Unit 5 Lesson 2 Investigation 1

Transcription

1 Name: Investigation 1 Modeling Rigid Transformations CPMP-Tools Computer graphics enable designers to model two- and three-dimensional figures and to also easil manipulate those figures. For eample, interior design software permits users to select images of various tpes and sizes of furniture and position them at different places in a room laout b sliding and rotating the shapes. utomotive design software permits users to create smmetric components of vehicles b reflecting and/or rotating basic design elements. Complete models of vehicles can be rotated and viewed from different angles. In this investigation, ou will eplore how coordinates are used in computer graphics software to transform the position of shapes in a plane. s ou work on the following problems, look for answers to these questions: How can coordinates be used to describe a sliding motion or translation? How can coordinates be used to describe a turning motion or rotation? How can coordinates be used to describe a mirror or line reflection? Unit 5 Lesson 2 Investigation 1 1 Interactive geometr software provides tools to reposition a shape b translation, rotation about a point, and reflection across a line. Other software will have similar commands. s a class or in pairs, eperiment with the first three commands in the Transform menu and the corresponding functions in the menu bar. a. Begin b eploring how the Translate command can be used to transform shapes in a plane. Draw a shape or select the Humanoid from Sample Sketches. Translate the shape to a different position. Observe how the original shape and its image appear to be related. Repeat for at least three other translations, including: one that slides the shape horizontall; one that slides the shape verticall; and one that slides the shape in a slanted direction. b. Net, eplore how the Rotate command can be used to transform shapes in a plane. Draw a figure or select the Humanoid shape from Sample Sketches. Rotate the figure counterclockwise about the origin. Observe how the original figure and its image appear to be related. Repeat for at least three other counterclockwise rotations about the origin, including: a 90 rotation; a 180 rotation; and a 45 rotation. LESSON 2 Coordinate Models of Transformations 197

2 c. Now, eplore how the Reflect command can be used to transform a shape in a plane. Draw a shape or select the Humanoid shape from Sample Sketches. Reflect the shape across a line of our choice. Observe how the original shape and its image appear to be related. Repeat for at least three other line reflections, including: a reflection across the -ais; a reflection across the -ais; and a reflection across the line =. In each case, first clear the window and redraw our shape. d. How do ou think the software determines the position of the translated image of a shape? The rotated image of a shape? The reflected image of a shape? Translating Shapes translation, or sliding motion, is determined b distance and direction. B looking carefull at a simple shape and its translated image, ou can discover patterns relating the coordinates of the shape and the coordinates of its image. 2 On the screen below, a flag BCDE and its translated image B C D E are shown. Horizontal Translation D C D C E B E B a. Describe the translation as precisel as ou can. b. Eplain how the translated image of the flag could be produced using onl the translated images of points, B, C, D, and E. c. Under this translation, what would be the image of (0, 0)? Of (1, -5)? Of (-5, -4)? Of (a, b)? d. Write a rule ou can use to obtain the image of an point (, ) in the coordinate plane under this translation. State our rule in words and in smbolic form (, ) (, ). 198 UNIT 3 Coordinate Methods

3 3 The screens below show a flag BCDE and its image under two other translations. Vertical Translation Oblique Translation D C D C E D B C E B D E C B E B a. Describe the vertical translation as precisel as ou can. The diagonal (oblique) translation. b. Under the vertical translation, what would be the image of (0, 0)? Of (2, 5)? Of (4.1, -2)? Of (a, b)? c. Write a rule ou can use to obtain the image of an point (, ) under the vertical translation. State our rule in words and in smbolic form (, ) (, ). d. Under the oblique translation, what would be the image of (0, 0)? Of (2, 5)? Of (4.1, -2)? Of (a, b)? e. Write a rule ou can use to obtain the image of an point (, ) under the oblique translation. State our rule in words and in smbolic form. 4 Compare the transformation rules ou developed for Part d of Problem 2 and for Parts c and e of Problem 3. Write a general rule that tells how to take an point (, ) and find its translated image if the preimage is moved horizontall h units and verticall k units. Compare our rule with others and resolve an differences. You now have a rule ou can use to find the translated image of an point when ou know the components of the translation the horizontal and vertical distances and directions the point is moved (left or right, up or down). This is eactl the information a calculator or computer graphics program needs in order to displa a set of points and their translated images. 5 Use the following questions to help write an algorithm that would guide a programmer in the development of a translation program that displas the original figure (called the preimage) and its translated image and connects corresponding vertices of the two figures. What information would ou need to input? What formula or formulas could be used in the processing portion? What information should be displaed in the output? LESSON 2 Coordinate Models of Transformations 199

