Unit 10 Transformations. Lesson 10.1 Dilations Lesson 10.1 Objectives Define transformation (G3.1.1) Differentiate between types of transformations (G3.1.2)

Unit 10Unit 10TransformationsTransformations

Lesson 10.1Lesson 10.1DilationsDilations

Lesson 10.1 ObjectivesLesson 10.1 Objectives

Define transformation (G3.1.1)

Differentiate between types of transformations (G3.1.2)

Define dilation (G3.2.1)

Identify the characteristics of a dilation (G3.2.2)

Calculate the magnitude of a dilation (G3.2.2)

Define transformation (G3.1.1)

Differentiate between types of transformations (G3.1.2)

Define dilation (G3.2.1)

Identify the characteristics of a dilation (G3.2.2)

Calculate the magnitude of a dilation (G3.2.2)

Definition of Transformation

Definition of Transformation

A transformation is any operation that maps, or moves, an object to another location or orientation.

A transformation is any operation that maps, or moves, an object to another location or orientation.

Transformation Terms 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 typically using letters of the alphabet.

The image is named after the pre-image, by adding a prime symbol (apostrophe)

When performing a transformation, the original figure is called the pre-image.

The new figure is called the image.

Many transformations involve labels typically using letters of the alphabet.

The image is named after the pre-image, by adding a prime symbol (apostrophe)

A

A’

Types of TransformationsTypes of Transformations

There are 4 basic transformations:1. A dilation.

2. A reflection in a line.

3. A rotation about a point.

4. A translation.

There are 4 basic transformations:1. A dilation.

2. A reflection in a line.

3. A rotation about a point.

4. A translation.

Example 10.1Example 10.1

Identify the following transformations:

1.


1.

2.

2.

3.

3.

Rotation

Reflection

Translation

DilationDilation

A dilation is a transformation that will increase or decrease the size of the figure while following the rules of similarity.

A dilation has a center which is used much like a focal point to enlarge or reduce every figure from.

All dilations have the following properties:1. If point P is not at the center C, then the image P’ lies on

ray CP.2. If point P is at the center, then P = P’.

A dilation is a transformation that will increase or decrease the size of the figure while following the rules of similarity.

A dilation has a center which is used much like a focal point to enlarge or reduce every figure from.

All dilations have the following properties:1. If point P is not at the center C, then the image P’ lies on

ray CP.2. If point P is at the center, then P = P’.

CP

P’

Scale Factor of a DilationScale Factor of a Dilation

The scale factor of a dilation is a positive number found by taking the distance from the center to the image divided by the distance from the center to the pre-image. It is basically a multiplier of how much the figure has

been reduced or enlarged. We use the letter “k” to stand for the scale factor,

and it can be found by:

The scale factor of a dilation is a positive number found by taking the distance from the center to the image divided by the distance from the center to the pre-image. It is basically a multiplier of how much the figure has

been reduced or enlarged. We use the letter “k” to stand for the scale factor,

and it can be found by:

CP

P’3

12

-

image distancek

pre image distance

'CPk

CP

4k

12

k

3

Reduction or EnlargementReduction or Enlargement

A reduction is when the image is smaller than the pre-image. The scale factor will be

a number between 0 and 1.

0 < k < 1

A reduction is when the image is smaller than the pre-image. The scale factor will be

a number between 0 and 1.

0 < k < 1

An enlargement is when the image is

larger than the pre-image. The scale factor will be

a number greater than 1.

k > 1

An enlargement is when the image is

larger than the pre-image. The scale factor will be

a number greater than 1.

k > 1

C C

DD’G

G’


Tell whether the figure shows an enlargement or a reduction.And also find the scale factor.

1.

Tell whether the figure shows an enlargement or a reduction.And also find the scale factor.

1.

2. 2.

Enlargement

3

2k

12

k

8

Reduction

3

5k

18

k

30

1.5

0.6


1. What type of dilation has occurred?1. What type of dilation has occurred?

Enlargement

2. Find the scale factor.2. Find the scale factor.

3. Find x.3. Find x.

5

k

31.667

5 2

3 x

5 6x 6

5x 1.2

Scale Factor with CoordinatesCentered at the Origin

Scale Factor with CoordinatesCentered at the Origin

When the origin on a coordinate plane is used as the center of dilation, the scale factor is simply distributed to both the x and y values of each coordinate. So multiply k to both the x and y coordinates.

