1. Math II Unit 1 Transformations

2. Transformations
A transformation is a change in position, shape, or size of a figure.

3. Identifying Transformations
Figures in a plane can be
Reflected
Rotated
Translated
To produce new figures. The new figures is called the IMAGE. The original figures is called the PREIMAGE. The operation that MAPS, or moves the preimage onto the image is called a transformation.

4. Copy this down
Rotation about a point
Reflection in a line
Translation

5. Some notation
When you name an image, take the corresponding point of the preimage and add a prime symbol. For instance, if the preimage is A, then the image is A’, read as “A prime.”

6. Terminology
In a transformation, the given figure is called the preimage and the transformed figure is called the image. Points on the image that correspond to points on the preimage are labeled similarly but with primes. A transformation is said to map a figure onto its image.

7. ISOMETRY
An ISOMETRY is a transformation that preserves lengths. Isometries also preserve angle measures, parallel lines, and distances between points. Transformations that are isometries are called RIGID TRANSFORMATIONS.

8. Ex. 2: Identifying Isometries
Which of the following appear to be isometries?
This transformation appears to be an isometry. The blue parallelogram is reflected in a line to produce a congruent red parallelogram.
So reflections are isometric.

9. Ex. 2: Identifying Isometries
Which of the following appear to be isometries?
This transformation is not an ISOMETRY because the image is not congruent to the preimage
Dilations are not isometric

10. Ex. 2: Identifying Isometries
Which of the following appear to be isometries?
This transformation appears to be an isometry. The blue parallelogram is rotated about a point to produce a congruent red parallelogram.
Rotations are isometric.

11. Isometric ARE ISOMETRIC Translations Reflections Rotations
IS NOT ISOMETRIC
Dilation (because it does not preserve the size)

12. Rotations

13. Orientation
The orientation of an object refers to the order of its parts as you move around the object in a clockwise or a counter-clockwise direction.

14. Rotations
A rotation is a type transformation that where every point turns around a center.

15. Rotation Exploration
In order to visualize a rotation, try this. Trace the following letter F and its “string” on a piece of tracing paper.

16. Rotation Exploration
Your results will look something like this.

17. Rotation the Rules Counter wise || clock-clockwise
90/-270 P(x,y) →P’(-y, x)
Example P(3,5) to P’(-5,3)
180/-180 P(x,y) →P’(-x,-y)
Example P(3,5) to P’(-3,-5)
270/-90 P(x,y) →P’(y, -x)
Example P(3,5) to P’(5,-3)

18. Reflections

19. The Butterfly
Reflect the butterfly.

20. Solution

21. A Reflection
 A reflection is a transformation in which each point is mapped onto to its image over a line in such a way that the line is the perpendicular bisector of the line segment connecting the point and its image.

22. A Reflection - a more simple definition
A reflection is a type of transformation that acts likes a mirror with an image reflected in the line.
If you forget the rules then think of folding your paper.

23. The Rules of Reflection
Over x axis P(x,y) →P’(x,-y)
Example P(3,5) to P’(3,-5)
Over y axis P(x,y) →P’(-x,y)
Example P(3,5) to P’(-3,5)
Over y = x P(x,y) →P’(y,x)
Example P(3,5) to P’(5, 3)
Over y = -x P(x,y) →P’(-y,-x)
Example P(3,5) to P’(-5, -3)

24. Translations
 A translation is a transformation that “slides” a figure to a new location. Size and direction are preserved.

25. Translations
In order to translate a figure you need to know two things.
How far will it be translated?
In what direction will it be translated?

26. Fact A translation (or slide) preserves size, shape, distance, and
orientation.

27. Application of a translation

28. Frieze Patterns
A frieze pattern is a pattern that repeats itself along a straight line. The pattern may be mapped onto itself with a translation.
Wallpaper borders are practical applications of frieze patterns.
Frieze patterns can be found around the eaves of some old buildings.

29. Translations – the rules
Example
P(3,5) P(x – 3,y + 4) to P’(0,9)

31. Dilations
A dilation is a transformation (notation  ) that produces an image that is the same shape as the original, but is a different size.   A dilation stretches or shrinks the original figure.     The description of a dilation includes the scale factor (or ratio) and the center of the dilation.   The center of dilation is a fixed point in the plane about which all points are expanded or contracted.  It is the only invariant point under a dilation.
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32. Dilations continued
A dilation of scalar factor k whose center of dilation is the origin may be written: 
 Dk (x, y) = (kx, ky). If the scale factor, k, is greater than 1, the image is an enlargement (a stretch). If the scale factor is between 0 and 1, the image is a reduction (a shrink). (It is possible, but not usual, that the scale factor is 1, thus creating congruent figures.)

33. Review