Title: Transformations
1Chapter 7
2Chapter Objectives
- Identify different types of transformations
- Define isometry
- Identify reflection
- Identify rotations
- Identify translations
- Describe composition transformations
3Lesson 7.1
4Lesson 7.1 Objectives
- Identify basic rigid transformations
- Define isometry
5Definition of Transformation
- A transformation is any operation that maps, or
moves, an object to another location or
orientation.
6Transformation 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
7Types 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
8Definition 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.
9Homework 7.1
- 1-33, 36-39
- p399-401
- In Class 9, 13, 27, 33
- Due Tomorrow
10Lesson 7.2
11Lesson 7.2 Objectives
- Utilize reflections in a plane
- Define line symmetry
- Derive formulas for specific reflections in the
plane
12Reflections
- 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.
13Theorem 7.1Reflection Theorem
- A reflection is an isometry.
- That means a reflection does not change the shape
or size of an object!
14Line 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.
15Homework 7.2a
- 1-11, 22-29
- p407-408
- In Class 7, 23
- Due Tomorrow
16Reflection 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)
17Homework 7.2b
- 12-14, 18-21, 50-51
- p407-410
- In Class 19
- Due Tomorrow
18Lesson 7.3
19Lesson 7.3 Objectives
- Utilize a rotation in a plane
- Define rotational symmetry
- Observe any patterns for rotations about the
origin
20Definitions 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 (-)
21Theorem 7.2Rotation Theorem
- A rotation is an isometry.
22Rotational 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.
23Homework 7.3a
- 1-19
- p416
- In Class 6, 11, 13
- Due Tomorrow
24Rotating 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!
25Theorem 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.
26Homework 7.3b
- 25-35, 45-50, 54
- p417-419
- In Class 25, 35
- Due Tomorrow
- Quiz Wednesday
- Lessons 7.1-7.4
27Lesson 7.4
28Lesson 7.4
- Define a translation
- Identify a translation in a plane
- Use vectors to describe a translation
- Identify vector notation
29Translation 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.
30Theorem 7.4Translation Theorem
- A translation is an isometry.
31Theorem 7.5Distance of Translation Theorem
- The distance of the translation is twice the
distance between the reflecting lines.
x
2x
32Coordinate 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
33Vectors
- 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.
34Component 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
35Use 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)
36Homework 7.4
- 1-30, 44-47
- p425-427
- In Class 3,7,17,25,45
- Due Tomorrow
- Quiz Tomorrow
- Lessons 7.1-7.4
37Lesson 7.5
- Glide Reflections
- and
- Compositions
38Lesson 7.5 Objectives
- Identify a glide reflection in a plane
- Represent transformations as compositions of
simpler transformations
39Glide 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!
40Compositions 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.
41Theorem 7.6Composition Theorem
- The composition of two (or more) isometries is an
isometry.
42Homework 7.5
- 1-8, 9-21, 23-24, 26-30
- skip 16, 28
- p433-435
- In Class 9,13,19
- Due Tomorrow
43Lesson 7.6
44Lesson 7.6 Objectives
- Identify a frieze pattern by type
- Visualize the different compositions of
transformations
45Frieze 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.
46Classifying 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
47Examples
TR
TG
TV
THG
TRVG
TRHVG
48Homework 7.6
- 2-23
- p440-441
- In Class 9,13,17,21
- Due Tomorrow
- Quiz Tuesday
- Lessons 7.5-7.6