Geometry Notes Transformations - Miami Arts Charter

Transformations Geometry

Preimage ? the original figure in the transformation of a figure in a plane.

Image ? the new figure that results from the transformation of a figure in a plane.

Example:

If function h : x 2x 3 , find the Image of 8 and the Preimage of 11.

Solution:

What would the outcome be if x 8?

What value of x would give an outcome of 11?

h :8 28 3 13,

h : x 2x 3 11

The image of 8 is 13.

2x 14

x 7. The preimage of 11 is 7.

Isometry ? a transformation that preserves length.

Mapping ? an operation that matches each element of a set with another element, its image, in the same set.

Transformation ? the operation that maps, or moves, a preimage onto an image. Three basic transformations are reflections, rotations, and translations.

The three main Transformations are:

Reflection

Flip!

Rotation

Turn!

Translation

Slide!

Reflection ? type of transformation that uses a line that acts like a mirror, called a line of reflection, with a preimage reflected over the line to form a new image. {Flip}

A reflection is a FLIP over

a line.

Every point is the same distance from the central line! The reflection has the same size as the original image.

Line of reflection ? the mirror line.

A reflection is an isometry.

Example: Given rectangle ABCD, write the coordinates of each of the points by reflection in:

7

D

6 5 4 3 2

1 A

2

C

B

4

6

a) The x-axis

A 1, 1, B 5,1, C 5, 6, D 1, 6

As the rectangle is reflected over the x -axis:

A'1, 1, B '5, 1, C '5, 6, D'1, 6.

b) The y-axis

A 1, 1, B 5,1, C 5, 6, D 1, 6

As the rectangle is reflected over the y-axis:

A'1,1, B '5,1, C '5, 6, D'1, 6.

c) The line y = x.

A 1, 1, B 5,1, C 5, 6, D 1, 6

As the rectangle is reflected over the y=x:

A'1,1, B '1, 5, C '6, 5, D'6,1.

Notice the patterns: reflection over the x-axis change the sign of the y- coordinate, (x, y) (x, -y)

reflection over the y-axis change the sign of the x-coordinate, (x, y) (-x, y)

reflection over y = x switch the order of the x and y-coordinates. (x, y) (y, x)

And if all else fails, just fold your sheet of paper along the mirror line and hold it up to the light!

Line of symmetry ? a line that a figure in the plane has if the figure can be mapped onto itself by a reflection in the line.

Examples: Determine and draw all lines of symmetry in the following figures.

Example: Reflect over the y-axis:

Solution:

Example: Notice that some letters possess vertical line symmetry, some possess horizontal line symmetry, and some possess BOTH vertical and horizontal line symmetry.

A point reflection exists when a figure is built around a single point called the center of the figure. It is a direct isometry.

P x,y x,y .

Rotation ? a type of transformation in which a figure is turned about a fixed point, called a center of rotation. {Turn} Center of rotation ? the fixed point. Angle of rotation ? the angle formed when rays are drawn from the center of rotation to a point and its image. Counterclockwise rotation is considered positive and clockwise is considered negative.

"Rotation" means turning around a center. The distance from the center to any point on the shape stays the same.

Every point makes a circle around the center. A rotation is an isometry.

Examples: O is the center of regular pentagon ABCDE. State the images of

A, B, C, D, and E under each rotation.

a) RO,144

A

Rotate counterclockwise

b) RO,-72

Rotate clockwise

144 about point O .

72 about point O .

E

72?

B

A C

A E

O

B D

B A

C E

C B

D

C

D A

E B.

D C E D.

Examples: a) Each of these figures has rotation symmetry. Estimate the center of rotation

and the angle of rotation?

a) For each shape, the center of rotation is the center of the figure. The angles of rotation, from left to right, are 120?, 180?, 120?, and 90?.

b) Do the regular polygons have rotation symmetry? For each polygon, what are the center and angle of rotation?

b) For each shape, the center of rotation is the center of the figure. The angles of rotation, from left to right, are 120?, 180?, 120?, and 90?.

c) Name the vertices of the image of KLM after a rotation of 90?. K(4, 2), L(1, 3), and M(2, 1).

c) K'(-2, 4), L'(-3, 1), and M'(-1, 2).

Notice the patterns: rotation of 90? change the sign of the y-coordinate then switch the x and y-coordinates, (x, y) (-y, x)

rotation of 180? change the signs of the x and ycoordinates, (x, y) (-x, -y)

rotation of 270? change the sign of the x-coordinate then switch the x and y-coordinates. (x, y) (y, -x)

Translation ? a type of transformation that maps every two points P and Q in the plane to points P' and Q', so that the following two properties are true. 1. PP' = QQ'. 2. PP ' QQ ' or PP ' and QQ ' are collinear. {Slide}

When you are sliding down a water slide, you are experiencing a translation. Your body is moving a given distance (the length of the slide) in a given direction. You do not change your

size, shape or the direction in which you are facing.

"Translation" simply means Moving ... ... without rotating, resizing or anything else, just moving.

Every point of the shape must move: the same distance,

in the same direction.

Theorem: A translation is an isometry.

Examples: Explain the meaning of: a) (x, y) (x ? 7, y + 3) Slide the original point(s) by moving it left 7 units

and up 3 units. {mapping notation} b) T(-7, 3) or T-7, 3(x, y) Slide the original point(s) by moving it left 7 units

and up 3 units. {symbol notation}

Examples: a) Which of the following lettered figures are translations of the shape of the

purple arrow? Chose ALL that apply. Solution: a, c, and e only.

b) Which of the following translations best describes the diagram at the left? i. 3 units right and 2 units down. ii. 3 units left and 2 units up. iii. 3 units left and 2 units down. Solution: (i) 3 right and 2 down.

Glide reflections ? a transformation in which every point P is mapped onto a point P" by the following two steps. 1. A translation maps P to P'.

2. A reflection in a line k parallel to the direction of the translation maps P' to P".

When a translation (a slide or glide) and a reflection are performed one after the other, a transformation called a glide reflection is produced. In a glide reflection, the line of reflection

is parallel to the direction of the translation. It does not matter whether you glide first and then reflect, or reflect first and then glide. This transformation is commutative.

Since translations and reflections are both isometries, a glide reflection is also an isometry.

Examples: a) Does this paw print illustration depict a glide reflection?

Solution: Yes.

b) Examine the graph below. Is triangle A"B"C" a glide reflection of triangle ABC? Solution: Yes.

c) A triangle has vertices A(3,2), B(4,1) and C(4,3). What are the coordinates of point B under a glide reflection, T0,1 rx? Solution: After the reflection, B'(-4, 1). After the translation, B"(-4, 2).

d) Given triangle ABC: A(1,4), B(3,7), C(5,1). Graph and label the following composition: T5,2 rxaxis.

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