14-1 Mappings and Functions

Transformational Geometry

One branch of geometry, known as transformational geometry, investigates how one geometric figure can be transformed into another. In transformational geometry we are required to reflect, rotate, and change the size of the figures.

Mapping

Image and Preimage

Mappings and Functions

• Mapping Geometry: Correspondence between a set of points.

• Function Algebra: Correspondence between sets of numbers.

One-to-one

• A mapping (or a function) from set A to set B is called a one-to-one mapping (or function) if every member of B has exactly one preimage in A.

y = x2 is not a one-to-one function

9 has two preimages, 3 and -3

Example 1

• Function k maps every number to a number that is two less than one third of the number.– Express this fact using function notation– Find the image of 9– Find the preimage of 16

Example 2

• Mapping T maps each point (x,y) to the point (x+2, 3y)– Express this fact using mapping notation– Find P’ and Q’ the images of P(2,4) and Q(-2,6)– Decide whether T maps M, the midpoint of PQ

to M’ the midpoint of P’Q’.– Decide whether PQ = P’Q’

Transformation

• A one-to-one mapping from the whole plane to the whole plane.– Reflection– Translation– Glide Reflection– Rotation– Dilation

Isometry

• If a transformation maps every segment to a congruent segment

• “Preserves distance”

Theorem

• An isometry maps a triangle to a congruent triangle

Corollary

• An isometry maps an angle to a congruent angle

Corollary

• An isometry maps a polygon to a polygon with the same area.

Example 3

• Mapping S maps each point (x,y) to and image point (x,-2y). Given A(-3,1) B(-1,3) C(4,1) and D(2,-1)– Decide whether S is an isometry

14-2 Reflections

Reflection

A reflection is another type of geometric transformation.

A reflection is a mirror image that is created when a figure is flipped over a line.

Example: Reflection Image About Line m

m

m

m

Line m is called the line of reflection

We call A’ the reflection image of the point A

Reflections

A

m

A’

The dashed line shows that the points are images of each other under this transformation.

Line m is perpendicular to the line segment AA’ and also bisects it.

A

m

A’

• We say A is reflected in line m to A’

• To abbreviate this “reflection in line m” we write Rm:AA’ or

Rm:(A) = A’A

m

A’

Theorem 14-2

• A reflection in a line is an isometry

Isometry

• Preserves distance

• Preserves angle measure

• Preserves area of a polygon

Invariant

• Another way to say that the distance, angle measure and area are preserved when doing a reflection, is to say

– Distance, angle measure and area are invariant under a reflection.

Triangle ABC has vertices A(2,4), B(0,6), and C(-2,2). Graph the figure and its reflected image over the x-axis. Then find the coordinates of the reflected image.

B

A

C

Triangle ABC has vertices A(2,4), B(0,6), and C(-2,2). Graph the figure and its reflected image over the x-axis. Then find the coordinates of the reflected image.

B

A

C

C’

A’

B’

Quadrilateral RSTV has vertices R(2,3), S(-1,5), T(-3,0), V(3,-4). Graph the figure and its reflected image over the y-axis. Then find coordinates of the reflected image.

Triangle ABC has the vertices A(-6,-1) B(-2,-1) C(-5,-6). Graph the figure and its reflected image over the line y=x. Then find coordinates of the reflected image.

Triangle ABC has the vertices A(-6,-1) B(-2,-1) C(-5,-6). Graph the figure and its reflected image over the line y=x. Then find coordinates of the reflected image.