This will not work directly for a dilation centered at any other location.

When the origin on a coordinate plane is used as the center of dilation, the scale factor is simply distributed to both the x and y values of each coordinate. So multiply k to both the x and y coordinates.

This will not work directly for a dilation centered at any other location.

4k

2,6A ' 2 4,6 4A ' 8, 24A


Draw the dilation of ABCD under a scale factor of k = ½ centered at the origin.

Draw the dilation of ABCD under a scale factor of k = ½ centered at the origin.

2,4A

4,4B

4,1C

2, 1D

' 1, 2A

' 2, 2B

' 2,0.5C

' 1, 0.5D

A’ B’

C’

D’

Lesson 10.1 HomeworkLesson 10.1 Homework

Lesson 10.1 – DilationsDue Tomorrow

Lesson 10.1 – DilationsDue Tomorrow

Lesson 10.2Lesson 10.2ReflectionsReflections


Define an isometry (G3.1.2)

Define a reflection (G3.1.1)

Identify characteristics of a reflection

Define line of symmetryUtilize properties of reflections to

perform bank shots

Define an isometry (G3.1.2)

Define a reflection (G3.1.1)

Identify characteristics of a reflection

Define line of symmetryUtilize properties of reflections to

perform bank shots

ReflectionsReflections 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

1. If P is not on line m, then m is the perpendicular bisector of PP’.

2. If P is on line m, then P = P’.

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

1. If P is not on line m, then m is the perpendicular bisector of PP’.

2. If P is on line m, then P = P’.


Give the image of the following reflections:1. K(4,5) in the y-axis

2. W(2,-7) in the x-axis

3. A(7,3) in the y-axis

4. L(-3,5) in the x-axis

5. I(-1,-3) in the y-axis

6. G(-4,-2) in the x-axis

7. N(5,1) over the line y = x.

Give the image of the following reflections:1. K(4,5) in the y-axis

2. W(2,-7) in the x-axis

3. A(7,3) in the y-axis

4. L(-3,5) in the x-axis

5. I(-1,-3) in the y-axis

6. G(-4,-2) in the x-axis

7. N(5,1) over the line y = x.

Reflection FormulaReflection Formula

There is a formula to all reflections. It depends on which type of a line are you reflecting

in. The formulas below will map the original

coordinates of (x,y) to:

There is a formula to all reflections. It depends on which type of a line are you reflecting

in. The formulas below will map the original

coordinates of (x,y) to:Vertical:

y-axis

( -x, y)

Horizontal:

x-axis( x , -y)

y = x

( y , x)

( x , y)

x = a

( -x + 2a, y)

y = a

( x , -y + 2a)


Give the image of the following reflections:1. O(3,0) over the line y = x

2. L(-3,5) over the line y = x

3. C(-2,6) over the line x = 1

4. E(-4,-6) over the line x = -1

5. D(2,4) over the line y = 1

6. R(-1,-5) over the line y = -2

Give the image of the following reflections:1. O(3,0) over the line y = x

2. L(-3,5) over the line y = x

3. C(-2,6) over the line x = 1

4. E(-4,-6) over the line x = -1

5. D(2,4) over the line y = 1

6. R(-1,-5) over the line y = -2

IsometryIsometry

An isometry is a transformation that preserves the following:lengthangle measuresparallel linesdistance between points

An isometry is also called a rigid transformation.

An isometry is a transformation that preserves the following:lengthangle measuresparallel linesdistance between points

An isometry is also called a rigid transformation.

Theorem 7.1:Reflection Theorem

Theorem 7.1:Reflection Theorem

A reflection is an isometry.That means a reflection does not

change the shape or size of an object!

A reflection is an isometry.That means a reflection does not

change the shape or size of an object!

m


Use the diagram to name the image of 1 after the reflection

1. Reflection in the x-axis1. 4

2. Reflection in the y-axis2. 2

3. Reflection in the line y = x3. 3

4. Reflection in the line y = -x4. 1

5. Reflection in the y-axis, followed bya reflection in the x-axis.

5. 3

Use the diagram to name the image of 1 after the reflection

1. Reflection in the x-axis1. 4

2. Reflection in the y-axis2. 2

3. Reflection in the line y = x3. 3

4. Reflection in the line y = -x4. 1

5. Reflection in the y-axis, followed bya reflection in the x-axis.

5. 3

Line of SymmetryLine 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.

