Transformations, translations, reflections, rotations, dilatations

2. To transform something is to change.
The main transformations include:
•Translation
•Rotation
•Reflection
•Dilatation (Enlargement)

3. Renaming Transformations
• The original shape is
labelled with capital
letters:
• The transformed shape or
“image” is labelled with the
same letters with a “prime”
symbol attached.

4. A translation "slides" an object a fixed
distance in a given direction.
The original object and its image have the:
•Same shape
•Same size
•Same orientation
•Same translation
vector between them
t(4, -3)
t(x,y)  (x+4, y-3)

5. In this example, the
slide" moves the
figure 7 units to the
left and 3 units
down.
Vector t (-7,-3)
Mapping t
t(x,y)  (x-7, y-3)

6. Let's examine some
translations related to
coordinate geometry.
The example shows how
each vertex moves the
same distance in the
same direction.

7. Translation Vector
• What are the
coordinates for:
A, B, C?
A’, B’,C’?
Copy and fill out the chart.
Original (x,y) Image (x,y)
A A’
B B’
C C’

8. • What are the
coordinates for
• A, B, C, D?
• A’, B’, C’, D’?
• Book 1, p. 97
• Q. 1a, 2, a,e,i
Original (x,y) Image (x,y)
A A’
B B’
C C’
D D’

9. Rotations and Art
The concept of rotations can be seen in wallpaper
designs, fabrics, and art work.

10. ROTATION
A rotation is a transformation that turns a
figure about a fixed point.
It is described in terms of:
•The centre of rotation,
•The direction of rotation: cw (-) or ccw (+)
•The degrees of rotation
An original shape and its rotation image
have the:
•Same size
•Same shape,
•Same orientation

11. Rotation Rules

12. Direction
This rotation is 90°
clockwise about the point B.
Book 1, p. 97, Q. 1b, 2b,c,h
Clockwise Counterclockwise
A B
B’
C
C’
A’

13. REFLECTION
A reflection is described by the line of reflection.
An object and its reflection have the:
•Same shape
•Same size
•Opposite orientation

14. The line (where a mirror may be placed) is called
the line of reflection. The distance from a point
to the line of reflection is the same as the distance
from the point's image to the line of reflection.
A reflection can be thought of as a "flipping" of an
object over the line of reflection.
If you folded the two shapes together line of
reflection the two shapes would overlap
exactly!

15. What happens to points in a
Reflection?
• Name the points of
the original triangle.
• Name the points of
the reflected triangle.
• What is the line of
reflection?
• How did the points
change from the
original to the
reflection?

16. Reflection Rules
Page, 97, Q. 1c, 2d, f, g

17. DILATATION
A dilatation is a transformation that produces an image
that has the
•Same shape
•Same orientation
•Different size.
•A centre of Dilatation
•A scale factor
A dilatation used to create an image larger than the
original is called an enlargement.
A dilatation used to create an image smaller than the
original is called a reduction.

18. Dilatation

19. Dilations always involve a change in size.
N.B. EVERY
(x,y) coordinate
of the original
triangle has
been multiplied
by the scale
factor (x2).

20. Dilatation Rules
• Given an enlargement with a scale
factor S, and the center of dilatation at
the origin,
• (x,y)  (Sx, Sy)
• E.g. Centre at (0,0), Scale Factor 2
• (x,y)  (2x, 2y)
• (3,4)  (2x3, 2x4) = (6,8)
• Page 109, Q. 1a, c, f

21. Stretch Rules
• A stretch is a dilatation in which only one of
the coordinates is affected.
• E.g. the x value is stretched by a factor of 2
• (x,y)  (2x, y)
• (5,8)  (10, 8)
• E.g. the y value is compressed by a factor
of 4
• (x,y)  (x, y/4)
• (5,8)  (5, 2) Page 111, Q. 5 a,b,c

22. Stretch

23. REVIEW: Answer each
question………………………..
Does this picture
show a
translation,
rotation, dilation,
or reflection?
How do you
know?
Rotation

24. Does this picture
show a translation,
rotation, dilation, or
reflection?
How do you know?
Dilation

25. Does this picture show a translation,
rotation, dilation, or reflection?
How do you know?
(Line) Reflection

26. Which of the following lettered figures are translations of the
shape of the purple arrow? Name ALL that apply.
Explain your thinking.
Letters a, c, and e are translations
of the purple arrow.

27. Has each picture been rotated in a clockwise
or counter-clockwise direction?
The birds were rotated clockwise and
the fish counterclockwise.

28. Basically, a tessellation is a way to tile a floor
(that goes on forever) with shapes so that
there is no overlapping and no gaps.

29. Dutch graphic artist M. C. Escher (1898-1972) is known for
his creative use of tessellations in his work. What
transformations can you see in this picture?
The birds and fish have been translated here.

30. What
transformations
can you see in this
Escher print?
Some birds
have been
translated
and some
have been
rotated.

31. Can you name examples in real life of each
transformation?
•Translation
•Rotation
•Reflection
•Dilation

32. Check out these sites:
http://www.farraguttn.com/fhs/math/nctm/index.htm
http://www.mathsnet.net/transformations/index.html
http://www.mcescher.com/