Geometry Unit 3: Transformations Geometry Unit 3 ... · Geometry Unit 3: Transformations Ms....

Geometry Unit3:Transformations

Ms.Talhami 1

GeometryUnit3:Transformations

Name_________________


Ms.Talhami 2

HelpfulVocabulary

Word Definition/Explanation Examples/HelpfulTips


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Whatisatransformationingeometry?__________________________________________________________________________________________ImportantVocabularyWords

Word Definition/Explanation/Example

Pre-Image

Image

Input

Output

Orientation

RigidMotion

Isometry/Isometric(directvsopposite)

Prime’Double-Prime’’TriplePrime’’’etc.


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UnderstandingWhataTransformationRuleCouldLookLike

1)T(–4,5)àT’(,) 2)T(,)àT’(9,–7) 3)T(2,–8)àT’(,) 4)T(,)àT’(–1,–3)TypesofTransformations

Transformation Explanation Example


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PracticeTransformationRules

A T(x,y)à(x,y+7) A(–4,9) A’(,) B(,) B(5,0)

B S(x,y)à(–y,x) A(–4,9) A’(,) B(,) B(9,7)

C F(x,y)à(5x,3y) A(–4,9) A’(,) B(,) B(–5,12)

D G(x,y)à(–x,–3x) A(–4,9) A’(,) B(,) B(–8,–24)

E H(x,y)à(2x–1,y–3) A(–4,9) A’(,) B(,) B(31,15)

F P(x,y)à(x+3,2y) A(–4,9) A’(,) B(,) B(5,8)

WhatisaMapping?

Word Definition/Explanation

Mapping

MappingPracticeDeterminewhichofthefollowingcouldbeclassifiedasamapping.

Mapping?YESorNO Mapping?YESorNO Mapping?YESorNO


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IsometryPractice


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Determineifthepre-imageandimageareisometricandalsowhichtransformationproducedtheimage.

1)

2)

3)

4)

5)

6)


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Determineifthepre-imageandimageareisometricandalsowhichtransformationproducedtheimage.

1)

2)

3)

4)

5)

6)


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Determinethecoordinatesoftheimage,plottheimageanddetermineifitisanisometrictransformation.


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Determinethecoordinatesoftheimage,plottheimageanddetermineifitisanisometrictransformation.


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SymmetryWord Example Non-Example

Symmetry

LineSymmetry

PointSymmetry

RotationalSymmetryOrderofRotationAngleofRotation


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SymmetryPracticeForeachfigurebelow,drawallthelinesofsymmetry,statewhetherthefigurehaspointssymmetry,andstatewhetherthefigurehasrotationalsymmetry.Ifthefigurehasrotationalsymmetry,statetheorderandangleofrotation.

square

rectangle

eclipse

circle

Point?YorNRotational?YorNOrder:Angle:




rhombus

parallelogram

kite





pentagon

hexagon

octagon

decagon





equilateral triangle






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Reflections

Notation

Examples

IsometricPropertiesPreserved

PropertiesNotPreserved

Practice1. – 8. Use the to reflect the following figures over their respective line m. Label the image. 1.

2.

3.

4.


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5.

6.

7.

8.

9. – 14. Determine the line of reflection for the following pre-image and images. 9.

10.

11.

12.

13.

14.


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15. Determine the pre-image coordinates, then reflect it, and determine the image coordinates.

a) A = ( ____ , ____) ( )x axisR A A ‘ = ( ____ , ____)

b) B = ( ____ , ____) ( )y axisR B B ‘ = ( ____ , ____)

c) C = ( ____ , ____) ( )mR C C ‘ = ( ____ , ____)

d) D = ( ____ , ____) ( )x axisR D D ‘ = ( ____ , ____)

e) E = ( ____ , ____) ( )x axisR E E ‘ = ( ____ , ____)

f) F = ( ____ , ____) ( )nR F F ‘ = ( ____ , ____)

g) G = ( ____ , ____) ( )y axisR G G ‘ = ( ____ , ____)

16. Determine the name of the point that meets the given conditions.

a) ( )mR A = ________ b) ( )hR C = ________

c) ( )hR D = ________ d) (______)gR = B

e) ( )nR D = ________ f) ( )nR B = ________

g) ( )mR D = ________ h) (______)mR = C17. Determine the name of the point that meets the given conditions.

a) ( )mR A = ________ b) ( )FCR A = ________

c) ( )hR B = ________ d) (______)ADR = B

e) ( )nR D = ________ f) ( )nR B = ________

g) ( )FCR D = ________ h) (______)BER = C

h

g

nmCB

A D

h

n

m

A

F E

D

B C


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18. The vertices of ΔABC are A(3, 0), B(3, 6) and C(0, 6). Find the coordinates of the images of the vertices under the given reflection:Given the x-axis the y-axis the line y = xA(3, 0) B(3, 6) C(0, 6)

19. The vertices of ΔABC are A(1, 4), B(3, 0) and C(-3, -4). Find the coordinates of the images of the vertices under the given reflection:Given the x-axis the y-axis the line y = xA(1, 4) B(3, 0) C(-3, -4)

Find the image of the point under a reflection in the x-axis20. (5, 7) 21. (6, 2) 22. (-1, -4) 23. (3, 0)

Find the image of the point under a reflection in the y-axis24. (5, 7)

25. (-4, 10)

26. (0, 6)

27. (-1, -6)

Find the image of the point under a reflection in the line y = x28. (5, 7)

29. (-3, 8)

30. (0, -2)

31. (6, 6)

