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).

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