Chapter 2: Transformational Geometry Assignment Sheet

Transcription

1 hapter : Transformational Geometry ssignment Sheet # Name omplete? 1 Functions Review Video : Transformations 3 Generic Transformations and Isometries 4 Symmetry 5 Dilations and Translations 6 Lab: Reflections and Rotations 7 Reflections and Rotations 8 Video: onstructing Transformations 9 Investigating Transformations 10 onstructing Transformations 11 Video: omposite Transformations 1 omposite Transformations 13 hapter Review 14 hapter Test

2 hapter (October) hapter 1 Test WR#3 Due 6 #1 Function Review No School For Students 7 No School olumbus Day 10 #3 Generic Transformations and Isometries 11 #4 Symmetry 1 HW: Video for Sheet # WR#4 Due 13 #5 Dilations and Translations 14 #6 Lab: Reflections and Rotations HW: Video for Sheet # WR#5 Due 0 1 Skills Quiz/PSM #7 Reflections and Rotations #9 Investigating Transformations #10 onstructing Transformations #1 omposite Transformations HW: Video for #8 HW: Video for #11 4 Skills Quiz/PSM Progress heck 5 Due #13 hapter Review hapter Test 6 WR#6 Due 7 8

3 Name Function Review Geometry hapter Wksht #1 In algebra we studied functions extensively and during that time we learned how to solve for variables using function notation. Let me show you a few examples to remind you. What is the value of the function f ( x) x 3 when x = -? What is the value of the function f ( x) x 7 when x = 5? What is the value of the 3 function f ( x) x when x = 1? 4 f ( x) x 3 f ( x) x f ( x) x f ( x) (1) f ( ) ( ) 3 f (5) f ( ) 43 1 f (5) 1 36 f( x) 9 4 When x = -, then y = 1 (-,1) When x = 5, then y = 1 (5,1) When x = 1, then y = -9 (1,-9) NYTS (Now You Try Some) 1. What is the value of the function f ( x) x x when x = 4?. What is the value of the function f ( x) 5x1 when x = 11? 3. What is the value of the function f ( x) x when x = 5? 5 In all of these examples we were given the x value and then asked to solve for the y value (the value of the function). In the next examples we will be given the y value (the value of the function) and then asked to work backwards to determine the x value that would have produced that result. Look closely at these examples. What is the value of x when f ( x) 4x 1 & f( x) 3? What is the value of x when f ( x) x 5 & f( x) 4? What is the value of x 5x when f( x) & f( x) 10? 1 f ( x) 4x 1 f ( x) x 5 5x 5x f( x) x 1 4x x 9 x 3 x 1 x 10 5x 4 x When y = - 3, then x = 1 (1,-3) When y = 4, then x = 3 ( 3,4) When y = 10, then x = 4 (4,10) NYTS (Now You Try Some) 4. What is the value of x when f ( x) 8x 5 & f( x) 9? 5. What is the value of x 3 when f ( x) x 14 & f( x) 13? 6. What is the value of x 1x when f( x) & f( x) 15? 3

4 7. Given functions, f ( x) x 1, g( x) x x, and x hx ( ), determine value of the function for: 5 a) f(-5) = b) g(-3) = c) h(0) = d) g(1) = e) f( 3 4 ) = f) h( )= 4 5 x 8. Given functions, rx ( ), 3 s( x) 4 x, and t( x) 4x 8, determine value of the function for: a) rx ( ) x = b) sx ( ) 64 x = c) tx ( ) 8 x = d) sx ( ) 1 x = e) rx ( ) x = f) tx ( ) 4 5 x = 9. Using the function machine to determine the missing output or input values. a) Input = -6 b) Input = 5 c) Input = d) Input = Output = Output = Output = -47 Output = 1

5 Name Video: Transformations Geometry hapter Wksht # What is a transformation? Transformations Transformations as a Function: T(x, y) (x + 3, y 6) (x, y) (y, x 3) Translation Reflection Dilation Rotation Invariant- Isometery- Line Symmetry- Direct Isometry- Point Symmetry-

