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See Fig. 18-2.
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4, y 2, y 6, y
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Fig. 18-2
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CHAPTER 18 Transformations
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(a) Ar( 1 4, 4 3) Ar(3, 7), Br( 1 Dr(3 4, 4 3) Dr(7, 7) 4, 3 3) Br(3, 6), Cr(3 4, 3 3)
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Cr(7, 6), and
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(b) As( 1 2, 4 5) As(1, 1), Bs( 1 2, 3 5) Bs(1, 2), Cs(3 2, 3 5) Cs(5, 2), and Ds(3 2, 4 5) Ds(5, 1) (c) A-( 1 6, 4 2) A-( 7, 2), B-( 1 C-(3 6, 3 2) C-( 3, 1), and D-(3 6, 3 6, 4 2) 2) B-( 7, 1), D-( 3, 2)
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Recognizing a translation Name the translation that takes ^ABC to (a) ^ArBrCr, (b) ^AsBsCs , and (c) ^A-B-C- as illustrated in Fig. 18-3.
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Fig. 18-3
Solutions
(a) P(x, y) A Pr(x (b) P(x, y) A Ps(x (c) P(x, y) A P-(x 4, y 2, y 6, y 2) 5) 3)
Naming translations Name the translation that moves everything: (a) Up 6 spaces (b) Down 1 space (c) To the right 2 spaces (d) To the left 10 spaces (e) Up 5 spaces and to the right 3 spaces (f) Down 7 spaces and to the right 4 spaces (g) 6 spaces to the left and 4 spaces up
CHAPTER 18 Transformations
Solutions
(a) P(x, y) A Pr(x, y (b) P(x, y) A Pr(x, y (c) P(x, y) A Pr(x (d) P(x, y) A Pr(x (e) P(x, y) A Pr(x (f) P(x, y) A Pr(x (g) P(x, y) A Pr(x 6) 1) 2, y) 10, y) 3, y 4, y 6, y 5) 7) 4)
18.4 Reflections
A transformation that flips everything over is called a reflection. This is because the image of an object in a mirror looks flipped over, as illustrated in Fig. 18-4.
Fig. 18-4
The reflection in Fig. 18-4 is a reflection across the y-axis because the edge of the mirror is pressed against the y-axis. The line where the mirror meets the plane is called the axis of symmetry. The reflection across the vertical line x a is given by P(x, y) A Pr(2a x, y). The reflection across the horizontal line y a is given by P(x, y) A Pr(x, 2a y).
SOLVED PROBLEMS
Performing reflections Let triangle ABC be formed by A( (a) Reflection across the x axis (y (b) Reflection across the line x (c) Reflection across the line y
Solutions
See Fig. 18-5. (a) Ar( 1, (b) As(8 1), Br(0, ( 1), 1) 1) 3), and Cr(3, As(9, 1), Bs(8
1, 1), B(0, 3), and C(3, 1). Graph ^ABC and its image under: 0), P(x, y) A Pr(x, y) 4, P(x, y) A Ps(8 x, y) y) 5, P(x, y) A P-(x, 10
1) 0, 3) 3) Bs(8, 3), and Cs(8 3, 1) Cs(5, 1) 1) C-(3, 9)
(c) A-( 1, 10
A-( 1, 9), B-(0, 10
B-(0, 7), and C-(3, 10
CHAPTER 18 Transformations
Fig. 18-5
Recognizing reflections Name the reflection that takes ^ABC to (a) ^ArBrCr, (b) ^AsBsCs , and (c) ^A-B-C- as illustrated in Fig. 18-6.
Fig. 18-6
Solutions
(a) Reflection across the y axis, P(x, y) A Pr( x, y) (b) Reflection across the line y (c) Reflection across the line x 1, P(x, y) A Ps(x, 2 4, P(x, y) A P-(8 x, y) y)
Naming reflections Name the transformation that (a) Reflects across x (b) Reflects across y 2 6
CHAPTER 18 Transformations
(c) Reflects across x (d) Reflects across y
Solutions
(a) P(x, y) A Pr(4 (b) P(x, y) A Pr(x, 12 (c) P(x, y) A Pr( 20 (d) P(x, y) A Pr(x, 1
x, y) y) x, y) y)
18.4A Reflectional Symmetry
A figure has reflectional symmetry if it looks the same after being flipped across an axis of symmetry that runs through its center. As illustrated in Fig. 18-7, a figure can have (a) one, (b) several, or (c) no axes of symmetry.
Fig. 18-7
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