Lesson 12.2Translations and

Reflectionspp. 504-508

Lesson 12.2Translations and

Reflectionspp. 504-508

Objectives:1. To define and perform translations

and rotations.2. To illustrate translations and

rotations as compositions of reflections.

3. To define the identity transformation.

Objectives:1. To define and perform translations

and rotations.2. To illustrate translations and

rotations as compositions of reflections.

3. To define the identity transformation.

Any time you perform two or more transformations on a geometric figure, you are

performing a composition of transformations.

Any time you perform two or more transformations on a geometric figure, you are

performing a composition of transformations.

A A translationtranslation is a transformation is a transformation formed by the composition of two formed by the composition of two reflections in which the lines of reflections in which the lines of reflection are parallel lines. A reflection are parallel lines. A translation can be thought of as a translation can be thought of as a sliding movement of the plane.sliding movement of the plane.

DefinitionDefinitionDefinitionDefinition

B

AC

DB

AC

D

l1l2

A A rotationrotation is a transformation is a transformation formed by the composition of two formed by the composition of two reflections in which the lines of reflections in which the lines of reflection intersect.reflection intersect.

DefinitionDefinitionDefinitionDefinition

XXhh

kk

HH JJ

IIHHII

JJ

HHII

JJX is the center of the rotation.X is the center of the rotation.The direction of this rotation is clockwise.The direction of this rotation is clockwise.

XXhh

kk

HH JJ

IIHHII

JJ

HHII

JJThe magnitude of the rotation is twice the measure of the acute or right angle between the lines of reflection.

The magnitude of the rotation is twice the measure of the acute or right angle between the lines of reflection.

If mHXH is 95°, the magnitude of the rotation is 95° and the angle between the lines of reflections is 47.5°.

If mHXH is 95°, the magnitude of the rotation is 95° and the angle between the lines of reflections is 47.5°.

XXhh

kk

HH JJ

IIHHII

JJ

HHII

JJ

The identity transformation is a transformation that maps each point of a geometric figure onto itself.

The identity transformation is a transformation that maps each point of a geometric figure onto itself.

Homeworkpp. 506-508Homeworkpp. 506-508

►B. Exercises13. If the magnitude of a rotation is 80°,

what is the measure of the acute angle between the lines of reflection?

►B. Exercises13. If the magnitude of a rotation is 80°,

what is the measure of the acute angle between the lines of reflection?

►B. Exercises15. Draw an acute triangle and rotate it

70° clockwise about point O. Then rotate the image 70° counterclockwise about point O. What is the

composition of these rotations called?

►B. Exercises15. Draw an acute triangle and rotate it

70° clockwise about point O. Then rotate the image 70° counterclockwise about point O. What is the

composition of these rotations called?

►B. Exercises16. Repeat exercise 15, using two

different centers. What is the composition?

►B. Exercises16. Repeat exercise 15, using two

different centers. What is the composition?

►B. Exercises17. If l and m intersect at point P to form

a 40° angle, then what is the composite of the reflections in l and m? Give its center and magnitude.

►B. Exercises17. If l and m intersect at point P to form

a 40° angle, then what is the composite of the reflections in l and m? Give its center and magnitude.

►B. Exercises17.►B. Exercises17. ll

mm

PP

40°40°

►B. Exercises18. If R is the reflection in l, and T is the

reflection in m, does R ◦ T = T ◦ R?

►B. Exercises18. If R is the reflection in l, and T is the

reflection in m, does R ◦ T = T ◦ R?

■ Cumulative Review23. Decide which numbers are greater

than others and put them in increasing order (Hint: decimals).

■ Cumulative Review23. Decide which numbers are greater

than others and put them in increasing order (Hint: decimals).

, 3.14, 10, 32, (1.1)12, , 3.14, 10, 32, (1.1)12, 33227

227

■ Cumulative Review24. Graph the set on the number line:

{-2, - , 2, , 4.1}

■ Cumulative Review24. Graph the set on the number line:

{-2, - , 2, , 4.1}3232

-2 -1 0 1 2 3 4 5-2 -1 0 1 2 3 4 5

-2-23232

--22 4.14.1

■ Cumulative ReviewGive the area and perimeter of each figure.

■ Cumulative ReviewGive the area and perimeter of each figure.

Figure Perimeter Area

25. Circle

26. Rectangle

27. Reg. Polygon

Figure Perimeter Area

25. Circle

26. Rectangle

27. Reg. Polygon

Analytic Geometry

Translating Conic Sections

Analytic Geometry

Translating Conic Sections

Circle

standard position x2 + y2 = r2

translated position (x-h)2 + (y-k)2 = r2

with center (h, k)

Circle

standard position x2 + y2 = r2

translated position (x-h)2 + (y-k)2 = r2

with center (h, k)

Parabola

standard position y = ax2

translated position y - k = a(x - h)2

or: y = a(x - h)2 + kwith vertex (h, k)

Parabola

standard position y = ax2

translated position y - k = a(x - h)2

or: y = a(x - h)2 + kwith vertex (h, k)

Graph x2 + (y - 1)2 = 4Graph x2 + (y - 1)2 = 4

Graph y = (x - 2)2 - 3Graph y = (x - 2)2 - 3

Write the equation that describes the graph.Write the equation that describes the graph.

(x - 4)2 + (y + 1)2 = 9(x - 4)2 + (y + 1)2 = 9

►ExercisesGraph.

1. (x + 2)2 + y2 = 4

►ExercisesGraph.

1. (x + 2)2 + y2 = 4

►ExercisesGraph.

2. y = x2 + 1

►ExercisesGraph.

2. y = x2 + 1

►ExercisesGraph.

3. x2 + (y - 2)2 = 1

►ExercisesGraph.

3. x2 + (y - 2)2 = 1

►ExercisesGraph.

4. y = 2(x + 1)2 + 4

►ExercisesGraph.

4. y = 2(x + 1)2 + 4

►ExercisesGraph.

5. (x - 4)2 + (y - 3)2 = 25

►ExercisesGraph.

5. (x - 4)2 + (y - 3)2 = 25