6 Name Generic Transformations and Isometries Geometry hapter Wksht #3 1. Use the given coordinate rules to solve missing coordinates. a) T (x,y) > (x, y + 7) (-4,9) (, ) (, ) (5,0) b) S (x,y) > (-y, x) (-4,9) (, ) (, ) (9,7) c) F (x,y) > (5x, 3y) (-4,9) (, ) (, ) (-5,1) d) G (x,y) > (-x, -3x) (-4,9) (, ) (, ) (-8,-4) e) H (x,y) > (x - 1, y - 3) (-4,9) (, ) (, ) (31,15) f) P (x,y) > (x + 3, y) (-4,9) (, ) (, ) (5,8). ircle which of the following are isometric transformations. Pre-Image a) b) c) d) e) f) 3. Jane claims that any two circles are always isometric because the shape never changes. Is she correct? YES or NO Explain your answer.

7 4. Determine if the pre-image and image are isometric and also which transformation produced the image. PRE-IMGE ircle nswer ircle nswer IMGE Isometry Not Isometry Rotation Reflection Translation Dilation Stretch Other Isometry Not Isometry Rotation Reflection Translation Dilation Stretch Other Isometry Not Isometry Rotation Reflection Translation Dilation Stretch Other Isometry Not Isometry Rotation Reflection Translation Dilation Stretch Other Isometry Not Isometry Isometry Not Isometry Rotation Reflection Translation Dilation Stretch Other Rotation Reflection Translation Dilation Stretch Other

8 Name Symmetry Geometry hapter Wksht #4 1. Draw in the lines of symmetry for each of the shapes. If none, leave the diagram blank. a) b) c) d) e) f) g) h) (Parallelogram) (Regular Hexagon). Use the diagrams from question #1 to determine the order and angle of rotation symmetry for the following shapes. If none, write none. a) Order = ngle = b) Order = ngle = c) Order = ngle = d) Order = ngle = e) Order = ngle = f) Order = ngle = g) Order = ngle = h) Order = ngle = 3. Provided is half of a shape and the line of reflection. a) omplete drawing the shape. b) Using dashes marks to show equal sides label each of the sides to show who is equal to who in the shape. c) Do the same for angles, label which angles are equal to each other in the shape using matching symbols. d) Finally, what do you notice about a shape that has one line of symmetry? 4. Given a regular hexagon, how can you alter it so that instead of having six lines of reflection it only has two. Draw the altered hexagon and draw in the two lines of symmetry.

9 5. Given, determine the following. a) Does it have point symmetry? b) How many lines of symmetry does it have? c) Draw in the line(s) of symmetry. d) What is the unique name for the one of the lines of symmetry? 6) What is rotational symmetry order? 7. These two shapes have both rotational and reflectional symmetry. What do they have in common? 8. Determine the following symmetry characteristics for these REGULR polygons. Lines of Symmetry Lines of Symmetry Lines of Symmetry Lines of Symmetry Rotational Order Rotational Order Rotational Order Rotational Order

10 Name Dilations and Translations Geometry hapter Wksht #5 Dilation Examples: 1.) D 3 (, 6).) D 1 (6, 18) 3 3.) D 4 (5,-) 4.) D (1, 5) 5.) D 1 (1,4) 6.) 4 Translation D 1 (35, 80) 5 To find the image of a point under a translation, you add each coordinate to the given factor for x and y a) 3, 4 T ( 3 x,4 y) b) T 3, 4 ( 3 x, 4 y) Examples: 7.), 7 T (,6) 8.) T 3, (6, 18) 9.) 4, 5 T (5,-) 10.) T, 8 (1, 5) 11.) T 3, 9 (1,4) 1.) T 5, 6 (35, 80)

11 Practice.) Use the diagram to the right a rule for the Translation Draw the image of the figure using the transformation shown. 3.) 4.) D 1