Using the given coordinates, draw ΔABC and ΔA`B`C` on one set of axes and then find the equation of the line of reflection.32. ΔABC: A(2, 4), B(2, 1) and C(-1, 1) ΔA`B`C`: A`(4, 4), B`(4, 1) and C`(7, 1)

33. ΔABC: A(1, 3), B(2, 5) and C(5, 3) ΔA`B`C`: A`(1, 1), B`(2, -1) and C`(5, 1)


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Translations

Notation

Examples



34. ΔABC: A(1, 4), B(2, 1) and C(4, 2) ΔA`B`C`: A`(-5, 4), B`(-6, 1) and C(`-8, 2)

35. ΔABC: A(4, 2), B(6, 2) and C(2, -1) ΔA`B`C`: A`(2, 4), B`(2, 6) and C`(-1, 2)

36. ΔABC: A(-1, 0), B(0, 2) and C(4, 1)ΔA`B`C`: A`(-1, 8), B`(0, 6) and C`(4, 7)

37. ΔABC: A(2, -1), B(4, 2) and C(-1, 2)ΔA`B`C`: A`(1, -2), B`(-2, -4) and C`(-2, 1)


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Practice1. Use the grid or patty paper to translate the following figures. Label the image.

2. Determine the translation coordinate rule from the vector. a) T (x,y) ----> (_______,________)

b) T (x,y) ----> (_______,________)

c) T (x,y) ----> (_______,________)

3. Determine the translation rule from the pre-image and image.

a) A (3,5) A’ (-1,3) T (x,y) ------------- > (___________ , ___________)

b) A (-4,11) A’ (3,0) T (x,y) ------------- > (___________ , ___________)

c) A (0,-8) A’ (-1,-3) T (x,y) ------------- > (___________ , ___________)

d) A (8,3) A’ (11,3) T (x,y) ------------- > (___________ , ___________)


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4. Given a translation rule, determine the missing point.

a) T (x,y) ------- > (x + 3, y – 5) A (-4,7) A’ (_______ , ______) b) T (x,y) ------- > (x -7, y – 1) A (9,1) A’ (_______ , ______) c) T (x,y) ------- > (x + 1, y + 6) A (_______ , ______) A’ (4,-1) d) T (x,y) ------- > (x, y + 4) A (8,-4) A’ (_______ , ______) e) T (x,y) ------- > (x + 3, y + 1) A (_______ , ______) A’ (-4,1) f) T (x,y) ------- > (x - 8, y - 5) A (_______ , ______) A’ (-3,-3)

5. Convert between vector component form and coordinate form. a) 5,2 ( )T A<− > = T (x,y) ------- > (______________ , ______________) b) 0, 12 ( )T A< − > = T (x,y) ------- > (______________ , ______________) c) 1.5, 7 ( )T A<− − > = T (x,y) ------- > (______________ , ______________) 6. Write the coordinate rule that matches the description. a) 4 down and 3 right T (x,y) ------- > (______________ , ______________) b) left 7 and down 2 T (x,y) ------- > (______________ , ______________) c) right 1 T (x,y) ------- > (______________ , ______________) 7. What is the resultant translation of Point A after mapping T (x,y) followed by R (x,y). a) A (-4,8) T (x,y) ------- > (x + 3, y – 7) A’ (_____ , _____) R (x,y) ------- > (x – 8, y – 2) A’’ (_____ , _____)

b) A (2,0) T (x,y) ------- > (x - 1, y) A’ (_____ , _____) R (x,y) ------- > (x - 3, y + 3) A’’ (_____ , _____)

c) A (5,-11) T (x,y) ------- > (x +7, y - 11) A’ (_____ , _____) R (x,y) ------- > (x - 9, y + 9) A’’ (_____ , _____)

8. Can you find a shortcut to doing two translations? 9. What is the pre-image of A’(-5,4) mapped by translation T (x,y) ------ > (x – 5, y + 11)?


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10. Given ΔABC shown at the right with vertices at A(−3, 8) B(− 6, -3) , and C(4, 2) , find and plot its image after a translation 4 units to the right and 7 units down. Label the image ΔA’ B’ C’ and state its coordinates. Sometimes this transformation will be symbolized by T4, -7 .

11. Given the two triangles shown in the image below, describe a sequence of rigid motions that would map ΔABC into ΔA’B’ C’ . There are many different, correct answers. Use the rule (x, y) (x + 2, y + 3) to find the image of the given point: 12. (3, -1)

13. (2, 6)

14. (0, -8)

15. (-5, -3)

Use the rule (x, y) (x - 4, y + 9) to find the image of the given point: 16. (4, 4)

17. (2, 0)

18. (-3, -10)

19. (-5, 9)

Find the image of the point (2, 7) under the given translation: 20. T1, 2

21. T3, -5

22. T-4, 0

23. T-2, -6


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Find the rule for the translation so the image of A is A` 24. A(3, 8) → A`(4, 6)

25. A(1, 0) → A`(0, 1)

26. A(2, 5) → A`(-1, 1)

27. A(-1, 2) → A`(-2, -3)

28. A(0, -3) → A`(-7, -3)

29. A(4, -7) → A`(4, -2)

30. Given ΔHOT whose vertices are H(-2, 0), O(0, 0) and T(0, 4): a) Give the coordinates of ΔHÒ`T` under the image T2, -3 of ΔHOT b) Give the coordinates of ΔH`Ò``T`` under the image of T1, 2 of ΔHÒ`T` c) Name a single transformation that is equivalent to T2, -3 followed by T1,2

31. Given quadrilateral WARM whose vertices are W(-4, 3), A(0, 1), R(2, 5) and M(0, 5) a) Give the coordinates of ΔWÀ`R`Mùnder the image T5, -2 of WARM b) Give the coordinates of ΔW`À``R``M`` under the image of T0, 2 of ΔWÀ`R`M` c) Name a single transformation that is equivalent to T5, -2 followed by T0,2

GlideReflections

Examples


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Rotations

Notation

Examples



Practice1. – 6. Use the grid to rotate the following figures. Label the image. 1.