4 Rotating Shapes Rotations about the origin have similar coordinate models. rotation, or turning motion, is determined b a point called the center of the rotation and a directed angle of rotation. flag BCDE and its images under counterclockwise rotations of 90, 180, and 270 about the origin are shown below. Rotations bout the Origin D C E B B C E D 6 Consider flag BCDE above and its image under a 90 counterclockwise rotation about the origin. a. On a cop of the table below, record the coordinates of the images of the five points on the flag under a 90 counterclockwise rotation about the origin. Eplain wh the rotated image of the flag could be produced using onl the rotated images of points, B, C, D, and E. Preimage 90 Counterclockwise Rotated Image (0, 0) (, ) B(3, 3) B (, ) C(5, 5) C (, ) D(7, 3) D (, ) E(5, 1) E (, ) b. Use an patterns ou see between preimage and image points in our completed table to help plot the points (-2, -5), (-4, 1), (5, -3), and their images under a 90 counterclockwise rotation about the origin on a new coordinate grid. i. For each preimage point, use dashed segments to connect the preimage to the origin and the origin to the image. ii. Connect each preimage segment to its image segment with a turn arrow that shows the directed angle of rotation. c. Write a rule relating the coordinates of an preimage point (, ) and its image point under a 90 counterclockwise rotation about the origin. State our rule in words and in smbolic form. 200 UNIT 3 Coordinate Methods

5 d. ccording to our rule, what is the image of (0, 0)? Wh does this image make sense? e. How should the slope of the line through a preimage point and the origin be related to the slope of the line through the origin and the image point? Verif our idea b computing and comparing slopes. f. Write an algorithm to guide the development of a program for a 90 counterclockwise rotation about (0, 0) that displas the preimage and image figures. 7 s ou probabl epect, counterclockwise rotations of 180 and 270 about the origin also have predictable coordinate patterns. Use a cop of the screen at the top of page 200 showing flag BCDE and its images to eplore these patterns. a. Investigate patterns in the coordinates of the preimage and image pairs when points are rotated 180 about the origin. b. Write a rule relating the coordinates of an preimage point (, ) and its image point under a 180 rotation about the origin. State our rule in words and in smbols. c. How is the slope of the line through two preimage points related to the slope of the line through the images of those points? What does this tell about a line and its image under a 180 rotation? d. Similarl, search for patterns in the coordinates of the preimage and image pairs when points are rotated 270 counterclockwise about the origin. e. Write a rule relating the coordinates of an preimage point (, ) and its image point under a 270 counterclockwise rotation about the origin. State our rule in words and in smbols. f. Describe how ou could modif the algorithm in Part f of Problem 6 so that it would guide development of a program to rotate a point 180 or 270 counterclockwise about the origin instead of 90 counterclockwise. Reflecting Shapes Line reflections can also be epressed using coordinates. line reflection is determined b a mirror line (or line of reflection) that is the perpendicular bisector of the segment connecting a point and its reflected image. point on the line of reflection is its own image. In the following problems, ou will build coordinate models for reflections across vertical and horizontal lines, as well as across the lines = and = -. LESSON 2 Coordinate Models of Transformations 201

