Unit 6: Transformations

Translations

Objectives: SWBAT translate point and images on a coordinate plane

A. Identifying Transformations

Translation ~ (AKA a slide) moves all points of an image the same distance in the

same directions

Preimage [A] ~ an image before

any transformations occur.

Image [A’]~ the image after any

transformations occurs

Isometry ~ study of congruent transformation

Also known as rigid transformations.

Coordinate Notation ~ (𝒙, 𝒚) → (𝒙 + 𝒂, 𝒚 + 𝒃) where x and y are the original

Coordinates, and the a, and b are the shifts

Consider the translation that is defined by the coordinate notation (x, y) (x + 2, y – 3)

1. What is the image of (1, 2)? (𝟏, 𝟐) → (𝟏 + 𝟐, 𝟐 − 𝟑) 𝑨𝒏𝒔𝒘𝒆𝒓: (𝟑, −𝟏)

2. What is the image of (3, -4)? (𝟑, −𝟒) → (𝟑 + 𝟐, −𝟒 − 𝟑) 𝑨𝒏𝒔𝒘𝒆𝒓: (𝟓, −𝟕)

3. What is the pre-image of (5, 8)? (𝟓, 𝟖) → (𝟓 − 𝟐, 𝟖 + 𝟑) 𝑨𝒏𝒔𝒘𝒆𝒓: (𝟑, 𝟏𝟏)

4. What is the pre-image of (0, -4)? (𝟎, −𝟒) → (𝟎 − 𝟐, −𝟒 + 𝟑) 𝑨𝒏𝒔𝒘𝒆𝒓: (−𝟐, −𝟕)

Draw the following segments after the translation.

6. (x, y)(x + 1, y – 5) 7. (x, y)(x + 5, y + 3)

Blue is pre-image, move right 1, Blue is pre-image, move right 5,

Down 5 to get to the image. Up 3 to get to the image

8. Graph the image after a translation of ⟨−1 , −4 ⟩ 9. Find LMN , where

𝑺𝒂𝒎𝒆 𝑨𝒔 (𝒙, 𝒚) → (𝒙 − 𝟏, 𝒚 − 𝟒) with a translation of (3, 1) ,

and then graph the image.

Black is pre-image, move left 1, The given information is the image

Down 4 to get to the image. We need to do the opposite of the

Opposite of the Translation to find

the pre image so subtract 3 and 1.

' 0,4 , ' 2, 1 , ' 2,0L M N

Reflections Objectives: SWBAT find the reflections of images and points over lines and axis

Reflection~

Flip an image over a line called the Line of Reflection.

Each point of the preimage and its image are

the equidistant from the line of the reflection.

Line of reflection:

A line where an image is reflected over.

Transformations in the Coordinate Plane Rules

Reflection over the 𝑥 𝑎𝑥𝑖𝑠: (𝒙, 𝒚) → (𝒙, −𝒚)

Reflection over the 𝑦 𝑎𝑥𝑖𝑠: (𝒙, 𝒚) → (−𝒙, 𝒚)

Reflection over the line 𝑦 = 𝑥 (𝒙, 𝒚) → (𝒚, 𝒙)

1. Draw the image of segment that is a:

Use the rules above to draw the segments. Start at the Red.

a. Reflection in the 𝑥 − 𝑎𝑥𝑖𝑠

Green Line

b. Reflection in the 𝑦 − 𝑎𝑥𝑖𝑠

Orange Line

c. Reflection in the line 𝑦 = 𝑥

Magenta Line

d. Reflection in the 𝑦 − 𝑎𝑥𝑖𝑠, followed by a reflection in the 𝑥 − 𝑎𝑥𝑖𝑠

Magenta Line

e. Reflection in the 𝑥 − 𝑎𝑥𝑖𝑠, followed by a reflection in the 𝑦 − 𝑎𝑥𝑖𝑠.

Magenta Line

f. Reflection in the line 𝑦 = 𝑥, followed by a Reflection in the x-axis.

Orange Line

A. Name the point. then plot its’ reflection over the x axis, y axis, and y=x.

Use the rules above to draw the segments.

