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 One-to-one Mappingy x
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 = x2 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. 
                                                                                
Output
Input
P’ (P Prime)
(P)
(y)
(x)

8. Mappings and Functions
M : P  P’
Function
f : x  x2
Name

9. Mappings and Functions
M : P  P’
Function
f : x  x2
Pre image

10. Mappings and Functions
M : P  P’
Function
f : x  x2
Image

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)
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
Describe the translation
What are we telling the (x) and (y) to do?
Ex. T: (1,3)  (5,1)
(x (+/-) __ , y (+/-) __ )
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
T: (x,y)  (x +2, y-1)
Then reflect over the y-axis
Breakdown
Step 1: Perform the “glide” – A’ = (-1,6)
Step 2: Perform the “reflection” – A” = (1, 6)

26. WHITEBOARDS
CLASSROOM EXERCISES
#1, 2, 8

27. 14-4 ROTATIONS

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

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