Transformations3-6, 3-7, & 3-8

Transformation•Movements of a figure in a plane

•May be a SLIDE, FLIP, or TURN

•a change in the position, shape, or size of a figure.

ImageThe figure you get after a

translation

Original Image

SlideA A’

B B’C C’

The symbol ‘ is read “prime”. ABC has been moved to A’B’C’. A’B’C’ is the image of ABC.

To identify the image of point A, use prime To identify the image of point A, use prime notation Anotation All. .

You read AYou read All as “A prime”. as “A prime”.

Translation•a transformation that moves

each point of a figure the same distance and in the same direction.

AKA - SLIDEA

BC

A’

C’ B’

Writing a Rule for a Translation

Finding the amount of

movement LEFT and RIGHT and UP and DOWN

Writing a Rule9

8

7

6

5

4

3

2

1

0 1 2 3 4 5 6 7 8 9

Right 4 (positive change in x)

Down 3 (negative change in y)

A

A’

B

B’

C

C’

Writing a RuleCan be written as:

R4, D3(Right 4, Down 3)

Rule: (x,y) (x+4, y-3)

Translations

Example 1: If triangle ABC below is translated 6 units to the right and 3 units down, what are the coordinates of point Al.

A (-5, 1) B (-1, 4) C (-2, 2)-First write the rule and then translate each point.

Al = (1, -2)

Rule (x+6, y-3)Rule (x+6, y-3)

Bl = (5, 1) Cl = (4, -1)-Now graph both triangles and see if your image points are

correct.

A

B

C

A’

B’

C’

Example 2: Triangle JKL has vertices J (0, 2), K (3, 4), L (5, 1). Translate the triangle 4 units to the left and 5 units up. What are the new coordinates of Jl?

-First graph the triangle and then translate each point.

Jl = (-4, 7)-You can use arrow notation to describe a translation.

For example: (x, y) (x – 4, y + 5) shows the ordered pair (x, y)

and describes a translation to the left 4 unit and up 5 units.

Kl = (-1, 9) Ll = (1, 6)

J

K

L

J’

K’

L’

You try some:

Graph each point and its image after the given translation.

a.) A (1, 3) left 2 units b.) B (-4, 4) down 6 unitsAl (-1, 3)

Bl (-4, -2)

AAl

B

Bl

Example 3: Write a rule that describes the translation below

Point A (2, -1) Al (-2, 2)

Point B (4, -1) Bl (0, 2)

Point C (4, -4) Cl (0, -1)

Point D (2, -4) Dl (-2, -1)

Rule (x, y) (x – 4, y + 3)

Example 4: Write a rule that describes each translation below.

a.) 3 units left and 5 units up b.) 2 units right and 1 unit down

Rule (x, y) (x – 3, y + 5)Rule (x, y) (x + 2, y – 1)

Reflection

Another name for a FLIP

A A’

C C’B B’

Reflection

Used to create SYMMETRY on the coordinate

plane

SymmetryWhen

one side of a

figure is a

MIRROR IMAGE of the other

Line of ReflectionThe line

you reflect a figure across

Ex: X or Y axis

X - X - axisaxis

In the diagram to the left you will notice that triangle ABC is reflected over the y-axis and all of the points are the same distance away from the y-axis.

Therefore triangle AlBlCl is a reflection of triangle ABC

Example 1: Draw all lines of reflection for the figures below. This is a line where if you were to fold the two figures over it they would line up. How many does each figure have?

a.) b.)

1 6

Example 2: Graph the reflection of each point below over each line of reflection.

a.) A (3, 2) is reflected over the x-axis

b.) B (-2, 1) is reflected over the y-axis

AA

AAll

BB BBll

Example 3: Graph the triangle with vertices A(4, 3), B (3, 1), and C (1, 2). Reflect it over the x-axis. Name the new coordinates.