2.

3.

4.

5.

6.


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7. – 9. Circle the center of rotation for the following pre-image and images. 7. A rotation of 90ο

8. A rotation of 180ο

9. A rotation of 180ο

10. Determine the pre-image coordinates, then rotate it, and determine the image coordinates.

,90 ( )OR ABC° Δ

a) A = ( ____ , ____) ,90 ( )OR ABC° Δ A ‘ = ( ____ , ____)

b) B = ( ____ , ____) ,90 ( )OR ABC° Δ

B ‘ = ( ____ , ____) c) C = ( ____ , ____) ,90 ( )OR ABC° Δ

C ‘ = ( ____ , ____)

,90 ( )OR DFE° Δ

d) D = ( ____ , ____) ,90 ( )OR DFE° Δ D ‘ = ( ____ , ____)

e) E = ( ____ , ____) ,90 ( )OR DFE° Δ

E ‘ = ( ____ , ____) f) F = ( ____ , ____) ,90 ( )OR DFE° Δ

F ‘ = ( ____ , ____)

11. Determine the name of the point that meets the given conditions.

a) ,60 ( )GR A° = ________ b) ,180 ( )GR B° = ________

c) ,300 ( )GR D° = ________ d) , 120 (______)GR − ° = B

e) ,240 ( )GR E° = ________ f) , 240 ( )GR F− ° = ________

g) ,60 ( )AR B° = ________ h) ,120 ( )CR D° = ________

A = A'

B

B'

E'B'

BE

D = D'

B'

E'

B

E

C

A B

DE

F

O

60°

GA

F E

D

B C


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Using the rule (x, y) → (-y, x) and give the coordinates of the image ΔA`B`C` and draw ΔABC and ΔA`B`C 12. ΔABC: A(1, 1), B(1, 5) and C(4, 5) (x, y) (-y, x)R90 A(1, 1) B(1, 5) C(4, 5)

13. ΔABC: A(4, 3), B(4, -2) and C(1, -1) (x, y) (-y, x)R90 A(4, 3) B(4, -2) C(1, -1)

14. ΔABC: A(1, 2), B(1, 6) and C(3, 2) (x, y) (-y, x)R90 A(1, 2) B(1, 6) C(3, 2)

15. ΔABC: A(1, 3), B(3, 5) and C(3, 0) (x, y) (-y, x)R90 A(1, 3) B(3, 5) C(3, 0)


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16. ΔABC: A(2, 1), B(3, -1) and C(-2, -1) (x, y) (-y, x)R90 A(2, 1) B(3, -1) C(-2, -1)

17. ΔABC: A(0, 0), B(2, 1) and C(-2, 4) (x, y) (-y, x)R90 A(0, 0) B(2, 1) C(-2, 4)

18. ΔABC: A(-2, 2), B(0, -4) and C(4, -4) (x, y) (-y, x)R90 A(-2, 2) B(0, -4) C(4, -4)

19. ΔABC: A(2, 5), B(0, 1) and C(-3, 7) (x, y) (-y, x)R90 A(2, 5) B(0, 1) C(-3, 7)


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Dilations

Notation

Examples



A dilation in which the scale factor, k, is: larger than 1 will make the image ______________________ between 0 and 1 will make the image ______________________ negative will cause the image to be reflected over the ___________________Practice1. Given the point (2,4), find: a) 3D b) 5D c)

21D

d) 2−D

2. Find the image of the point (0, 6) under a dilation of 4.

3. If the image of the point (6, 3) becomes (18, 9) after being dilated, what is the scale factor?

4. If the image of the point (8, 4) becomes (2, 1) after being dilated, what is the scale factor?

5. If a dilation maps (2, 5) onto (14, y), what is the value of y?

6. If a dilation maps (-9, 6) onto (x, 2), what is the value of x?

7. Under a dilation with respect to the origin, the image of A(-20, 8) is A’(-5, 2). What is the constant of dilation?

8. If the image of B(-2, 3) under a dilation is B’(-10, 15), what is the constant of dilation?


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9. Under a dilation with respect to the origin, the image of A(1, 3) becomes A’(4, 12). What are the coordinates of B’, the image of B(1,5) under the same dilation?

10. In which quadrant would the image of point (5,–3) fall after a dilation using a factor of -3? (1) I (3) III (2) II (4) IV

11. Triangle A’B’C’ is the image of ∆ABC under a dilation such that A’B’ = 3AB. Triangles ABC and A’B’C’ are (1) congruent but not similar (2) similar but not congruent (3) both congruent and similar (4) neither congruent nor similar

12. Which transformation represents a dilation? (1) (8,4) (11,7)→ (3) )8,4()4,8( −−→ (2) (8,4) ( 8,4)→ − (4) )2,4()4,8( →

13. Under the transformation ( , ) ( , ),x y x y→ 2 2 which property is not preserved? (1) distance (3) parallelism (2) orientation (4) angle measure

14. In which quadrant would the image of point (-6,5) fall after a dilation using a factor of 21 ?

(1) I (3) III (2) II (4) IV

15. The image of point A after a dilation of 3 is (6,15). What was the original location of point A? (1) (2,5) (3) (9,18) (2) (3,12) (4) (18,45)

16. On the accompanying set of axes, graph ∆ABC with coordinates A(–1,2), B(0,6), and C(5,4). Then graph and state the coordinates of triangle A’B’C’, the image of ∆ABC after a dilation of 2.


