Tracing Paper Transformations

Activity Activity Lab Lab Tracing Paper Transformations

Tracing Paper FOR USE WITH LESSON 9-3 Transformations In Lesson 9-1, you learned how to describe a translation using variables. In the Students use tracing paper to Activity on page 477, you drew a reflection in a plane by using tracing paper. perform vector translations, Now, you will use tracing paper to perform translations and rotations. You will rotations, and reflections. also discover ways to describe certain rotations and reflections using variables. Compositions of rotations, compositions of reflections, and how these compositions are 1 ACTIVITY related are also examined. You can use the vector arrow shown in the diagram y C؅ S + + to represent the translation (x, y) (x 4, y 2). 5 A؅ The translation shifts #ABC with A(-3, 3), C Guided Instruction B(-1, 1), and C(1, 4) to #A9B9C9 with A9(1, 5), A B؅ B9(3, 3), and C9(5, 6). You can see this translation L1 Special Needs using tracing paper as follows: B x Discuss how vectors differ from • Draw #ABC and the vector arrow on graph Ϫ2 O 24 rays that continue indefinitely. paper. Also, show the line containing the arrow. Ask: What two qualities are • Trace #ABC and the vector arrow. y associated with vectors? magnitude (size) and direction • Move your tracing of the vector arrow along the 5 vector line until the tail of the tracing is on the C Visual Learners head of the original vector arrow. Your tracing A Using a colored pencil, have of #ABC should now be at A9(1, 5), B9(3, 3), students draw the vector that 9 and C (5, 6). B x connects each preimage vertex to Ϫ2 O 2468 its image vertex. Exercises 1-3 Point out that Use tracing paper. Find the translation image of each triangle for the given vector. students should draw the lines 1.y 2.y 3. y that contain each vector. E 2 A9(0, 1) D9(3, –1) 2 A9(3, –1) D O B x B9(1, 3) F x E9(4, 1) B x B9(4, 1) - - - Resources 2 C9(2, 0) 22O F9(5, –2) 2 O C9(5, –2) -2 A -2 -2 A Students use graph paper, tracing C C paper, and a straightedge. 4. Show that the composition of the translation in Exercise 1 followed by the translation in Exercise 2 gives you the translation in Exercise 3. See margin.

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To rotate a figure 90° about the origin, trace the figure, one y C axis, and the origin. Then turn your tracing paper A counterclockwise, keeping the origin in place and aligning the 2 traced axis with the other original axis. B x 5. Use #ABC from Activity 1 with A(-3, 3), B(-1, 1), and O 2 4 # C(1, 4). What is the image of ABC for a 90° turn about Ϫ2 the origin? A9(–3, –3), B9(–1, –1), C9(–4, 1) Ϫ4

490 Activity Lab Tracing Paper Transformations

4. kABC after <–1, 3> is Ar(0, 1), Br (1, 3), Cr (2, 0). kArBrCr after <3, –2> is As(3, –1), Bs(4, 1), Cs(5, –2), which is the same as kArBrCr from Exer. 3 490 6. Copy and complete the table. Use tracing paper to find point Activity 2 P PЈ P9, the image of point P, for a turn of 90° about the origin. ؊4, 3) Teaching Tip) (4 ,3) 7. Study the pattern in your table. Complete this rule for a Have students mark a small (Ϫ3, 4) (؊4, ؊3) turn of 90° about the origin. segment for the positive x-axis. Ϫ Ϫ ؊ (x, y) S (■, ■) (–y, x) ( 3, 4) (4, 3) When the figure is rotated, the Ϫ segment can be aligned with the # (3, 4) (4, 3) 8. Test your rule with tracing paper. For TRN with T(0, 2), y-axis to measure a 90° rotation. R(3, 0), and N(4, 5), a 90° turn about the origin should result (3, 0) (0, 3) !in #T9R9N9 with T9(-2, 0), R9(0, 3), and N9(-5, 4). Does it? (0, 4) (؊4, 0) Error Prevention yes 9. In parts (a) and (b) below, what should be the result of In Exercise 9b, make sure students each composition? understand that the x and y a. a 90° turn about the origin followed by a 90° turn about the origin variables have been inter- 180° turn about origin b. (x, y) S (-y, x) followed by (x, y) S (-y, x) (x, y) S (–x, –y) changed. Encourage them to c. Use tracing paper. Test your conjectures from parts (a) perform the substitutions. and (b) on #ABC from Activity 1. Check students’ work. Activity 3 3 ACTIVITY

To reflect a figure across an axis using tracing paper, trace the figure, the axis, and Teaching Tip the origin. Then turn over your tracing Point out that it does not matter paper, keeping the origin in place and aligning the traced axis y which way the tracing paper is C flipped over (top into the page or with the original axis. A top away from the page). # 2 10. What is the reflection image of ABC across the x-axis? Contrast this with rotations that 9 9 9 Across the y-axis? A (–3, –3), B (–1, –1), C (1, –4); B O x are always counterclockwise. A9(3, 3), B9(1, 1), C9(–1, 4) Ϫ Ϫ 2 4 11. Copy and complete the table. Use tracing paper to find points 4 2 Ϫ Px and Py, the reflection images of point P across the x-axis 2 and y-axis, respectively.

Study the patterns in the table. Complete these rules for reflections across the axes. P Px Py (x-axis: (x, y) S (■, ■) 13. y-axis: (x, y) S (■, ■) (3, 4) (3, ؊4) (؊3, 4 .12 – – (x, y)(x, y) (Ϫ3, 4) (؊3, ؊4) (3, 4) 14. Test your rules on #KLM with K(-2, 3), L(3, 1), (Ϫ3, Ϫ4) (؊3, 4) (3, ؊4) and M(1, -2). K9(–2, –3), L9(3, –1), M9(1, 2); yes Ϫ ؊ ؊ a. Across the x-axis, #KLM should map to 9. Does it? (3, 4) (3, 4) ( 3, 4) (b. Across the y-axis, #KLM should map to 9. Does it? (3, 0) (3, 0) (؊3, 0 K9(2, 3), L9(–3, 1), M9(–1, –2); yes (In parts (a) and (b) below, what should be the result of each (0, 4) (0, ؊4) (0, 4 .15 composition? a. (x, y) S (x, -y) followed by (x, y) S (-x, y) (x, y) S (–x, –y) b. a reflection across the x-axis followed by a reflection across the y-axis 180° turn about origin c. Use tracing paper. Test your conjectures from parts (a) and (b) on #ABC from Activity 1. Check students’ work.

EXERCISES 16. Use tracing paper to find a rule for a reflection across the line y = x. Test your rule on #ABC from Activity 1. (x, y) S (y, x) 17. Compare the results of Exercises 9 and 15. Make a conjecture about the compositions suggested by each. A 180° turn about the origin is the same as a reflection in the x-axis followed by a reflection in the y-axis. 18. What other transformation is equivalent to a reflection across the line y = x followed by a reflection across the line y =-x? Explain.180° turn about origin

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