12 Name Lab: Reflections and Rotations Geometry hapter Wksht #6 Materials: Sheets of Graph Paper, 6 different colored markers, pen/pencil Goals: To be able to rotate and reflect on object with/without a rule on the coordinate plane To understand the difference between the image and pre-image of an object To create rules to reflect and Rotate and object on the coordinate plane Rotations Directions: 1. Plot and label the following 6 points (use a different color marker for each point). (, 3) (5, 7) ( 8, ) D (1, 9) E(, 7) F(4, ). Rotate your paper 90 o by turning it to the left one turn. (FYI: Geometrical notation for this is R 90 ) Once you have done this and are looking at your transformed points, fill out the chart below based on your NEW points given the NEW orientation of you graph. (I started the first row for you) Pre-Image (Starting Point) (, 3) (-3, ) D E F Image (New point) 3. Do you see a pattern between the Pre-Image points and the Image points? ould you come up with a rule to Rotate a coordinate 90 o using x and y? 4. Rotate your paper 180 o by turning it to the left one turn. If you already moved it back to the original position, you will have to turn it to the left two turns. (FYI: Geometrical notation for this is R 180 ) Once you have done this and are looking at your transformed points, fill out the chart below based on your NEW points given the NEW orientation of you graph. (I started the first row for you) Pre-Image (Starting Point) (, 3) (-,-3) Image (New point)

13 D E F 5. Do you see a pattern between the Pre-Image points and the Image points? ould you come up with a rule to Rotate a coordinate 180 o using x and y? 6. Rotate your paper 70 o by turning it to the left one turn. If you already moved it back to the original position, you will have to turn it to the left three turns. (FYI: Geometrical notation for this is R 70 ) Once you have done this and are looking at your transformed points, fill out the chart below based on your NEW points given the NEW orientation of you graph. (I started the first row for you) Pre-Image (Starting Point) (, 3) (3,-) D E F Image (New point) 7. Do you see a pattern between the Pre-Image points and the Image points? ould you come up with a rule to Rotate a coordinate 70 o using x and y? 8. What do you think a Rotation of 360 o would be the same as? 9. What do you think a Rotation of -90 o would be the same as?

14 Reflections Directions: 1. On the other graph paper, plot and label the following points (use a different color marker for each point). (1, 0) (3, 1) ( 5, ) D (8, 4) E( 4, 3) F(6, 5). Reflect the point over the x-axis by folding the piece of paper at the x-axis. (FYI: Geometrical notation for this is r x-axis ) Once you have done this and are looking at your transformed points, fill out the chart below based on your NEW points given the NEW orientation of you graph. (I started a row for you). Pre-Image (Starting Point) (1,0) Image (New point) D (8, 4) D (8, -4) E F 3. Do you see a pattern between the Pre-Image points and the Image points? ould you come up with a rule to reflect a coordinate over the x-axis? 4. Reflect the point over the y-axis by folding the piece of paper at the y-axis make sure you have opened your graph paper back up first. (FYI: Geometrical notation for this is r y-axis ) Once you have done this and are looking at your transformed points, fill out the chart below based on your NEW points given the NEW orientation of you graph. (I started a row for you). Pre-Image (Starting Point) (1,0) Image (New point) D (8, 4) D (-8, 4) E F 5. Do you see a pattern between the Pre-Image points and the Image points? ould you come up with a rule to reflect a coordinate over the x-axis?

15 6. The line y=x contains all coordinates where x and y are the same ex (-4, -4), (-3,-3), (-, -), (-1, -1), (0, 0), (1, 1), (, ), (3,3), (4,4) Graph this line using a PENIL and a STRIGHTEDGE. Reflect the points over the line y=x by folding the piece of paper at the line you drew make sure you have opened your graph paper back up first. (FYI: Geometrical notation for this is r y=x ) Once you have done this and are looking at your transformed points, fill out the chart below based on your NEW points given the NEW orientation of you graph. (I started a row for you). Pre-Image (Starting Point) (1,0) Image (New point) D (8, 4) D (4, 8) E F 7. Do you see a pattern between the Pre-Image points and the Image points? ould you come up with a rule to reflect a coordinate over the line y=x? 8. The line y=-x contains all coordinates where x and y are the same but one is negative ex (-4, 4), (-3,3), (-, ), (-1, 1), (0, 0), (1, -1), (, -), (3,-3), (4,-4) Graph this line using a PENIL and a STRIGHTEDGE. Reflect the points over the line y=-x by folding the piece of paper at the line you drew make sure you have opened your graph paper back up first. (FYI: Geometrical notation for this is r y=-x ) Once you have done this and are looking at your transformed points, fill out the chart below based on your NEW points given the NEW orientation of you graph. (I started a row for you). Pre-Image (Starting Point) (1,0) Image (New point) D (8, 4) D (-4, -8) E F 9. Do you see a pattern between the Pre-Image points and the Image points? ould you come up with a rule to reflect a coordinate over the line y=-x?