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25. (5, 7)

26. (-3, 8)

27. (0, -2)

28. (6, 6)

MixedTransformations1. – 4.Which transformation has taken place? 1. ________________

2. _______________

3. _______________

4. ________________

2. Complete the chart.

Using the rule (x, y) → (4x, 4y), find the image of the given point: 17. (3, -1)

18. (2, 6)

19. (0, -8)

20. (-5, -3)

Find the image of the given point under a dilation of 5 21. (5, 7)

22. (6, 2)

23. (-1, -4)

24. (3, 0)

Under D-3 find the image of the given point

Write a single rule for a dilation by which the image of A is A` 29. A(2, 5) → A`(4, 10)

30. A(3, -1) → A`(21, -7)

31. A(10, 4) → A`(5, 2)

32. A(-20, 8) → A`(-5, 2)

33. A(4, 6) → A`(6, 9)

34. A(4, -3) → A`(-8, 16)

35. A(-2, 5) → A`(8, -20)

36. A(-12, 9) → A`(-8, 6)

37. A(0, 9) → A`(0, -3)


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3. Using your rules above, determine the location of the missing location.

a) ,90 ( 6,4) (_______, _______)OR ° − = b) 0 ( 3, 5) (_______, _______)xR = − − =

c) ,270 (_______, _______) ( 3, 4)OR ° = − − d) (7, 8) (_______, _______)x axisR − =

e) (_______, _______) (8,4)y axisR = f) , 270 ( 5, 8) (_______, _______)OR − ° − − =

g) 0 (2,8) (_______, _______)yR = = h) ,180 (_______, _______) (0,9)OR ° =

i) (_______, _______) (3, 5)x axisR = − j) ,180 ( 4,2) (_______, _______)OR ° − =

k) (_______, _______) ( 4,2)y axisR = − l) ( 2,15) (_______, _______)y axisR − =

m) ,90 (_______, _______) (9, 3)OR ° = − n) , 90 (10,3) (_______, _______)OR − ° =

CompositionofTransformations

Notation

Examples

Composition of transformations is not commutative.


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Practice1. a) Given ΔABC with coordinates A(-1, -2), B(0, 4) and C(3, -1) b) Graph and label ΔA`B`C` the image of ΔABC after a reflection in the line y = -x.

2. a) Given ΔABC with coordinates A(2, 3), B(0, 6) and C(2, 6) b) Graph and label ΔA`B`C` the image of ΔABC after a translation T4, -3 c) Graph and label ΔA``B``C`` the image of ΔA`B`C` after a reflection in the y = x.


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3. a) Given ΔABC with coordinates A(0, 9), B(-3, 0) and C(-6, 9) b) Graph and label ΔA`B`C` the image of ΔABC after a reflection through the origin c) Graph and label ΔA``B``C`` the image of ΔA`B`C` after a dilation of 1

3.

d) Graph and label ΔA```B```C``` the image of ΔA``B``C`` after a translation T5, 4

4. a) Given ΔABC with coordinates A(2, 3), B(0, 6) and C(2, 6) b) Graph and label ΔA`B`C` the image of ΔABC after a reflection in the y-axis. c) Graph and label ΔA``B``C`` the image of ΔA`B`C` after a reflection y = x d) Graph and label ΔA```B```C``` the image of ΔA``B``C`` after a rotation clockwise about the origin90o


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5. a) Given ΔABC with coordinates A(1, 2), B(0, 5) and C(5, 4) b) Graph and label ΔA`B`C` the image of ΔABC after (x, y) → (x – 6, y + 3) c) Graph and label ΔA``B``C`` the image of ΔA`B`C` after a reflection x-axis d) Graph and label ΔA```B```C``` the image of ΔA``B``C`` after a reflection in the origin.

6. a) Given ΔABC with coordinates A(1, 2), B(4, -2) and C(5, 4) b) Graph and label ΔA`B`C` the image of ΔABC after a reflection x-axis c) Graph and label ΔA``B``C`` the image of ΔA`B`C` after a reflection in the origin. d) Graph and label ΔA```B```C``` the image of ΔA``B``C`` after a dilation of 2


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7. a) Given ΔABC with coordinates A(-1, 2), B(5, 2) and C(3, 4) b) Graph and label ΔA`B`C` the image of ΔABC after the composition R90

o ο rx-axis c) Graph and label ΔA``B``C`` the image of ΔABC after D2

8. a) Given ΔABC with coordinates A(4, 2), B(8, 2) and C(2, 6) b) Graph and label ΔA`B`C` the image of ΔABC after the composition R180

o ο T2, 4 c) Graph and label ΔA``B``C`` the image of ΔABC after D1/2


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9. a) Given ΔABC with coordinates A(1, 0), B(6, 3) and C(4, 5) b) Graph and label ΔA`B`C` the image of ΔABC after the composition D2 ο r0

10. a) Given ΔABC with coordinates A(-1, 4), B(3, 7) and C(5, 1) b) Graph and label ΔA`B`C` the image of ΔABC after the composition R90

o ο rx-axis


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11. a) Given ΔABC with coordinates A(2, -3), B(1, 1) and C(-2, -1) b) Graph and label ΔA`B`C` the image of ΔABC after the composition D2/3 ο D3 c) Graph and label ΔA``B``C`` the image of ΔABC after T8, -4

12. a) Given ΔABC with coordinates A(-1, 3), B(3, 7) and C(0, -6) b) Graph and label ΔA`B`C` the image of ΔABC after the composition ry-axis ο ry = x c) Graph and label ΔA``B``C`` the image of ΔA`B`C` after T0, -5