𝑥 𝑎𝑥𝑖𝑠 𝑦 𝑎𝑥𝑖𝑠 𝑦 = 𝑥

1. A (3, 2) (𝟑, −𝟐) (−𝟑, 𝟐) (𝟐, 𝟑)

2. B (5, 0) (𝟓, 𝟎) (−𝟓, 𝟎) (𝟎, 𝟓)

3. C (-3, -1) (−𝟑, 𝟏) (𝟑, −𝟏) (−𝟏, −𝟑)

4. Plot each point, then plot its’ reflection in the line 𝑦 = 2. Name the point.

a. M (4, 4) 𝑀′(𝟒, 𝟎)

b. N (-5, 2) 𝑁′(−𝟓, 𝟐)

c. P (-2, -4) 𝑃′(−𝟐, 𝟖)

3. Describe the composition of the transformation using coordinate notation for

the above translations .

a. M (4, 4) (𝒙, 𝒚) → (𝒙, 𝒚 − 𝟐)

b. N (-5, 2) (𝒙, 𝒚) → (𝒙, 𝒚)

c. P (-2, -4) (𝒙, 𝒚) → (𝒙, 𝒚 + 𝟏𝟐)

Rotations Objectives: SWBAT find and rotate images and points over center of rotation

Rotation~ A “turn”

is a transformation around a fixed point called the center of rotation, through a

specific angle, and in a specific direction.

Each point of the preimage and its image are the same distance from the center.

Center of Rotation:

The fixed point a rotation turns around

Angle of Rotation:

The amount of turning of a rotation

Clockwise~

same direction as a clock's hands

Counter Clockwise~

Opposite same direction as a clock's hands

Complete the following sentences.

1. A clockwise rotation of 45 around P maps A onto _B_.

2. A clockwise rotation of 90 around P maps C onto _ E_.

3. A clockwise rotation of 180 around P maps _ B __ onto F.

4. A counterclockwise rotation of 90 around P maps H onto _ F _.

5. A counterclockwise rotation of 45 around P maps __F _ onto E.

6. A counterclockwise rotation of 135 around P maps _ B__ onto A.

p

A

B

C

DE

F

G

H

45

Rotations can be counterclockwise and Clockwise so be careful.

Convert the following clockwise measurements to counterclockwise measurements.

A) 90 degrees is _____270__________ counterclockwise

B) 180 degrees is ____180___________ counterclockwise

C) 270 degrees is ____90___________ counterclockwise

Coordinate Plane and Rotation

Special Rotation Rules

90° (𝒙, 𝒚) → (−𝒚, 𝒙)

180° (𝒙, 𝒚) → (−𝒙, −𝒚)

270° (𝒙, 𝒚) → (𝒚, −𝒙)

Counter Clockwise ONLY. If you have clockwise you must convert to counter.

Examples

1. Given XYZ with vertices X(2, 0), Y( 7, 0), and Z ( 2, 6). Then rotate it 180

degrees around the origin.

𝑿(𝟐, 𝟎)

𝒀( 𝟕, 𝟎) 𝒁 ( 𝟐, 𝟔)

𝑿′(−𝟐, 𝟎)

𝒀′( −𝟕, 𝟎) 𝒁′ (− 𝟐, −𝟔)

x axis

y axis

(x,y)

(x',y')

60°

Rotate 90ABC counterclockwise around the origin. The Vertices are

A( 0, 2), B ( 3, 1) and C( 4, 3).

𝑨( 𝟎, 𝟐)

𝑩( 𝟑, 𝟏) 𝑪( 𝟒, 𝟑)

𝑨′( −𝟐, 𝟎)

𝑩′( −𝟏, 𝟑) 𝑪′(−𝟑, 𝟒)

3. Rotate 270MNO counterclockwise around the origin. The Vertices are

M( 0, 4), N ( 4, 0) and O( 4, 2).