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.


Determine the number of lines of symmetry each figure has.1.

Determine the number of lines of symmetry each figure has.1.

2.

2.

3.

3.

4.

4.

5.

5.

6.

6.

One

Two

Four

None

Three

One

Bank ShotsBank Shots

In Physics, the Law of Reflection states that any object that strikes a flat surface will bounce off at the same angle.

FYI: That angle is measured to a normal line, which is drawn perpendicular to the surface.

That concept can be confirmed by utilizing the definition of reflection and applying it with an everyday task such as a bank shot.

Thus the Geometry of a bank shot can be accomplished using the following process:

1. Reflect the target location over the wall to be used for the bank shot.

2. Draw a segment that connects the ball and the image of the target after reflection.

3. The point of intersection of the segment and the wall should be the aiming point to perform the desired bank shot.

In Physics, the Law of Reflection states that any object that strikes a flat surface will bounce off at the same angle.

FYI: That angle is measured to a normal line, which is drawn perpendicular to the surface.

That concept can be confirmed by utilizing the definition of reflection and applying it with an everyday task such as a bank shot.

Thus the Geometry of a bank shot can be accomplished using the following process:

1. Reflect the target location over the wall to be used for the bank shot.

2. Draw a segment that connects the ball and the image of the target after reflection.

3. The point of intersection of the segment and the wall should be the aiming point to perform the desired bank shot.

Geometry of a Bank ShotGeometry of a Bank Shot

Double Bank ShotDouble Bank Shot


Lesson 10.2 – ReflectionsDue Tomorrow

Lesson 10.2 – ReflectionsDue Tomorrow

Lesson 10.3Lesson 10.3RotationsRotations


Define a rotation (G3.1.1)

Identify characteristics of rotation Define rotational symmetry Recognize patterns for rotations around

the origin of a coordinate plane

Define a rotation (G3.1.1)

Identify characteristics of rotation Define rotational symmetry Recognize patterns for rotations around

the origin of a coordinate plane

Definitions of RotationsDefinitions 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. The rotation with either be in a clockwise direction or

a counterclockwise direction. At higher levels of math/science, a clockwise rotation is

said to be negative in direction. So, a counterclockwise rotation is positive in direction.

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. The rotation with either be in a clockwise direction or

a counterclockwise direction. At higher levels of math/science, a clockwise rotation is

said to be negative in direction. So, a counterclockwise rotation is positive in direction.

Theorem 7.2:Rotation Theorem

Theorem 7.2:Rotation Theorem

A rotation is an isometry.A rotation is an isometry.


Solve for x and y.1.

Solve for x and y.1.

2.

2.

Rotating About the OriginRotating About the Origin

Recall that a reflection in the line y = x will swap x and y. (x , y) (y , x)

Now, recall what happens when you perform a reflection in the y-axis.

(x , y) (-x , y) Rotating about the origin in 90o counterclockwise turns is like

reflecting in the line y = x and in the y-axis at the same time! (x , y) (y , x) (-y , x)

To rotate 180o would require you to do that process twice. Rotating about the origin in 90o clockwise turns will do the same,

only the second reflection would be in the x-axis.

Recall that a reflection in the line y = x will swap x and y. (x , y) (y , x)

Now, recall what happens when you perform a reflection in the y-axis.

(x , y) (-x , y) Rotating about the origin in 90o counterclockwise turns is like

reflecting in the line y = x and in the y-axis at the same time! (x , y) (y , x) (-y , x)

To rotate 180o would require you to do that process twice. Rotating about the origin in 90o clockwise turns will do the same,

only the second reflection would be in the x-axis.


Rotate the figure the given number of degrees around the origin. List the coordinates of the image.

1. 90o CCW

Rotate the figure the given number of degrees around the origin. List the coordinates of the image.

1. 90o CCW 2. 180o

CCW 2. 180o

CCW 3. 270o

CCW3. 270o

CCW

Rotational SymmetryRotational Symmetry

A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or less. Notice, the rotation of 180° or less could go either

clockwise or counterclockwise. The rotation must occur around the center of the

object.

A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or less. Notice, the rotation of 180° or less could go either

clockwise or counterclockwise. The rotation must occur around the center of the

object.

This figure has 90o and

180o rotational symmetry


Determine if the figure has rotational symmetry.If so, describe any rotations that would map the figure onto itself.

1.