A

B

C

A’ (4,-3)

B’ (3,-1)

C’ (1,-2)

Symmetry of the Alphabet

• Sort the letters of the alphabet into groups according to their symmetries

• Divide letters into two categories:•symmetrical•not symmetrical

Symmetry of the Alphabet

• Symmetrical: A, B, C, D, E, H, I, K, M, N, O, S, T, U, V, W, X, Y, Z

• Not Symmetrical: F, G, J, L, P, Q, R

Rotation

Another name for a TURN

B

B’

C

C’

A

A’

Rotation

A transformation that turns about

a fixed point

Center of Rotation

The fixed point

(0,0)

AA’

C

C’

B

B’

Rotating a Figure

Measuring the degrees of

rotation

90 degreesA

A’

C

C’

B

B’

Rotations in a Coordinate PlaneRotations in a Coordinate Plane

In a coordinate plane, sketch the quadrilateral whose vertices are A(2, –2), B(4, 1), C(5, 1), and D(5, –1). Then, rotate ABCD 90º counterclockwise about the origin and name the coordinates of the new vertices. Describe any patterns you see in the coordinates.

SOLUTION

Plot the points, as shown in blue. Use a protractor, a compass, and a straightedge to find the rotated vertices. The coordinates of the preimage and image are listed below.

Figure ABCD Figure A'B'C'D'

A(2, –2) A '(2, 2)

B(4, 1) B '(–1, 4)

C(5, 1) C '(–1, 5)

D(5, –1) D '(1, 5)

In the list, the x-coordinate of the image is the opposite of the y-coordinate of the preimage. The y-coordinate of the image is the x-coordinate of the preimage.

This transformation can be described as (x, y) (–y, x).

Rotational Rotational symmetry can symmetry can be found in be found in many objects many objects that rotate that rotate about a about a centerpoint.centerpoint.

A. Determine the angle of rotation for each hubcap. Explain how you found the angle.

B. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

Hubcap 1

A. Determine the angle of rotation for each hubcap. Explain how you found the angle.

B. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

Hubcap 1

There are 5 lines of symmetry in this design.

360 degrees divided by 5 =

Hubcap 1

The angle of rotation is 72º.

72º

Hubcap 2

There are NO lines of symmetry in this design.

Hubcap 2

The angle of rotation is 120º.

(360 / 3)

There are NO lines of symmetry in this design.

120º

Hubcap 3

A. Determine the angle of rotation for each hubcap. Explain how you found the angle.

B. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

Hubcap 3

There are 10 lines of symmetry in this design.

360 / 10 = 36However to make it look

exactly the same you need to rotate it 2 angles.

36 x 2 = 72

Hubcap 3

A.The angle of rotation is 36º.

B.There are 10 lines of symmetry in this design.

36º

Hubcap 4

A. Determine the angle of rotation for each hubcap. Explain how you found the angle.

B. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

Hubcap 4

A. .

B.There are 9 lines of symmetry in this design.

Hubcap 4

A.The angle of rotation is 40º.

B.There are 9 lines of symmetry in this design.

40º

Think About it:

Is there a way to determine the angle of rotation for a particular design without actually measuring it?

When there are lines of symmetry 360 ÷ number of lines of symmetry = angle of rotation

When there are no lines of symmetry: 360 ÷ number of possible rotations around the circle.

5 lines of symmetry 3 points to

rotate it to

Homework• Pg 138 #8, 12, 18, & 22• Pg 143 #8, 10, 16, & 18• Pg 148 #6, 8, 10

Tessellation

A design that covers a plane with NO GAPS

and NO OVERLAPS

Tessellation

Formed by a combination of TRANSLATIONS, REFLECTIONS,

and ROTATIONS

Pure Tessellation

A tessellation that uses only

ONE shape

Pure Tessellation

Pure Tessellation

Semiregular Tessellation

A design that covers a plane

using more than one shape

Semiregular Tessellation

Semiregular Tessellation

Semiregular Tessellation

Semiregular Tessellation

Tessellation

Used famously in artwork by M.C. Escher

Group Activity

• Choose a letter (other than R) with no symmetries

• On a piece of paper perform the following tasks on the chosen letter:

• rotation• translation•Reflection