1. Chapter 14 Transformations

2. 14.1 Mappings and Functions

3. Transformational Geometry One branch of geometry, known as transformational geometry, investigates how one geometric figure can be transformed into another. In transformational geometry we are required to reflect, rotate, and change the size of the figures.

4. Mappings = Functions Mapping  Geometry: Correspondence between a set of points. Function  Algebra: Correspondence between sets of numbers.

5. One-to-one A mapping (or a function) from set A to set B is called a one-to-one mapping (or function) if every member of B has exactly one pre image in A.

6. Mappings = Functions One to one Function y = x -3 f(x) = x – 3 M: x  x-3 One-to-one Mapping M : P  P’ Not one-to-one y = x 2 P P’

7. Think of when you copy/paste a picture on your computer. The original figure is called the pre-image; the new (copied) picture is called the image of the transformation. Input Output (P) P’ (P Prime) (x) (y)

8. Mappings and Functions Mapping M : P  P’ Function f : x  x 2

9. Mappings and Functions Mapping M : P  P’ Function f : x  x 2

10. Mappings and Functions Mapping M : P  P’ Function f : x  x 2

11. Example 1 Function k maps every number to a number that is 6 more than its double. –Express this fact using function notation –Find the image of 9 –Find the pre image of 16

12. Example 2 Mapping T maps each point (x,y) to the point (x+2, 3y) –Express this fact using mapping notation –Find P’ and Q’ the images of P(2,4) and Q(-2,6)

14. Transformation A one-to-one mapping from the whole plane to the whole plane. –Reflection –Translation –Glide Reflection –Rotation

15. Isometry A transformation that maps every segment to a congruent segment “Preserves distance” –Every point is the same distance from the other points as it was in the original image –This is how we can always check if we have an isometry

16. Theorem An isometry maps a triangle to a congruent triangle

17. Corollary An isometry maps an angle to a congruent angle

18. Corollary An isometry maps a polygon to a polygon with the same area.

19. Mapping S maps each point (x,y) to and image point (x,-2y). Given A(-3,1) B(-1,3) C(4,1) and D(2,-1) –Decide whether S is an isometry

20. 14-3 Translations

21. Brightstorm intro

22. WHICH DIRECTION DO WE SLIDE? Are we adding or subtracting from our x and y coordinates?  x + a we move to the right  x - a we move to the left  y + a we move up  y - a we move down

23. Translations: Types of Problems 1.Describe the translation –What are we telling the (x) and (y) to do? –Ex. T: (1,3)  (5,1) –(x (+/-) __, y (+/-) __ ) 2.Use image to work backwards to find Pre-image –Use the same example as above… –Ex. T: (__, __ )  (2,8) –What did x have to be to get to 2? What did y have to be to get to 8?

24. Glide Reflections Brightstorm intro

25. Glide Reflection Breakdown A (-3, 7) T: (x,y)  (x +2, y-1) Then reflect over the y-axis Breakdown 1.Step 1: Perform the “glide” – A’ = (-1,6) 2.Step 2: Perform the “reflection” – A” = (1, 6)

26. WHITEBOARDS CLASSROOM EXERCISES –#1, 2, 8

27. 14-4 ROTATIONS

28. 3 things…. 1.Point of rotation – A fixed point from which we will rotate (this is were the pointed end of the compass would go) 2.Angle of rotation – How many degrees around the point of rotation am I going? 3.Direction 1.Counterclockwise (positive) 2.clockwise (negative)

29. Half – Turn H o : (x, y)  (-x, -y) This is a special type of rotation in which each point is rotated 180 degrees