𝑴( 𝟎, 𝟒)

𝑵( 𝟒, 𝟎) 𝑶( 𝟒, 𝟐)

𝑴′( 𝟒, 𝟎)

𝑵′( 𝟎, −𝟒) 𝑶′( 𝟐, −𝟒)

The point 𝑃(−2, −5) is rotated 90° counter clockwise about the origin, and then the image is

reflected across the line 𝑥 = 3. What are the coordinates of the final image 𝑃" ?

Make sure you go in order of the transformations and record each step so you don’t get

lost.

p

P’

Rotation

P’’

Reflection

𝑷(−𝟐, −𝟓)

𝑷′(𝟓, −𝟐)

𝑷′′( 𝟏, −𝟐)

x

y

G

K

H

What are the coordinates for the image of ∆𝐺𝐻𝐾 after a rotation 90° clockwise about

the origin and a translation of (𝑥, 𝑦) → (𝑥 + 3, 𝑦 + 2) ?

All the rules are for clockwise, so we need to convert the 90 degree clockwise to

270 degrees counterclockwise.

Pre-Image

Rotation

Translation

𝑮(−𝟐, −𝟑)

𝑮′(−𝟑, 𝟐)

𝑮′′(𝟎, 𝟒)

𝑯(𝟏, −𝟓)

𝑯′(−𝟓, −𝟏)

𝑯′′(−𝟐, 𝟏)

𝑲(𝟐, −𝟏)

𝑲′(−𝟏, −𝟐)

𝑲′′(𝟐, 𝟎)

Rotating around a point other than the origin

1. Write down your objects key points (if it is a triangle, put down the 3 vertices)

2. Subtract the point you are rotating around from each point (mind your PEMDAS)

3. Apply correct rotation rule

4. Add the point you are rotating back

5. Plot your points.

Figure ABC is rotated 90° clockwise about the point (2, 0). What are the coordinates of 𝐴′ after

the rotation?

Points

6,3

3, 2

4,5

A

B

C

*

int

6,3

2,0

4,3

Subtract the Rotation Po

A

A

*

int

3,2

2,0

1,2

Subtract the Rotation Po

B

B

*

int

4,5

2,0

2,5

Subtract the Rotation Po

C

C

*

*

*

,

4,3 3, 4

1,2 2, 1

2,5 5, 2

Apply Rotation Rule

y x

A

B

C

int

3,4

2,0

' 5,4

Add Back the Rotation Po

A

A

int

2, 1

2,0

' 4, 1

Add Back the Rotation Po

B

A

int

5, 2

2,0

' 7, 2

Add Back the Rotation Po

C

C

7. Graph XYZ with vertices X(2, 0), Y( 7, 0), and Z ( 2, 6). Then rotate it 180 degrees around (2, -3).

*

*

*

Points

2,0 2, 3 0,3

7,0 2, 3 5,3

2,6 2, 3 0,9

Rotation

X X

Y Y

Z Z

*

*

*

0,

,

03

5,3

0,9

, 3

5, 3

0, 9

Apply Rotation Rule

x y

X

Y

Z

*

*

*

int

2, 3 ' 2, 6

2

0,

, 3 ' 3, 6

2, 3 ' 2, 12

3

5, 3

0, 9

Add Back the Rotation Po

X X

Y Y

Z Z

Dilations Objectives: SWBAT reduce or enlarge a figure by a scale factor of k.

Dilation~ enlarges or reduces the original figure proportionally

Multiply each part of the image by the scale factor to get the dilated image.

Scale Factor~ “k” the amount to enlarge each point

𝒌 > 𝟏 → 𝒆𝒏𝒍𝒂𝒓𝒈𝒆𝒎𝒆𝒏𝒕 𝒌 < 𝟏 → 𝒓𝒆𝒅𝒖𝒄𝒕𝒊𝒐𝒏

Examples

A. Graph AB with vertices (0,2)A and (2,1)B . Then find the coordinates of the dilation

image of AB with scale factor of 3, and graph its dilation image.

𝑨(𝟎, 𝟐) → 𝟑(𝟎, 𝟐) → 𝑨′(𝟎, 𝟔) 𝑩(𝟐, 𝟎) → 𝟑(𝟐, 𝟎) → 𝑩′(𝟔, 𝟎)

B. Graph DEF with vertices (3,3)D , (0, 3)E and ( 6,3)F . Then find the coordinates

of the dilation image with scale factor of 1

3 and graph its dilation image.