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13. a) Given ΔABC with coordinates A(-5, 1), B(-2, 5) and C(-7, 4) b) Graph and label ΔA`B`C` the image of ΔABC after the composition ro ο ry = x c) Graph and label ΔA``B``C`` the image of ΔA`B`C` after T2, 7

14. a) Given ΔABC with coordinates A(3, 4), B(1, 7) and C(3, 7) b) Graph and label ΔA`B`C` the image of ΔABC after the composition ry = x ο ry-axis c) Graph and label ΔA``B``C`` the image of ΔA`B`C` after T5, -1


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15. a) Given ΔABC with coordinates A(-1, 3), B(-6, 5) and C(-4, 7) b) Graph and label ΔA`B`C` the image of ΔABC after the composition ry = x ο rx-axis c) Graph and label ΔA``B``C`` the image of ΔABC after T8, 3

16. a) Given ΔABC with coordinates A(1, 0), B(7, 4) and C(5, 7) b) Graph and label ΔA`B`C` the image of ΔABC after the composition T1, 5 ο r0 c) Graph and label ΔA``B``C`` the image of ΔABC after ry-axis


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17. 2 5 ( )x xR R BC= − = − B’ = (__,__) B’’ =(__,__) C’ = (__,__) C’’ =(__,__)

Circle the resultant transformation from BC to '' ''B C ?

Rotation Reflection Translation

Describe the transformation. Specically its direction and distance from B to B’’ and from C to C’’. What does the distance of the transformation from B to B’’ have to do with the distance between the two parallel lines?

18. 5 2 ( )x xR R BC= − = − B’ = (__,__) B’’ =(__,__) C’ = (__,__) C’’ =(__,__)

Circle the resultant transformation from BC to '' ''B C ?

Rotation Reflection Translation Describe the transformation. Specically its direction and distance from B to B’’ and from C to C’’. What does the distance of the transformation from B to B’’ have to do with the distance between the two parallel lines?


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19. 4 1( )y yR R AB= = A’ = (__,__) A’’ =(__,__) B’ = (__,__) B’’ =(__,__)

Circle the resultant transformation from AB to '' ''A B ?


Describe the transformation. Specically its direction and distance from A to A’’ and from B to B’’. What does the distance of the transformation from A to A’’ have to do with the distance between the two parallel lines?

20. 1 4 ( )y yR R AB= = A’ = (__,__) A’’ =(__,__) B’ = (__,__) B’’ =(__,__)

Circle the resultant transformation from AB to '' ''A B ?


Describe the transformation. Specically its direction and distance from A to A’’ and from B to B’’. What does the distance of the transformation from A to A’’ have to do with the distance between the two parallel lines?

y = 1

y = 4

A (1,-1)B (5,-2)

y = 1

y = 4

A (1,-1)B (5,-2)


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21. ( )y axis x axisR R ABCΔ A’ = (__,__) A’’ =(__,__) B’ = (__,__) B’’ =(__,__) C’ = (__,__) C’’ =(__,__)

Circle the resultant transformation from ΔABC to ΔA’’B’’C’’? Rotation Reflection Translation How did you recognize which transformation mapped A to A’’, B to B’’ and C to C’’’? What does the angle of the transformation ∠AOA’’ have to do with the angle between the two intersecting lines?

22. ( )x axis y axisR R ABCΔ A’ = (__,__) A’’ =(__,__) B’ = (__,__) B’’ =(__,__) C’ = (__,__) C’’ =(__,__)


A (6,-1)

B (5,-7)

C (3,-4) A (-2,-2)

B (-7,-4)

C (-6,0)


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23. ( )y x x axisR R ABC= Δ A’ = (__,__) A’’ =(__,__) B’ = (__,__) B’’ =(__,__) C’ = (__,__) C’’ =(__,__)


24. ( )x axis y xR R ABC= Δ A’ = (__,__) A’’ =(__,__) B’ = (__,__) B’’ =(__,__) C’ = (__,__) C’’ =(__,__)


45°

A (6,-1)

B (7,-4)

C (3,-2)

45°

A (6,-1)

B (7,-4)

C (3,-2)


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MixedPractice5. The transformation R90 maps point

(5, 3) onto the point whose coordinates are 1) (5, -3) 2) (3, -5) 3) (3, 5) 4) (-3, 5)

6. What is the image of A(5, 2) under R90? 1) (-5, 2) 2) (5, -2) 3) (2, 5) 4) (-2, 5)

7. What are the coordinates of the image of P(-2, 5) after a clockwise rotation of 90º about the origin? 1) (-5, -2) 2) (-2, -5) 3) (2, 5) 4) (5, 2)

8. What are the coordinates of the image of (2, -5) after a counterclockwise rotation of 90º about the origin? 1) (-2, 5) 2) (2, 5) 3) (-5, -2) 4) (5, 2)

9. What is the image of the point (-3, 6) on rotation of 90° about the origin?

10. What is the image of the point (2, 3) under a clockwise rotation of 90° R-90 about the origin?

11. The point (-2, 1) is rotated 180º about the origin in a clockwise direction. What are the coordinates of its image?

1. If the letter is rotated 180 degrees, which is the resulting figure? 1) 2) 3) 4)

2. The accompanying diagram shows the starting position of the spinner on a board game.

How does this spinner appear after a 270° counterclockwise rotation about point P? 1)

2)

3)

ANS. 3 4)

3. If point (5, 2) is rotated counter-clockwise 90° about the origin, its image will be point 1) (2, 5) 2) (2, -5) 3) (-2, 5) 4) (-5, -2)

4. What are the coordinates of M`, the image of

M(2, 4), after a counterclockwise rotation of 90º about the origin? 1) (-2, 4) 2) (-2, -4) 3) (-4, 2) 4) (-4, -2)


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12. What is the image of R90(1, 2)?

