lesson 12.2 translations and reflections pp. 504-508
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Lesson 12.2 Translations and Reflections pp. 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. - PowerPoint PPT PresentationTRANSCRIPT
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