Lesson 12.2 Translations and Reflections pp. 504-508

1. Lesson 12.2 Translations and Reflections pp. 504-508

2. Objectives: 1. To define and perform translations and rotations. 2. To illustrate translations and rotations as compositions of reflections. 3. To define the identity transformation.

3. Any time you perform two or more transformations on a geometric figure, you are performing a composition of transformations.

4. Definition A translation is a transformation formed by the composition of two reflections in which the lines of reflection are parallel lines. A translation can be thought of as a sliding movement of the plane.

7. B A C D l1 l2 B A C D

8. Definition A rotation is a transformation formed by the composition of two reflections in which the lines of reflection intersect.

9. J H J I h X I H H I k J X is the center of the rotation. The direction of this rotation is clockwise.

10. J H J I h X I H H I k J The magnitude of the rotation is twice the measure of the acute or right angle between the lines of reflection.

11. J H J I h X I H H I k J If mHXH is 95°, the magnitude of the rotation is 95° and the angle between the lines of reflections is 47.5°.

12. The identity transformation is a transformation that maps each point of a geometric figure onto itself.

13. Homework pp. 506-508

14. ►B. Exercises 13. If the magnitude of a rotation is 80°, what is the measure of the acute angle between the lines of reflection?

15. ►B. Exercises 15. Draw an acute triangle and rotate it 70° clockwise about point O. Then rotate the image 70° counterclockwise about point O. What is the composition of these rotations called?

16. ►B. Exercises 16. Repeat exercise 15, using two different centers. What is the composition?

17. ►B. Exercises 17. If l and m intersect at point P to form a 40° angle, then what is the composite of the reflections in l and m? Give its center and magnitude.

18. ►B. Exercises 17. l m 40° P

19. ►B. Exercises 18. If R is the reflection in l, and T is the reflection in m, does R◦T = T◦R?

20. 22 7 3 , 3.14, 10, 32, (1.1)12,  ■ Cumulative Review 23. Decide which numbers are greater than others and put them in increasing order (Hint: decimals).

21. 3 2 -2  4.1 -2 -1 0 1 2 3 4 5 3 2 - 2 ■ Cumulative Review 24. Graph the set on the number line: {-2, - , 2, , 4.1}

22. ■ Cumulative Review Give the area and perimeter of each figure. Figure Perimeter Area 25. Circle 26. Rectangle 27. Reg. Polygon

23. Analytic Geometry Translating Conic Sections

24. Circle standard position x2 + y2 = r2 translated position (x-h)2 + (y-k)2 = r2 with center (h, k)

25. Parabola standard position y = ax2 translated position y - k = a(x - h)2 or: y = a(x - h)2 + k with vertex (h, k)

26. Graph x2 + (y - 1)2 = 4

27. Graph y = (x - 2)2 - 3

28. Write the equation that describes the graph.

29. (x - 4)2 + (y + 1)2 = 9

30. ►Exercises Graph. 1. (x + 2)2 + y2 = 4

31. ►Exercises Graph. 2. y = x2 + 1

32. ►Exercises Graph. 3. x2 + (y - 2)2 = 1

33. ►Exercises Graph. 4. y = 2(x + 1)2 + 4

34. ►Exercises Graph. 5. (x - 4)2 + (y - 3)2 = 25