example 5 What is the image of F(11,-2) under the translation -6,-8 Example 6 The point Q (7,-1) is the image under the translation 5,-2. What is the preimage Q? Example 7 B(-6,-9) maps to B (-12, 3). Describe the rule and vector. 8
9 True or False, if false, explain why. 1. A translation is a transformation. 2. Translations preserve angle measure. 3. Translations reverse orientation. 4. Under a translation, a line is parallel to its image. 5. Under every line reflection, a line is parallel to its image. 6. If P and Q are the translation images of P and Q, then PP = QQ. 7. If S and T are the reflection images of S and T, then SS = TT. 8. The image of a square under a translation is a square. 9. A point may be its own image under a translation. 10. A point may be its own image under a line reflection. 11. In the tessellation to the right, a translation maps A onto B. Name the image of : a. E c. B e. EFK b. K d. AE f. BCGF N P Q R J K L M 12. Using the same diagram. in a translation that maps A onto F, name the image of: a. a. E c. B e. EFK b. K d. AE f. BCGF E F G H A B C D In 14 25, refer to the graph. A translation maps A onto A. Name the coordinates of the translation image of each of the following points. 14. O 16. C 18. E 15. B 17. D 19. A Name the coordinates of the pre-image of each of the following points. C B 20. A 22. B 24. D 21. O 23. C 25. E A O A D E 9
10 The scale factor is usually called "k". A dilation is NOT an isometry. Dilations If 0 < k < 1, then the dilation is a. If k > 1 then the dilation is an. Mapping Rule: Dk (x,y) (kx, ky) k = k = Example 1 - The dashed triangle is a dilation image of the solid triangle. What is the scale factor? a) b) Example 2 - A dilation has center (0,0). Find the image of each point for the scale factor given. a) D(2, -5): 2 b) A(-6, -2): 1.5 c) T(0, 6): 3 d) C(-4, -7): 0.1 Example 3 - WXY is dilated by 2/3.W(2,5), X(6,2), Y (2,2) Find X. Example 4 - Find the vertices for the image of ABC with vertices A (4,-2), B (-2, 5), C (-3,-1). Use the rule (x,y) (2x, 2y). What is the scale factor? Example 5 - Write the rule to describe the dilation. A (-6, 9), B(3, 12), C(0,-6) What is the scale factor? A'(-2, 3), B'(1, 4), C'(0,-2) Example 6 - What is the ratio of side lengths between the preimage and image after the dilation D4? Example 7 - Square ABCD has vertices A(-4,4), B(4,4), C(4,-4), and D(-4,-4). Write the coordinates of the image after a dilation of the figure by a scale factor of 4 about the origin. Example 8 - Find the image after a dilation of 2 for ABC. A(3,-1), B(-5,4), and C(2,-6). 10
11 Example 9- a) Given A(0, 0) B(2, 2) C(-1, 2) D(-3, 0), find the image of the figure with a dilation of 3. Graph the preimage and the image. b) Find the image of the figure with a dilation of 1/2. Example 10: What is the rule for a scale factor of 3 and left 2 units? A. (x,y ) = (3x, 3y) B. (x, y ) = (3x + 2, 3y) C. (x, y ) = (3x, 3y 2) D. (x, y ) = (3x -2, 3y) Bonus: Given R(4,1), S(7,6), and T (9,3) If T(x, y ) = (2x 3, 2y + 4) Find R. 11
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