6 8 flag BCDE and its reflected image across the -ais are shown on the screen below. Reflected cross the -ais D C C E B B D E a. Investigate patterns in the coordinates of preimage and image pairs when points are reflected across the -ais. b. Eplain wh the reflected image of the flag could be produced using onl the reflected images of points, B, C, D, and E. c. Write a rule which tells how to take an point (, ) and find its reflected image across the -ais. State our rule in words and in smbols. d. On a cop of the diagram, use dashed segments to connect point to point and point D to point D. Use coordinates to verif that the -ais is the perpendicular bisector of and DD. 9 The table below shows coordinates of si preimage points and coordinates (a, b) of a general point. Plot each of the si points and its reflected image across the -ais. a. Record the coordinates of the image points in a table like the one below. Preimage Reflected Image cross -ais (-4, 1) (-4, -1) (3, -2) (-2, -5) (4, 5) (0, 1) (-3, 0) (a, b) b. What pattern relating coordinates of preimage points to image points do ou observe? Use the pattern to give the coordinates of the image of (a, b). 202 UNIT 3 Coordinate Methods

7 c. Write a rule that tells how to take an point (, ) and find its reflected image across the -ais. State our rule in words and in smbols. d. How is the -ais related to the segment determined b an point (a, b) not on the -ais and its reflected image? Justif our answer using coordinates. 10 Draw the graph of =. Plot each preimage point in the table below and its reflected image across that line. Connect each preimage/image pair with a dashed segment. a. Record the coordinates of the image points in a cop of the table below. Preimage Reflected Image cross = (-4, 1) (1, -4) (3, -2) (-2, -5) (4, 5) (0, 1) (-3, 0) (a, b) b. Describe a pattern relating coordinates of preimage points to image points. c. Write a rule relating the coordinates of an preimage point (, ) to its reflected image across the line =. State our rule in words and in smbols. d. How is the line of reflection, =, related to the segment determined b an point (a, b) not on the line and its image? Justif our answer. 11 Net, investigate patterns in the coordinates of the preimage and image pairs when points are reflected across the line = -. a. Draw the graph of = -. Then plot the si preimage points in the table in Problem 10 and their reflected images across the line. b. Describe a pattern relating coordinates of preimage points to coordinates of image points. c. Write a rule relating the coordinates of an preimage point (, ) and its reflected image across the line = -. State our rule in words and in smbols. d. How is the segment determined b a point and its reflected image related to the line = -? LESSON 2 Coordinate Models of Transformations 203

8 12 You now have coordinate models for the following line reflections. reflection across the -ais reflection across the -ais reflection across the line = reflection across the line = - Sharing the workload among our classmates, develop planning algorithms that would guide a programmer in the development of line reflection programs for each of these four line reflections. Identif the input, processing, and output portions of each of our algorithms. Summarize the Mathematics In this investigation, ou developed coordinate rules relating points and their images under different rigid transformations: translations, rotations about the origin, and line reflections. a translation is determined b a single point and its image. i. Suppose a translation slides the point O(0, 0) to the point (a, b). Write a smbolic rule (, ) (, ) that describes this translation. ii. Suppose a translation slides the point (a, b) to the point B(c, d). Write a smbolic rule (, ) (, ) that describes this translation. b Summarize the coordinate rules for these rotations about the origin. i. For a rotation of 90 counterclockwise: (, ) (, ) ii. For a rotation of 180 counterclockwise: (, ) (, ) iii. For a rotation of 270 counterclockwise: (, ) (, ) iv. For a rotation of 270 clockwise: (, ) (, ) c Summarize the coordinate rules for line reflections: i. cross the -ais: (, ) (, ) ii. cross the -ais: (, ) (, ) iii. cross the line = : (, ) (, ) iv. cross the line = -: (, ) (, ) Be prepared to eplain our coordinate rules and strategies ou could use to remember or redevelop them. 204 UNIT 3 Coordinate Methods