13. Write the coordinates of P`, the image of P(5, 1) after a clockwise rotation of 180° about the origin.

14. What is the image of (5, 1) under a counterclockwise rotation of 90°?

15. The point (-3, 4) is rotated 180º about the origin in a counterclockwise direction. What are the coordinates of its image?

16. What is the image of (6, 5) under a counterclockwise rotation of 180°?

17. Point A is rotated 180º in a counterclockwise direction about the origin. If the coordinates of A are (-1, 3), what are the coordinates of A`, its image?

18. If point P(3, -2) is rotated 90o about the origin, what is the image of P?

19. The coordinates of the vertices of ΔRST are R(-2, 3), S(4, 4) and T(2, -2). Triangle R`S`T` is the image of ΔRST after a rotation of 90° about the origin. State the coordinates of the vertices of ΔR`S`T`.

20. The coordinates of the vertices of ΔABC are A(1, 2), B(-4, 3) ans C(-3, -5). State the coordinates of ΔA`B`C`, the image of ΔABC after a rotation of 90º about the origin.

21. Point A is located at (4, -7). The point is reflected in the x-axis. Its image is located at 1) (-4, 7) 2) (-4, -7) 3) (4, 7) 4) (7, -4)

22. When the point (2, 5) is reflected in the x-axis, what are the coordinates of its image? 1) (-5, 2) 2) (-2, 5) 3) (2, 5) 4) (5, 2)

23. What is the image of point (-3, 7) after a reflection in the x-axis? 1) (3, 7) 2) (-3, -7) 3) (3, -7) 4) (7, -3)

24. What are the coordinates of point (2, -3) after it is reflected over the x-axis? 1) (2, 3) 2) (-2, 3) 3) (-2, -3) 4) (-3, 2)

25. Point (-2, 3) is reflected in the x-axis. In which quadrant does its image lie? 1) I 2) II 3) III 4) IV

26. Reflecting (5, 1) in the y-axis yields an image of 1) (5, -1) 2) (-5, -1) 3) (5, 1) 4) (-5, 1)


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27. The image of point (3, 4) when reflected in the y-axis is 1) (-3, -4) 2) (-3, 4) 3) (3, -4) 4) (4, 3)

28. What is the image of the point (2, -3) after the transformation ry-axis? 1) (2, 3) 2) (-2, -3) 3) (-2, 3) 4) (-3, 2)

29. What are the coordinates of point P, the image of point (3, -4) after a reflection in the line y = x? 1) (3, 4) 2) (-3, 4) 3) (4, -3) 4) (-4, 3)

30. What is the image of (5, -2) under the transformation ry = x? 1) (-5, 2) 2) (5, 2) 3) (2, 5) 4) (-2, 5)

31. If the point (2, -5) is reflected in the line y = x, then the image is? 1) (5, -2) 2) (-2, 5) 3) (-5, 2) 4) (-5, -2)

32. The coordinates of point A are (-3a, 4b). If point A' is the image of point A reflected over the line y = x, the coordinates of A' are 1) (4b, -3a) 2) (3a, 4b) 3) (-3a, -4b) 4)

(-4b, -3a)

33. A function, f, is defined by the set {(2, 3), (4, 7), (-1, 5)}. If f is reflected in the line y = x, which point will be in the reflection? 1) (5, -1) 2) (-5, 1) 3) (1, -5) 4) (-1, 5)

34. What is the image of point (-3, -1) under a reflection in the origin? 1) (3, 1) 2) (-3, 1) 3) (1, 3) 4) (-1, -3)

35. The point (-3, -2) is reflected in the origin. The coordinates of its image are 1) (-2, -3) 2) (3, 2) 3) (2, 3) 4) (-3, 2)

36. If M(-2, 8) is reflected in the y-axis, what are the coordinates of M', the image of M?

37. Find the image of (1,5) when it is reflected over the line y = x.

38. Find the image of P(2, -5) under the transformation ry = x.


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39. The coordinates of the endpoints of AB are A(0, 2) and B(4, 6). Graph and state the coordinates of A` and B`, the images of A and B after AB is reflected in the x-axis.

40. Triangle SUN has coordinates S(0,6), U(3,5), and N(3,0). On the accompanying grid, draw and label ΔSUN. Then, graph and state the coordinates of ΔS`U`N`, the image of ΔSUN after a reflection in the y-axis.

41. On the accompanying grid, draw and label

quadrilateral ABCD with points A(1, 2), B(6, 1), C(7, 6), and D(3, 7). On the same set of axes, plot and label quadrilateral A`B`C`D`, the reflection of quadrilateral ABCD in the y-axis. Determine the area, in square units, of quadrilateral A`B`C`D`.

42. On the accompanying set of axes, draw the reflection of ABCD in the y-axis. Label and state the coordinates of the reflected figure.


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43. Triangle ABC has coordinates A(2, 0), B(1, 7), and C(5, 1). On the accompanying set of axes, graph, label, and state the coordinates of ΔA`B`C`, the reflection of ΔABC in the y-axis.

44. Carson is a decorator. He often sketches his room designs on the coordinate plane. He has

graphed a square table on his grid so that its corners are at the coordinates A(2, 6), B(7, 8), C(9, 3) and D(4, 1). To graph a second identical table, he reflects ABCD over the y-axis. On the accompanying set of coordinate axes, sketch and label ABCD and its image A'B'C'D', which show the locations of the two tables. Then find the number of square units in the area of ABCD.