𝑫(𝟑, 𝟑) →𝟏

𝟑(𝟎, 𝟐) → 𝑫′(𝟏, 𝟏)

𝑬(𝟎, −𝟑) →𝟏

𝟑(𝟎, −𝟑) → 𝑬′(𝟎, −𝟏)

𝑭(−𝟔, 𝟑) →𝟏

𝟑(−𝟔, 𝟑) → 𝑭′(−𝟐, 𝟏)

C. Graph image with vertices '( 1,1)A , '( 2,2)B , '(2,2)C and '(1,1)D with 𝑘 = 0.5. Find

the coordinates of the dilation pre-image and graph its pre-image.

In this example, you get the image, work backwards to find the pre-image.

𝑨′(−𝟏, 𝟏) → 𝟐(−𝟏, 𝟏) → 𝑨(−𝟐, 𝟐)

𝑩′(−𝟐, 𝟐) → 𝟐(−𝟐, 𝟐) → 𝑩(−𝟒, −𝟐) 𝑪′(𝟐, 𝟐) → 𝟐(𝟐, 𝟐) → 𝑪(𝟒, 𝟒) 𝑫′(𝟏, 𝟏) → 𝟐(𝟏, 𝟏) → 𝑫(𝟐, 𝟐)

Apply the dilation 𝐷: (𝑥, 𝑦) → (4𝑥, 4𝑦) to the triangle given vertices. Which of the following

is the perimeter of the image ∆𝐴′𝐵′𝐶′ ?

In this example, 𝑫: (𝒙, 𝒚) → (𝟒𝒙, 𝟒𝒚) means the same thing as a 𝒌 = 𝟒

Perimeters of dilations are also proportional to the scale factor, so you have

two options, find the dilated sides, and add them up. Or find the perimeter

currently, and multiply it by the scale factor.

What is the scale factor for the dilation of Δ𝐴𝐵𝐶 to image Δ𝐴′𝐵′𝐶′ ?

𝒌 = 𝟒

y

x

A'

A

B'

C'

C

B

x

y

C

A B

Transformation Rules

REFLECTION

Explanation

OVER

𝒚 = 𝒙

OVER

𝒚 𝒂𝒙𝒊𝒔

OVER

𝒙 𝒂𝒙𝒊𝒔 LINE

𝒚 = 𝑨 𝑵𝒖𝒎𝒃𝒆𝒓 LINE

𝒙 = 𝑨 𝑵𝒖𝒎𝒃𝒆𝒓

A mirror image over a

line

(𝒙, 𝒚) → (𝒙, −𝒚) (𝒙, 𝒚) → (−𝒙, 𝒚) (𝒙, 𝒚) → (𝒚, 𝒙) Must Count to

the line

Must Count to

the line

When can’t I count?

Can’t count with you go over the line 𝒚 = 𝒙

ROTATION

Explanation

90 Degrees Counterclockwise

180 Degrees Counterclockwise

270 Degrees Counterclockwise

Spinning around a fixed point

(𝒙, 𝒚) → (−𝒚, 𝒙) (𝒙, 𝒚) → (−𝒙, −𝒚) (𝒙, 𝒚) → (𝒚, −𝒙)

What do I do when

I don’t rotate around the origin

Subtract the Point of Rotation Do the Rule

Add the Rule Back In

DILATION

Explanation

SHRINK GROW Coordinate

Form

Shrinking or

Enlarging an image

K value is in-between 0 and 1

𝟎 < 𝒌 < 𝟏

K value is greater than 1

𝒌 > 𝟏

(𝒙, 𝒚) → (𝒌𝒙, 𝒌𝒚)

TRANSLATION

Explanation

Coordinate Form Vector Form

A slide

Shifting the image left / right, up / down

(𝒙, 𝒚) → (𝒙 ± 𝒂, 𝒚 ± 𝒃) (𝒂, 𝒃) or ⟨𝒂, 𝒃⟩