Determine if the figure has rotational symmetry.If so, describe any rotations that would map the figure onto itself.

1.

2.

2.

3.

3.


Lesson 10.3 – RotationsDue Tomorrow

Lesson 10.3 – RotationsDue Tomorrow

Lesson 10.4Lesson 10.4TranslationsTranslations


Define a translation (G3.1.1)

Describe a translation using coordinate notation

Define a translation (G3.1.1)

Describe a translation using coordinate notation

Translation DefinitionTranslation Definition

A translation is a transformation that maps an object by shifting, or sliding, the object and all of its parts in a straight line. A translation must also move the entire

object the same distance.

A translation is a transformation that maps an object by shifting, or sliding, the object and all of its parts in a straight line. A translation must also move the entire

object the same distance.

Coordinate FormCoordinate Form

Every translation has a horizontal amount of movement and a vertical amount of movement.

A translation can be described in coordinate notation.

Every translation has a horizontal amount of movement and a vertical amount of movement.

A translation can be described in coordinate notation.

, ( , )x y x a y b

A translation in the horizontal direction will change x.

A translation in the vertical direction will change y.

Adding values to thex-coordinate will move the object to the right

Subtracting values from thex-coordinate will move the object to the left

Subtracting values from they-coordinate will move the object to down

Adding values to they-coordinate will move the object to up


Describe the translation in words, and using coordinate notation.

1.

Describe the translation in words, and using coordinate notation.

1. 2. 2. 3. 3.

Right 1 and Up 3

, ( 1, 3)x y x y

Left 5 and Up 3

, ( 5, 3)x y x y

Right 1 and Down 2

, ( 1, 2)x y x y


Find the coordinates of the image after performing the given translation.

1. R(3,5)(x,y) (x+2 , y+7)

1. R’(5,12)2. A(-1,-6)

(x,y) (x+4 , y-3)2. A’(3,-9)

3. C(2,-4)(x,y) (x-8 , y+1)

3. C’(-6,-3)4. E(4,-2)

(x,y) (x-4 , y-5)4. E’(0,-7)

Find the coordinates of the image after performing the given translation.

1. R(3,5)(x,y) (x+2 , y+7)

1. R’(5,12)2. A(-1,-6)

(x,y) (x+4 , y-3)2. A’(3,-9)

3. C(2,-4)(x,y) (x-8 , y+1)

3. C’(-6,-3)4. E(4,-2)

(x,y) (x-4 , y-5)4. E’(0,-7)


Find the coordinates of the image of each vertex of the polygon by performing the given translation.

1. (x,y) (x+3 , y-2)

Find the coordinates of the image of each vertex of the polygon by performing the given translation.

1. (x,y) (x+3 , y-2) 2. (x,y) (x-5 , y+6)2. (x,y) (x-5 , y+6)

Theorem 7.4:Translation Theorem

Theorem 7.4:Translation Theorem

A translation is an isometry.A translation is an isometry.


Lesson 10.4 – TranslationsDue Tomorrow

Lesson 10.4 – TranslationsDue Tomorrow

Lesson 10.5Lesson 10.5VectorsVectors


Utilize vectors to perform translations (L1.2.3)

Identify properties of vectors (L1.2.3)

Perform vector addition/subtraction (L1.2.3)

Utilize vectors to perform translations (L1.2.3)

Identify properties of vectors (L1.2.3)

Perform vector addition/subtraction (L1.2.3)

VectorsVectors

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.

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.

Component Form of VectorsComponent Form of Vectors

Component form of a vector is a way of combining the individual horizontal and vertical movements of a vector into a more simple form. <x , y>

Component form works the same as coordinate notation

Naming a vector is the same as naming a ray :

Component form of a vector is a way of combining the individual horizontal and vertical movements of a vector into a more simple form. <x , y>

Component form works the same as coordinate notation

Naming a vector is the same as naming a ray :

, ( 5, 3)x y x y is now just 5,3

FGOOOOOOOOOOOOOO


Name the given vector and give the component form of the vector.

1.

Name the given vector and give the component form of the vector.

1.

4,3

CDOOOOOOOOOOOOOO

2. 2.

10, 2

ABOOOOOOOOOOOOOO

3. 3.

4,4

ABOOOOOOOOOOOOOO

4. 4.

7, 3

DROOOOOOOOOOOOOO

Magnitude & Directionof a Vector

Magnitude & Directionof a Vector

Recall that every vector has a magnitude and direction.