16 Name Reflections and Rotations Geometry hapter Wksht #7 1. Determine the line of reflection for the following pre-image and images. a) b) c) d) e) f). Determine the pre-image coordinates, then reflect it, and determine the image coordinates. a) = (, ) rx axis ( ) = (, ) m b) = (, ) ry axis ( ) = (, ) c) = (, ) rm ( ) = (, ) D d) D = (, ) rx axis ( D ) D = (, ) E e) E = (, ) rx axis ( E ) E = (, ) f) F = (, ) rn ( F ) F = (, ) G F n g) G = (, ) ry axis ( G ) G = (, )

17 3. ircle the center of rotation for the following pre-image and images. a) rotation of 90 b) rotation of 180 c) rotation of 180 E ' D = D' E E' = ' ' E' ' 4. Determine the pre-image coordinates, then rotate it, and determine the image coordinates. Patty paper will help determine the image coordinates. (Patty paper might be helpful here.) R ( ) O,90 a) = (, ) R ( ),90 = (, ) O b) = (, ) R ( ),90 = (, ) O c) = (, ) R ( ),90 = (, ) O R ( ) O,90 DFE d) D = (, ) R ( ),90 DFE D = (, ) O D E O e) E = (, ) R ( ),90 DFE E = (, ) O f) F = (, ) R ( ),90 DFE F = (, ) O F 5. Determine the name of the point that meets the given conditions. a) R ( ),60 R G b) ( ) G,180 c) R ( ) G,300 D d) R ( ) G, 10 e) R ( ) G,40 E f) R ( ) G, 40 F g) R ( ),60 h) R ( ),10 D

18 Name Video: onstructing Transformations Geometry hapter Wksht #8 1. onstructing a reflection using a compass and straightedge. m (1)Placing the compass pointer at point make a circle of any radius as long as it intersects line m twice. In this case, name them points and D. () Leaving the compass radius the same - place the compass pointer at and make an arc on the opposite side of. Do that as well from D. The intersection of these two arcs is. (3) Placing the compass pointer at make a circle of any radius as long as it intersects line m twice. In this case, points F and G. (4) Leaving the compass radius the same - place the compass pointer at F and make an arc on the opposite side of. Do that as well from G. The intersection of these two arcs is (5) Use the straightedge to create. ' '. onstructing a translation using a compass and straightedge. (1) Translate by Vector '. () Place the compass pointer on point and measure the length of '. (3) Using the compass length of, place the compass pointer at point and create an arc in the general direction of the translation. Do the same thing from point. ' (4) Measure the length of. Place the compass pointer at and create an arc in the general direction of the arc from. The intersection of these two arcs is. (5) Measure the length of. Place the compass pointer at and create an arc in the general direction of the arc from. The intersection of these two arcs is. (6) Use the straightedge to form the translated triangle.

