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Transformations

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Title: Transformations


1
Chapter 7
  • Transformations

2
Chapter Objectives
  • Identify different types of transformations
  • Define isometry
  • Identify reflection
  • Identify rotations
  • Identify translations
  • Describe composition transformations

3
Lesson 7.1
  • Rigid Motion in a Plane

4
Lesson 7.1 Objectives
  • Identify basic rigid transformations
  • Define isometry

5
Definition of Transformation
  • A transformation is any operation that maps, or
    moves, an object to another location or
    orientation.

6
Transformation Terms
  • When performing a transformation, the original
    figure is called the pre-image.
  • The new figure is called the image.
  • Many transformations involve labels
  • The image is named after the pre-image, by adding
    a prime symbol (apostrophe)
  • A A A
  • We say it as A prime

7
Types of Transformations
Types Reflection Rotation Translation
Characteristics
Orientation
Pictures
Flips object over line of reflection
Turns object using a fixed point as center or
rotation
Slides object through a plane
Stays same just tilted
Order in which object is drawn is reversed
Stays same and stays upright
8
Definition of Isometry
  • An isometry is a transformation that preserves
    length.
  • Isometry also preserve angle measures, parallel
    lines, and distances between points.
  • If you look at the meaning of the two parts of
    the word, iso- means same, and metry- means meter
    or measure.
  • So simply stated, isometry preserves size.
  • Any transformation that is an isometry is called
    a Rigid Transformation.

9
Homework 7.1
  • 1-33, 36-39
  • p399-401
  • In Class 9, 13, 27, 33
  • Due Tomorrow

10
Lesson 7.2
  • Reflections

11
Lesson 7.2 Objectives
  • Utilize reflections in a plane
  • Define line symmetry
  • Derive formulas for specific reflections in the
    plane

12
Reflections
  • A transformation that uses a line like a mirror
    is called a reflection.
  • The line that acts like a mirror is called the
    line of reflection.
  • When you talk of a reflection, you must include
    your line of reflection
  • A reflection in a line m is a transformation that
    maps every point P in the plane to a point P, so
    that the following is true
  • If P is not on line m, then m is the
    perpendicular bisector of PP.
  • If P is on line m, then PP.

13
Theorem 7.1Reflection Theorem
  • A reflection is an isometry.
  • That means a reflection does not change the shape
    or size of an object!

14
Line of Symmetry
  • A figure in the plane has a line of symmetry if
    the figure can be mapped onto itself by a
    reflection in a line.
  • What that means is a line can be drawn through an
    object so that each side reflects onto itself.
  • There can be more than one line of symmetry, in
    fact a circle has infinitely many around.

15
Homework 7.2a
  • 1-11, 22-29
  • p407-408
  • In Class 7, 23
  • Due Tomorrow

16
Reflection Formula
  • There is a formula to all reflections.
  • It depends on which type of a line are you
    reflecting in.
  • vertical
  • horizontal
  • y x

Vertical y-axis x a
Horizontal x-axis y a
y x
( x , y)
( -x 2a , y)
( x , -y 2a)
( y , x)
17
Homework 7.2b
  • 12-14, 18-21, 50-51
  • p407-410
  • In Class 19
  • Due Tomorrow

18
Lesson 7.3
  • Rotations

19
Lesson 7.3 Objectives
  • Utilize a rotation in a plane
  • Define rotational symmetry
  • Observe any patterns for rotations about the
    origin

20
Definitions of Rotations
  • A rotation is a transformation in which a figure
    is turned about a fixed point.
  • The fixed point is called the center of rotation.
  • The amount that the object is turned is the angle
    of rotation.
  • A clockwise rotation will have a negative
    measurement.
  • A counterclockwise rotation will have a positive
    measurement.

clockwise or negative (-)
21
Theorem 7.2Rotation Theorem
  • A rotation is an isometry.

22
Rotational Symmetry
  • A figure in a plane has rotational symmetry if
    the figure can be mapped onto itself by a
    rotation of 180 or less.
  • A square has rotational symmetry because it maps
    onto itself with a 90 rotation, which is less
    than 180.
  • A rectangle has rotational symmetry because it
    maps onto itself with a 180 rotation.

23
Homework 7.3a
  • 1-19
  • p416
  • In Class 6, 11, 13
  • Due Tomorrow

24
Rotating About the Origin
  • Rotating about the origin in 90o turns is like
    reflecting in the line y x and in an axis at
    the same time!
  • So that means to switch the positions of x and y.
  • (x,y) ? (y,x)
  • Then the original x-value changes sign, no matter
    where it is flipped to.
  • So overall the transformation can be described by
  • (x,y) ? (-y,x)
  • Every time you 90o you repeat the process.
  • So going 180o means you do the process twice!