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45. Triangle ABC has vertices A(-2, 2), B(-1, -3), and C(4, 0). Find the coordinates of the vertices of ΔA`B`C`, the image of ΔABC after the transformation r x-axis.

46. Triangle XYZ, shown in the diagram below, is reflected over the line x = 2. State the coordinates of ΔX`Y`Z`, the image of ΔXYZ.

47. Two parabolic arches are to be built. The equation of the first arch can be expressed as y = -x2 + 9, with a range of 0 ≤ y ≤ 9, and the second arch is created by the transformation T7, 0. On the accompanying set of axes, graph the equations of the two arches. Graph the line of

symmetry formed by the parabola and its transformation and label it with the proper equation.


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48. In the accompanying diagram of circle O, diameter AB is perpendicular to chord CD at

point E. What is the image of AC in AB?

1) AD 2) BD 3) ED 4) AE

49. The parabola shown in the accompanying diagram undergoes a reflection in the y-axis.

What will be the coordinates of the turning point after the reflection? 1) (3, -1) 2) (3, 1) 3) (-3, 1) 4) (-3, -1)

50. The accompanying graph shows the relationship between kinetic energy, y, and velocity, x.

The reflection of this graph in the line y = x is 1) 2) 3) 4)

51. What is the image of the point (-5, 2) under the translation T3, -4? 1) (-9, 5) 2) (-8, 6) 3) (-2, -2) 4) (-15, -8)

52. A translation moves P(3, 5) to P1(6, 1). What are the coordinates of the image of point (-3, -5) under the same translation? 1) (0, -9) 2) (-5, -3) 3) (-6, -1) 4) (-6, -9)


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53. The image of point (-2, 3) under translation T is (3, -1). What is the image of point (4, 2) under the same translation? 1) (-1, 6) 2) (0, 7) 3) (5, 4) 4) (9, -2)

54. The image of the origin under a certain translation is the point (2, -6). The image of point (-3, -2) under the same translation is the point 1) (-6, 12) 2) (-5, 4) 3)

− 32, 13

⎛⎝⎜

⎞⎠⎟

4) (-1, -8)

55. Triangle ABC has vertices A(1, 3), B(0, 1), and C(4, 0). Under a translation, A~, the image point of A, is located at (4, 4). Under this same translation, point C` is located at 1) (7, 1) (x + 3, y + 1) ANS. 1 2) (5, 3) 3) (3, 2) 4) (1, -1)

56. A design was constructed by using two rectangles ABDC and A`B`C`D`. Rectangle A`B`C`D` is the result of a translation of rectangle ABDC. The table of translations is shown below. Find the coordinates of points B and D`.

Rect. ABCD Rect. A`B`C`D`

A (2, 4) A` (3, 1) B B` (-5, 1) C (2, -1) C` (3, -4) D (-6, -1) D`

57. Triangle TAP has coordinates T(-1, 4), A(2, 4), and P(2, 0). On the set of axes below, graph and label ΔT`A`P`, the image of ΔTAP after the translation (x, y) (x – 5, y – 1).

58. The transformation T-2, 3 maps the point (7,

2) onto the point whose coordinates are 1) (9, 5) 2) (5, 5) 3) (5, -1) 4) (-14, 6)

59. The image of A(-1, 3) under the translation T2, 1 is 1) (1, 4) 2) (-3, 2) 3) (-2, 3) 4) (0, 5)

60. What is the image of point (2, 4) under the translation T-6, 1? 1) (-4, 3) 2) (-4, 5) 3) (8, 3) 4) (8, 5)

61. If translation T maps point (-3, 1) onto point A`(5, 5), which is translation T? 1) T2, 4 2) T2, 6 3) T8, 6 4) T8, 4


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62. What is the image of (-2, 3) after the transformation T3, -1?

63. What is the image of the point (-3, 4) under the translation T-2, 0?

64. Find the coordinates of the image of (-3, 4) under the transformation T-2, 3.

65. Find the coordinates of P`, the image of P(-3, 4) under the translation T4, 1.

66. If the transformation Tx, y maps point A(1, -3) onto point A`(-4, 8), what is the value of x?

67. Translation T maps point (2, 6) to point (4, -1). What is the image of point (-1, 3) under translation T?

68. A translation maps P(3, -2) to P`(1, 1). Under the same translation, find the coordinates of Q`, the image of Q(-3, 2).

69. A translation maps P(4, 1) to P`(2, -1). What are the coordinates of Q`, the image of Q(1, 3) under the same translation?

70. A translation maps P(4, -3) onto P`(0, 0). Find the coordinates of Q’, the image of Q(2, 1) under the same translation.

71. A translation maps the origin to the point (5, -3). What is the image of the point (-3, 2) under the same translation?

72. Under a given translation, the origin maps onto the point (3, 5). What is the image of the point (7, -1) under this same translation?

73. A translation maps the point (5, -2) to a point (0, -2). What is the image of point (0, -2) under the same translation?