19 Name Investigating Transformations Geometry hapter Wksht #9 1. Which transformation has taken place? a) b) c) d) D D D F ' D' D' D' D' ' E E' ' ' ' D F' '. omplete the chart. Relationship between preimage and image ROTTION REFLETION TRNSLTION Distances SME OR DIFFERENT SME OR DIFFERENT SME OR DIFFERENT Orientation SME OR DIFFERENT SME OR DIFFERENT SME OR DIFFERENT Special Points 3. Given that was mapped to using a single transformation. a) Why couldn t this mapping have resulted by a single translation? ' b) What transformation must have mapped these two triangles? Explain your answer. ' ' 4. Given that was mapped to using a single transformation. a) Why couldn t this mapping have resulted by a single reflection? ' b) What transformation must have mapped these two triangles? Explain your answer. ' '

20 5. is congruent to. student tries to determine which of these single transformations mapped onto. She concludes that a reflection had to be involved and more than one transformation had to map these on two triangles. a) How can she conclude that a reflection was involved? ' ' b) How can she conclude that this wasn t just a single reflection? ' 6. Determine the location of Point, a) after a reflection =, where was point? b) after a rotation of 7 =, where was point? 7. fter a reflection = 4 cm, how far was away from the line of reflection? 8. If after a reflection = and = 6 cm. What is the relationship between and the line of reflection. Draw a diagram. 9. The distance from point to the line of reflection is 10 cm, and the distance from point to the line of reflection is also 10 cm. Jeffrey concludes that is the image of under a reflection. What do you think of this conclusion? 10. was translated by the arrow making ' ' and ' '. a) What other segments in the diagram are congruent? ' b) What other segments in the diagram are parallel? '

21 Name onstructing Transformations Geometry hapter Wksht #10 1. Use a compass and a straightedge to construct the following reflections. r m m r m m r m m r m m

22 . Determine the Line of Reflection a) In trying to find the line of reflection you need to work backwards through the definition of a reflection. ' onstruct the line of reflection of & ' How do you know that this is a reflection and not a rotation? ' ' b) What about this transformation tells you that it must be a reflection and not something else? ' onstruct the line of reflection of &. ' ' c) What in this diagram gives us a clue about where the line of reflection is? ' ' onstruct the line of reflection of &.

23 TIP. If you are trying to find the center of a circle we use the perpendicular bisector to help us. circle with no center Perform the perpendicular bisector of any two points on the circle. This line passes through the center. Perform it a second time and the intersection must be the center because both perpendicular bisectors split the circle in half. Now you have found the center of the circle. 4. a) onstruct the center of rotation and determine the angle of rotation. (Use a protractor to measure the angle size) b) onstruct the center of rotation of D & D. Use the tip above. pply it to find the ENTER of rotation. ' ' ' D ' D' c) onstruct the center of rotation of & and determine the angle of rotation. (Use a protractor to measure the angle size) Use the tip above. pply it to find the ENTER of rotation. ' ' '

24 5. Use a compass and a straightedge to construct the following translations. T ( ) ' ' T ( ) ' ' T ( ) ' ND THEN rm ( ' ' ') m '

25 Name Video: omposite Transformations Geometry hapter Wksht #11 Rules Dilation Translation Reflection Rotation D (3,5) T (3,5, 3 ) rx axis(3,5) R 90 ( 3,5) ry axis(3,5) R 180 ( 3,5) ry x (3,5) (3,5) R 70 ( 3,5) x ry ompositions Example: What is the image of R (3, -) under the composition below? R90 D Example: Given triangle with E(-,1) (0, -1) and M(-1, -3) state the coordinates of the image of triangle under the composition. (The use of the grid is optional) r y R 180 Example: Given triangle with (, 3) (-, 4) and (-1, 1) state the coordinates of the image of triangle under the composition. (The use of the grid is optional) r y axis T 3,4

26 Name omposite Transformations Geometry hapter Wksht #1 OMPOSITE TRNSFORMTIONS DOES ORDER MTTER Use the composite transformation to plot and 1a) T 3,5 r ( ) y axis b) r ( ) y axis T 3,5 (6,-1) (6,-1) (3,-4) (3,-4) (5,-7) (5,-7) c) Did doing the transformations in a different order matter? Explain why? a) r ( ) x axis R O,90 b) RO,90 rx axis ( ) (1,8) (1,8) (4,7) (4,7) (1,3) (1,3) c) Did doing the transformations in a different order matter? Explain why?