White Board Practice

1. Rm : stands for ?

White Board Practice

2. Rk :A ____

A

CD

B

U

S

T X

Y

W

k

j

White Board Practice

3. Rk (B) = ____

A

CD

B

U

S

T X

Y

W

k

j

White Board Practice

4. Rk AB ____

A

CD

B

U

S

T X

Y

W

k

j

White Board Practice

5. Rk (C) = ____

A

CD

B

U

S

T X

Y

W

k

j

White Board Practice

6. Rk :T = ____

A

CD

B

U

S

T X

Y

W

k

j

White Board Practice

7. Rk :BC = ____

A

CD

B

U

S

T X

Y

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k

j

White Board Practice

8. Rk :STU ____

A

CD

B

U

S

T X

Y

W

k

j

White Board Practice

9. Rj :(S) = ____

A

CD

B

U

S

T X

Y

W

k

j

White Board Practice

10. Rj :ST = ____

A

CD

B

U

S

T X

Y

W

k

j

White Board Practice

11. Rj : ( ) =XY

A

CD

B

U

S

T X

Y

W

k

j

White Board Practice

12. Rj : line k ______

A

CD

B

U

S

T X

Y

W

k

j

14-3 Translations and Glide Reflections

Translation

Translation

• A transformation glides all points of the plane the same distance in the same direction.

• A translation is a transformation that corresponds to physical sliding without turning.

A

C

B

A’

C’

B’

Vectors

Coordinates

• You don’t need to know the coordinates, you just need to know that if one point slides up 5 and to the right 3, then all points slide up 5 and to the right 3

If a transformation is a translation then all arrows

• Must be parallel and the same length

Example 1

• The translation T: (x,y)(x+3, y-1) maps triangle ABC to triangle A’B’C’. A(3,-1), B(0,2), C(2,-3)

(a) Graph triangle ABC and its image

(b) Draw arrows connecting A to A’, B to B’, and C to C’

(c) Are the arrows the same length and parallel?

Example 2

• If T: (2,2)(-2,-2), then

T: (4,4)( ? , ? )

Glide Reflection

• Glide reflection is a transformation where a translation is followed by a reflection in a line parallel to the direction of translation.

•The order of the two transformations (translation and reflection) is not important.•You will get the same result by first reflecting and then translating the image.

Example 3

• A glide reflection moves all points down 3 units and reflects all points in the x-axis. Find the image of A(2,-1), B(1,1) and C(3,3)

14.4 Rotations

To avoid confusion

• R (Reflection)

• RP,45° (Rotation)

A ROTATION of a geometric figure is the turn of the figure around a

fixed point.

Clockwise Negative

Counter-clockwise Positive

- 5

- 4

- 3

- 2

54321- 1- 1

- 2- 3- 4- 5

1

2

3

4

5

Rotate the figure 90 around the

origin.

A

BC

- 5

- 4

- 3

- 2

54321- 1- 1

- 2- 3- 4- 5

1

2

3

4

5

Rotate the figure clockwise 90

around the origin.

A

BCB’

C’A’

- 5

- 4

- 3

- 2

54321- 1- 1

- 2- 3- 4- 5

1

2

3

4

5

AB

CD

Rotate the figure -90around the origin.

- 5

- 4

- 3

- 2

54321- 1- 1

- 2- 3- 4- 5

1

2

3

4

5

AB

CD

D’

C’B’

A

Rotate the figure -90around the origin.

- 5

- 4

- 3

- 2

54321- 1- 1

- 2- 3- 4- 5

1

2

3

4

5

A

B C

Rotate the figure 180 around the

origin.

- 5

- 4

- 3

- 2

54321-1- 1

- 2- 3- 4- 5

1

2

3

4

5

A

B C

A’

B’C’

Rotate the figure 180 counter-

clockwise around the origin.

Theorem

• A rotation is an isomety

Special Rotations

• 360

• 180

• 390

360 rotation

• Rotates any point P around to itself.

180

• A rotation about point O of 180 is called a half turn.

• A Halfturn about the origin can be written Ho: (x,y)(-x,-y)

Rotation of 390 ??

• 360 + 30

Example 1

• State another name for each rotation

(a)Ro,-270°

(b)Ro,180°

(c) Ro,450°

(d)Ro,135°

Example 2

• The diagonals of square ABCD intersect at O. Complete each statement.

