1.2: Transformations (Mappings)

1. 1.2: Transformations(Mappings) Common Core State Standards for Mathematical Content Congruence G-CO Experiment with transformations in the plane G-CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions [Build on rigid motions as a familiar starting point for development of concept of geometric proof] G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent

2. Transformation Definition – anything that maps (or moves).

3. Transformation 1. Reflection: A transformation that creates a mirror image

5. The blue line down the center is the Line of Reflection The white line down the center is the Line of Symmetry

7. A line of symmetry: When a line of reflection can be drawn through some plane figure so that one side is a reflected image of the other side

8. Lines of symmetry

9. Equilateral Triangle Lines of symmetry

10. Line segment

11. A A’ Preimage: Original shape or object. Preimage A Image: Shape or object after it has been moved. Image A’ (A prime)

12. Reflection • Must draw the line of reflection A A’

13. What do you notice about the transformation between the Preimage of ∆FGH and the image ∆F’G’H’

14. Properties of Reflections • ∆FGH is congruent ∆F’G’H’ The reflected image is congruent to the original figure

15. 2. The orientation of the image is reversed

16. 3. The line of reflection is the perpendicular bisector of the segments joining the corresponding points

18. Reflections on the coordinate plane

20. Is the red figure a reflection of the blue figure?

21. Quick Review on Graphing lines Graph of a horizontal line y = b, where b is the y-intercept Graph of a vertical line x = a, where a is the x-intercept Line in slope intercept form y = mx + b, where m is the slope and b is the y-intercept

22. Graph the following lines: y = 2 x = -1 y = x y = -x x = -1 y = x y = 2 y = -x

23. Do activity worksheet

24. Reflections Summary: 3 properties of reflections: Congruency, orientation, line of reflection is a perpendicular bisector Reflecting over the x–axis (x, y)  ( , ) Reflecting over the y-axis (x, y)  ( , ) Reflecting over the line y = x (x, y)  ( , ) Reflecting over the line y = -x (x, y)  ( , )

25. y x Transfromations 1. Reflection Reflect the triangle using the line: x = 1

27. * Special mirror *

28. Reflect the triangle over The line y = -x * Special mirror *

30. Homework: Reflections Page 727: 34 - 39; 41, 42, 43 (you will graph paper)

32. Translation: a slide A A’

34. Preimage A transformation can be composed of one or more slides image

35. Notice: • The orientation of the image is NOT • reversed (Image was not flipped ) or rotated • Preimage and image are congruent

36. Is it a translation?

37. Describing a translation (x, y) (x + a, y + b) F F’ E E’

38. Describe the translation: (x, y) -> (x – 3, y - 3)

39. Translation in coordinate plane If ΔABC with A(-1,-3), B(1,-1), & C(-1,0), Find the coordinates of the image after the translation: (x,y) (x-3,y+4) Subtract 3 from all x’s Add 4 to all the y’s

40. ΔABC A (-1,-3) B (1,-1) C (-1,0) ΔA’B’C’ A’ ( ) B’ ( ) C’ ( ) Finding the new points (x,y) => (x-3,y+4)

41. Write the translation for this picture (x,y) ( x , y )

42. Find the coordinates of Under the translation of (x-1, y-3)

43. Composition of transformations Describe the transformation Preimage Image

44. Do activity worksheet - Translations

45. * Two different mirrors, reflect over x=1 first, then that reflection over x = 5

46. Homework: Translations Page 735: 22, 23, 24, 25 (you will need graph paper)

47. Rotation (turn) A transformation in which a figure is turned about a fixed point (center of rotation)

48. The fixed point is called the center of rotation

49. Preimage image • Notice: • Preimage and image are congruent • Only the location has changed

50. Notice: • The distance from the center of rotation to the preimage point is the same as the distance from the center of rotation to the corresponding image point.