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28 Name hapter Review Geometry hapter Wksht #13 Vocabulary: Reflection Rotation Translation Transformation Dilation Isometry Direct Isometry Invariant 1. What is the value of the function, f ( x) 3x 1 when x = -3? ) 8 ) - 10 ) -8 D) x. Given the function, f( x), what is the value of x when f( x) 16? 4 ) x = 4 ) x = -4 ) x = 64 D) x = student places x = 3 into the function, f ( x) 5x 7 and gets f( x) 8. f( x) 8 is the: ) Input ) Output ) Domain D) Maximum If you have a rotational symmetry angle of 18, then your order is: ) ) 5 ) 10 D) Which of the following has exactly one line of symmetry? ) Rectangle ) n Isosceles Trapezoid ) Rhombus D) Parallelogram This shape has: ) Only Rotational Symmetry ) Only Reflectional Symmetry 8. ) oth Rotational & Reflectional Symmetries D) Neither symmetry 9. is reflected to create image. Which statement is always true? ) ' ' ) ' ' ) ' ' D) ' ' Which transformation changes the orientation of the shape? ) Rotation ) Dilation ) Translation D) Reflection 11. Determine whether the following are (T)rue or (F)alse. 13. T( x, y) ( x 5, y) is an isometric transformation. T or F 14. T( x, y) (3 x, y ) is an isometric transformation. T or F 15. If a shape has a rotational order of 1, then it has NO rotational symmetry. T or F 16. rotational symmetry of order means that the angle of the order is 90. T or F 17. If point is.5 cm from the line of reflection, then = 5 cm. T or F 18. rotation of 180 reverses the orientation of the shape. T or F 19. When a rotation of 80 about point T is performed, all points in the plane move. T or F 0. The isometric transformations are rotation, reflection, dilation and translation. T or F 1.Given coordinate rulet( x, y) ( x, 5) determine the image of (-,)?. Given coordinate rule, T( x, y) ( x, y 6) determine the pre-image of (-,)?

29 is reflected over line g to create the image. What is the relationship between ', ' and '? 6. What are the coordinate rules for the given transformations? REFLETION RULES ROTTION RULES Ry axis ( x, y) (, ) RO,90 ( x, y) (, ) Rx axis ( x, y) (, ) RO,180 ( x, y) (, ) R ( x, y) (, ) R ( x, y) (, ) y x O,70 What are the characteristics of the given transformations? HRTERISTI REFLETION ROTTION TRNSLTION ISOMETRI? YES OR NO YES OR NO YES OR NO DISTNES (all points YES OR NO YES OR NO YES OR NO move the same distance) OREINTTION? SME OR REVERSE SME OR REVERSE SME OR REVERSE N =? YES OR NO YES OR NO YES OR NO 7. Determine whether the following are (T)rue or (F)alse. If a translation maps (-3,5) to (3, 5), then T(x,y) > (x 6,y) would be the rule. T or F F (x,y) > (x + 3,y) is a translation of 3 units to the right. T or F rotation of 70about the origin maps (1,-) to (-,-1) T or F R R R T or F O,180 x axis y axis Three reflections over parallel lines could also be described as a single translation. T or F composite transformation of a rotation followed by translation reverses the orientation of the shape. T or F translation of right and 5 down followed by a reflection on the x axis is T, 5 Rx axis T or F double reflection over y = 3 followed by y = 4, translates all points up units. T or F double reflection over y = -3 followed by y = -1, translates all points up 4 units. T or F R R T T or F y 11 y 6 0,10 8. Perform the following on the grid. a) R ( ),90 NOPQ M b) R ( ),180 GHI F c) R ( DE) d) T ( JKL) 3, G I F H E D J K 6 4 M N Q O P L 4

30 9. Use your rules to solve these. a) R (5,4) '(, ) c) x axis R R (, ) ''(4, 3) e) x axis y axis T, 1 Ry axis (5,3) ''(, ) g) RO,90 (3,) '(, ) b) R R ( 5,1) ''(, ) y axis x axis d) T 3,5 (, ) '(9, 3) f) R ( 1,6) '(, ) h) x axis RO,90 (, ) '(8,3) 30. onstruct the following R ( ) m and enter of Rotation for Triangle DEF m