(a)Ro,-90° :B(b)Ro,-270°:C

(c) Ro,180° :A

(d)RD,-90°:A

Page 589

• Classroom Exercises 1-11

14.5 Dilations

Isometries

• Reflection

• Translation

• Glide reflection

• Rotation

DilationsA dilation is a

transformation that changes the size but not the shape of an

object or figure.

Every dilation has a fixed point that is called the

center of dilation.

So a dilations is related to….

Do,k

• O is the center of dilation

• k is the scale factor

• If k>1, the dilation is called an expansion.– The shape will get bigger

•If k<1, the dilation is called an contraction.

–The shape will get smaller

DilationsTo dilate an object with a center of dilation of

the origin only:

1) Graph object if necessary.

2) Multiply the coordinates of the object by the scale factor.

3) Graph new coordinates.

Example 1

Do,2

D0,-1

Example 2

-1

Your turn:

A negative scale factor

• Changes the direction of the dilation

• It will create opposite rays

To do a dilation with a center of dilation not at the origin

• Measure from the center of dilation to a point.

• Multiply that distance by the absolute value of the scale factor.

• Measure from the center of dilation to a new point with your new distance.

Remember….

• If the scale factor is negative you would measure in the opposite direction.

Example 3

• Find the image of WXYZ under D0,1/2

X

ZW

Y

O

Example 4

• Find the image of RST under D0,3

O

S

R T

Theorem

• A dilation maps a triangle to a

similar triangle

Corollary

• A dilation maps an angle to a

Congruent angle

Corollary

• A dilation D0,k maps any segment to a parallel segment k times as long.

Corollary

• A dilation D0,k maps any polygon to a similar polygon whose area is k2times as large

14.6 Composites of Mappings

Theorem

• The composite of two isometries is an isometry.

Theorem

• A composite of reflections in two parallel lines is a translation. The translation glides all points through twice the distance from the first line of reflection to the second.

Theorem

• A composite of reflections in two intersecting lines is a rotation about the point of intersection of the two lines. The measure of the angle of rotation is twice the measure of the angle from the first line of reflection to the second.

Corollary

• A composite of reflections in perpendicular lines is a half turn about the point where the lines intersect.

White Board Practice

Page 602 # 3

14-7 Inverses and the Identity

T: glides every runner one place to the right

T2: glides every runner two places to the right

T-1: glides every runner one place to the left

The inverse of TWritten T-1

T-1 ° T:PP

• Keeps all points fixed

Identity

• The mapping that maps every point to itself is called the identity transformation.

• I is the identity

• T ° I = T and I ° T = T

Inverse

• The inverse of a transformation T is defined as the transformation such that T-1 ° T = I or T ° T-1 = I

Example 1

• The symbol 2-1 stands for the inverse of 2 or ½ . They multiple to be 1. Give the value of the following.

a) 3-1

b) 7-1

c) (4/5)-1

d) (2 -1)-1

Example 2

• Find the inverses of the following transformations.

a) Reflection Rx

b) Translation T: (x,y)(x-2, y+3)

c) Rotation R o,a

d) Dilation D o,3

Example 3

• Which pairs of transformations are inverses?a)R o,180 and R o,-180

b)R o,270 and R o,-90

c) T: (x,y) (x+1, y-2) and U: (x,y) (x-2, y-1)

d) Rx ° Ry and Ry ° Rx

14.8 Symmetry in the Plane and in Space

Symmetry

• A figure in the plane has symmetry if there is an isometry, other than the identity that maps the figure to itself.

Line SymmetryWhat is a line of

symmetry?

• A line on which a figure can be folded so that both sides match

Here are some examples of common geometric figures and their lines of symmetry.

Line symmetry

• is really reflecting

Point Symmetry

Point Symmetry

Point Symmetry

Point Symmetry

• is really half turns

Rotational Symmetry

Rotational Symmetry

Translational Symmetry

Glide reflection Symmetry