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COMPOSITION OF TRANSFORMATIONS

COMPOSITION OF TRANSFORMATIONS. Warm Up Determine the coordinates of the image of P (4, –7) under each transformation. . 1. a translation 3 units left and 1 unit up . (1, –6). 2. a rotation of 90° about the origin. (7, 4). 3. a reflection across the y -axis. (–4, –7).

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COMPOSITION OF TRANSFORMATIONS

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  1. COMPOSITION OF TRANSFORMATIONS

  2. Warm Up Determine the coordinates of the image of P(4, –7) under each transformation. 1. a translation 3 units left and 1 unit up (1, –6) 2. a rotation of 90° about the origin (7, 4) 3. a reflection across the y-axis (–4, –7)

  3. You drew reflections, translations, and rotations. • Draw glide reflections and other compositions of isometries in the coordinate plane. • Draw compositions of reflections in parallel and intersecting lines.

  4. Composite Photograph Composite photographs are made by superimposing one or more photographs.

  5. Morphing Morphing is a popular special effect in movies. It changes one image into another.

  6. Definition When a transformation is applied to a figure, and then another transformation is applied to its image, the result is called a composition of the transformations.

  7. Find a single transformation for a 75° counterclockwise rotation with center (2,1) followed by a 38° counterclockwise rotation with center (2,1) 113° counterclockwise rotation with center (2,1) 38° 75°

  8. Find a single transformation equivalent to a translation with vector <−2, 7> followed by a translation with vector <9, 3>. Translation with vector <7, 10>

  9. Quadrilateral BGTS has vertices B(–3, 4), G(–1, 3), T(–1 , 1), and S(–4, 2). Graph BGTS and its image after a translation along 5, 0 and a reflection in the x-axis. Step 1 translation along 5, 0 (x, y) → (x + 5, y) B(–3, 4) → B'(2, 4) G(–1, 3)→ G'(4, 3) S(–4, 2)→S'(1, 2) T(–1, 1)→T'(4, 1) Step 2 reflection in the x-axis (x, y)→(x, –y) B'(2, 4) → B''(2, –4) G'(4, 3)→ G''(4, –3) S'(1, 2)→ S''(1, –2) T'(4, 1)→ T''(4, –1)

  10. Quadrilateral RSTU has vertices R(1, –1), S(4, –2), T(3, –4), and U(1, –3). Graph RSTU and its image after a translation along –4, 1and a reflection in the x-axis. Which point is located at (–3, 0)? A. R' B. S' C. T' D. U'

  11. Definition An isometry is a transformation that preserves distance. Translations, reflections and rotations are isometries.

  12. The composition of two or more isometries – reflections, translations, or rotations results in an image that is congruent to its preimage. Glide reflections, reflections, translations, and rotations are the only four rigid motions or isometries in a plane.

  13. Two translations equal One translation

  14. Reflections over two parallel lines equals One translation

  15. Copy and reflect figure EFGH in line p and then line q. Then describe a single transformation that maps EFGH onto E''F''G''H''. Step 1 Reflect EFGH in line p. Step 2 Reflect E'F'G'H' in line q. Answer: EFGH is transformed onto E''F''G''H'' by a translation down a distance that is twice the distance between lines p and q.

  16. Reflections over two intersection lines equals One rotation

  17. Graph Other Compositions of Isometries ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph ΔTUV and its image after a translation along –1 , 5 and a rotation 180° about the origin. Step 1 translation along –1 , 5 (x, y) → (x + (–1), y + 5) T(2, –1) → T'(1, 4) U(5, –2) → U'(4, 3) V(3, –4) → V'(2, 1) Step 2 rotation 180 about the origin (x, y) → (–x, –y) T'(1, 4) →T''(–1, –4) U'(4, 3) → U''(–4, –3) V'(2, 1) → V''(–2, –1)

  18. A. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown. Step 1 A brick is copied and translated to the right one brick length. Step 2 The brick is then rotated 90° counterclockwise about point M, given here. Step 3 The new brick is in place.

  19. The symbol for a composition of transformations is an open circle.  The notation is read as a reflection in the x-axis following a translation of (x+3, y+4).  Be careful!!!  The process is done in reverse!!

  20. You may see various notations which represent a composition of transformations: could also be indicated by

  21. Symbology • Rotation of d degrees of the point (x,y): Rd(x,y) • Translation of vector of the point (x,y): Ta,b(x,y) • Reflexion across the x-axis of the point (x,y): rx-axis(x,y) • Reflexion across the y-axis of the point (x,y): ry-axis(x,y)

  22. Your Turn to Try a Few

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