The magnitude measures the size of the vector.

For translations, the magnitude will measure the distance between the pre-image and the image.

To find the magnitude of a vector, you must use Pythagorean Theorem with the component form of the vector.

Recall that every vector has a magnitude and direction.

The magnitude measures the size of the vector.

For translations, the magnitude will measure the distance between the pre-image and the image.

To find the magnitude of a vector, you must use Pythagorean Theorem with the component form of the vector.

The direction of the vector is measured as the angle made at the initial point of the vector.

This is typically the angle made with the positive or negative x-axis.

To find that angle, you must use inverse trigonometry to find the angle made at the initial point of the vector.

The direction of the vector is measured as the angle made at the initial point of the vector.

This is typically the angle made with the positive or negative x-axis.

To find that angle, you must use inverse trigonometry to find the angle made at the initial point of the vector.

2 2 2a b c 22 2x y Vector

1tany

x


Find the magnitude and direction of each vector.

1.

Find the magnitude and direction of each vector.

1.

2. 2.

3. 3. 12,5PQ QQQQQQQQQQQQQQ

22 2x y JM QQQQQQQQQQQQQQ

22 25 2 JM QQQQQQQQQQQQQQ

29 5.4JM QQQQQQQQQQQQQQ

1tany

x

1 2tan

5

o21.8

22 24 2 MT QQQQQQQQQQQQQQ

1 2tan

4

20 4.5MT

QQQQQQQQQQQQQQo26.6

22 212 5 PQ

QQQQQQQQQQQQQQ1 5

tan12

169 13PQ QQQQQQQQQQQQQQ o22.6

Vector AdditionVector Addition

Imagine a translation that maps a figure by moving 4 units to the right, 5 units up, 6 units to the right, and 2 units up. How many units will horizontally has the figured moved?

10 How many units vertically has the figured moved?

7

So the resulting translation would be described as adding the vectors <4,5> and <6,2> by adding like terms. <4+6,5+2> = <10,7>

The same can be done when subtracting vectors as well.

Imagine a translation that maps a figure by moving 4 units to the right, 5 units up, 6 units to the right, and 2 units up. How many units will horizontally has the figured moved?

10 How many units vertically has the figured moved?

7

So the resulting translation would be described as adding the vectors <4,5> and <6,2> by adding like terms. <4+6,5+2> = <10,7>

The same can be done when subtracting vectors as well.


Perform the stated vector operation.1.

Perform the stated vector operation.1.

2.

2.

3.

3.

3,9 5,7v wQQQQQQQQQQQQQ Q

4.

4.

10,12 3,2a b

8,3 1,2s t

4,2 2, 9m n QQQQQQQQQQQQQQQQQQQQQQQQQQQQ

5.

5. 11, 3 3,4 8, 2d e f

QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ


Lesson 10.5 – VectorsDue Tomorrow

Lesson 10.5 – VectorsDue Tomorrow

Lesson 10.6Lesson 10.6Glide Reflections and CompositionsGlide Reflections and Compositions


Define transformationDefine transformation

Compositions of TransformationsCompositions of Transformations

When two or more transformations are combined to produce a single transformation, the result is called a composition.

When two or more transformations are combined to produce a single transformation, the result is called a composition.

Order is ImportantOrder is Important

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.

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.



I.

I. Rotation

II.

II. Reflection

III.

III. Translation


I.

I. Rotation

II.

II. Reflection

III.

III. Translation



I.

I. Rotation

II.

II. Reflection

III.

III. Translation


I.

I. Rotation

II.

II. Reflection

III.

III. Translation

Theorem 7.6:Composition Theorem

Theorem 7.6:Composition Theorem

The composition of two (or more) isometries is an isometry.

The composition of two (or more) isometries is an isometry.

Glide Reflection DefinitionGlide 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!

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!



I.

I. Rotation

II.

II. Reflection

III.

III. Translation


I.

I. Rotation

II.

II. Reflection

III.

III. Translation


Lesson 10.6 – Glide Reflections & Compositions

Due Tomorrow

Lesson 10.6 – Glide Reflections & Compositions

Due Tomorrow

Documents

Unit 10 Transformations. Lesson 10.1 Dilations Lesson 10.1 Objectives Define transformation (G3.1.1) Differentiate between types of transformations (G3.1.2)