25
Theorem 7.3Angle of Rotation Theorem
  • The angle of rotation is twice as big as the
    angle of intersection.
  • But the intersection must be the center of
    rotation.
  • And the angle of intersection must be acute or
    right only.

26
Homework 7.3b
  • 25-35, 45-50, 54
  • p417-419
  • In Class 25, 35
  • Due Tomorrow
  • Quiz Wednesday
  • Lessons 7.1-7.4

27
Lesson 7.4
  • Translations
  • and
  • Vectors

28
Lesson 7.4
  • Define a translation
  • Identify a translation in a plane
  • Use vectors to describe a translation
  • Identify vector notation

29
Translation Definition
  • A translation is a transformation that maps an
    object by shifting or sliding the object and all
    of its parts in a straight light.
  • A translation must also move the entire object
    the same distance.

30
Theorem 7.4Translation Theorem
  • A translation is an isometry.

31
Theorem 7.5Distance of Translation Theorem
  • The distance of the translation is twice the
    distance between the reflecting lines.

x
2x
32
Coordinate form
  • Every translation has a horizontal movement and a
    vertical movement.
  • A translation can be described in coordinate
    notation.
  • (x,y) ? (xa , yb)
  • Which tells you to move a units horizontal and b
    units vertical.

Q
b units up
P
a units to the right
33
Vectors
  • Another way to describe a translation is to use a
    vector.
  • A vector is a quantity that shows both direction
    and magnitude, or size.
  • It is represented by an arrow pointing from
    pre-image to image.
  • The starting point at the pre-image is called the
    initial point.
  • The ending point at the image is called the
    terminal point.

34
Component Form of Vectors
  • Component form of a vector is a way of combining
    the individual movements of a vector into a more
    simple form.
  • ltx , ygt
  • Naming a vector is the same as naming a ray.
  • PQ

Q
y units up
P
x units to the right
35
Use of Vectors
  • Adding/subtracting vectors
  • Add/subtract x values and then add y values
  • lt2 , 6gt lt3 , -4gt
  • lt5 , 2gt
  • Distributive property of vectors
  • Multiply each component by the constant
  • 5lt3 , -4gt
  • lt15 , -20gt
  • Length of vector
  • Pythagorean Theorem
  • x2 y2 lenght2
  • Direction of vector
  • Inverse tangent
  • tan-1 (y/x)

36
Homework 7.4
  • 1-30, 44-47
  • p425-427
  • In Class 3,7,17,25,45
  • Due Tomorrow
  • Quiz Tomorrow
  • Lessons 7.1-7.4

37
Lesson 7.5
  • Glide Reflections
  • and
  • Compositions

38
Lesson 7.5 Objectives
  • Identify a glide reflection in a plane
  • Represent transformations as compositions of
    simpler transformations

39
Glide Reflection Definition
  • A glide reflection is a transformation in which a
    reflection and a translation are performed one
    after another.
  • The translation must be parallel to the line of
    reflection.
  • As long as this is true, then the order in which
    the transformation is performed does not matter!

40
Compositions of Transformations
  • When two or more transformations are combined to
    produce a single transformation, the result is
    called a composition.
  • So a glide reflection is a composition.
  • The order of compositions is important!
  • A rotation 90o CCW followed by a reflection in
    the y-axis yields a different result when
    performed in a different order.

41
Theorem 7.6Composition Theorem
  • The composition of two (or more) isometries is an
    isometry.

42
Homework 7.5
  • 1-8, 9-21, 23-24, 26-30
  • skip 16, 28
  • p433-435
  • In Class 9,13,19
  • Due Tomorrow

43
Lesson 7.6
  • Frieze Patterns

44
Lesson 7.6 Objectives
  • Identify a frieze pattern by type
  • Visualize the different compositions of
    transformations

45
Frieze Patterns
  • A frieze pattern is a pattern that extends to the
    left or right in such a way that the pattern can
    be mapped onto itself by a horizontal
    translation.
  • Also called a border pattern.

46
Classifying Frieze Patterns
  • The horizontal translation is the minimum that
    must exist.
  • However, there are other transformations that can
    occur.
  • And they can occur more than once.

Type Abbreviation Description





Translation
Horizontal translation left or right
T
180o Rotation
180o Rotation CW or CCW
R
Reflection either up or down in a horizontal line
Reflection in Horizontal Line
H
Reflection in Vertical Line
Reflection either left or right in a vertical line
V
Horizontal translation with reflection in a
horizontal line
Horizontal Glide Reflection
G
47
Examples
TR
TG
TV
THG
TRVG
TRHVG
48
Homework 7.6
  • 2-23
  • p440-441
  • In Class 9,13,17,21
  • Due Tomorrow
  • Quiz Tuesday
  • Lessons 7.5-7.6
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