74. A translation maps (2, 1) onto (-3, 2). Find the image of (4, -1) under the same translation.

75. A translation maps P(3, -2) onto P`(5, 0). Find the coordinates of the image of Q(4, -6) under the same translation.

76. The point (3, -2) is rotated 90º about the origin and then dilated by a scale factor of 4. What are the coordinates of the resulting image? 1) (-12, 8) 2) (12,-8) 3) (8, 12) 4) (-8, -12)

77. What is the image of point A(4, 2) after the composition of transformations defined by

R90 ο ry = x? 1) (-4, 2) 2) (4, -2) 3) (-4, -2) 4) (2, -4)


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78. What is the image of point (1, 1) under rx-axis ο R0 90? 1) (1, 1) 2) (1, -1) 3) (-1, 1) 4) (-1, -1)

79. What are the coordinates of point A`, the image of point A(-4, 1) after the composite transformation R90 ο ry = x where the origin is the center of rotation? 1) (-1, -4) 2) (-4, -1) 3) (1, 4) 4) (4, 1)

80. The coordinates of ΔJRB are J(1, -2), R(-3, 6), and B(4, 5). What are the coordinates of the vertices of its image after the transformation T2, -1 ο ry-axis? 1) (3, 1), (-1, -7), (6, -6) 2) (3, -3), (-1, 5), (6, 4) 3) (1, -3), (5, 5), (-2, 4) 4) (-1, -2), (3, 6), (-4, 5)

81. If the coordinates of point P are (2, -3), then R90 ο R180 is 1) (-2, 3) 2) (-2, -3) 3) (3, -2) 4) (-3, -2)

82. Find the coordinates of A` ry-axis ο ry = x if the coordinates of A are (6, 1).

83. Find the coordinates of the image of (2, 4) under the transformation ry-axis ο T3, -5.

84. What is the image that results from this composition of transformations rx-axis ο Ro 90 of (-3, 0)?

10. 85. Find the coordinates of point N(-1, 3) under the composite ry-axis ο T-2, 4.

86. If the coordinates of A are (2, 3), what are the coordinates of A`, the image of A after Ro 90 ο ry-axis ?

12. 87. If the coordinates of B are (1, -5), what are the coordinates of B’, the image of B after

R90 ο rx-axis?

88. Find the image of point A(3, -2) under the composition of translations T2, 1 ο T-6, -4.

15. 89. Write a single translation that is equivalent to T3, -1 followed by T-5, 5.

90. Which transformation is equivalent to the composite line reflections ry-axis ! ry = x? 1) a rotation 2) a dilation 3) a translation 4) a glide reflection


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91. The accompanying graph represents the figure I .

Which graph represents I after a transformation defined by ry = x ο R 90?

1) 2) 3) 4)

92. Given point A(-2, 3). State the coordinates of the image of A under the composition T-3, -4 ο rx-axis.

93. On the accompanying grid, graph and label AB, where A is (0, 5) and B is (2, 0). Under the transformation rx-axis ο ry-axis, A maps to A``, and B maps to B``. Graph and label A``B``. What single transformation would map AB to A``B``?


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94. On the accompanying grid, graph and label ΔABC with vertices A(3, 1), B(0, 4), and C(-5, 3). On the same grid, graph and label ΔA`B`C`. the image of ΔABC after the transformation rx-axis ο ry = x.

95. The coordinates of the vertices of ΔABC are A(1, 6), B(2, 9), and C(7, 10).

a) On the graph below, draw and label ΔABC. b) Graph and state the coordinates of ΔA`B`C`, the image of ΔABC after a reflection over the line y = x. c) Graph and state the coordinates of ΔA``B``C``, the image of ΔA`B`C after a reflection in the x-axis. d) Graph and state the coordinates of ΔA```B```C``, the image of ΔA``B``C`` after the

transformation (x, y) → (x – 5, y + 3).


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96. Given: ΔABC with coordinates A(1, 2), B(0, 5), and C(5, 4). a) On the graph below, draw and label ΔABC. b) Graph and state the coordinates of ΔA`B`C, the image of ΔABC after the translation T-6, 3. c) Graph and state the coordinates of ΔA``B``C``, the image of ΔA`B`C` after a reflection in the x-axis. d) Graph and state the coordinates of ΔA```B```C```, the image of ΔA``B``C`` after a reflection in the origin.

97. Triangle ABC has coordinates A(-1, 3), B(3, 7), and C(0, 6).

a) On the graph below, draw and label ΔABC. b) Graph and state the coordinates of ΔA`B`C, the image of ΔABC after a reflection in the line y = x. c) Graph and state the coordinates of ΔA``B``C``, , the image of ΔA`B`C following ry-axis. d) Graph and state the coordinates of ΔA```B```C```, , the image of ΔA``B``C``, after a translation that maps P(0, 0) onto P`(0, -5).


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98. Triangle ABC has coordinates A(-1, 2), B(6, 2), and C(3, 4). a) On the grid below, draw and label ΔABC. b) Graph and state the coordinates of ΔA`B`C`, the image of ΔABC after the composition ΔABC R90 ο rx-axis. c) Write a transformation equivalent to R90 ο rx-axis.

99. On the graph below, draw and label ΔPQR, whose vertices are P(3, 5), Q(9, 5), and R(7, 7). On

the same set of axes, graph and state the coordinates of a) ΔP`Q`R `, the image of ΔPQR after R90. b) ΔP``Q``R``, the image of ΔP`Q`R` after rx-axis. c) ΔP```Q```R```, the image of ΔP``Q``R`` after ry-axis. Based upon these graphs, write a single transformation that shows the composition ry-axis ο rx-axis ο R90


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100. Given ΔABC with points A(4, 3), B(4, -2), and C(2, 3). On the grid below, sketch ΔABC. On the same set of axes, graph and state the coordinates of ΔA`B`C`, the image of ΔABC after a reflection in the line y = x. On the same set of axes, graph and state the coordinates of ΔA``B``C``, the image of ΔA`B`C` after the translation T-4, 3.

111. The coordinates of the vertices of parallelogram ABCD are A(-2, 2), B(3, 5), C(4, 2), and D(-1, -1). State the coordinates of the vertices of parallelogram A``B``C``D`` that result from

the transformation ry-axis ο T2, -3.


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112. Given triangle ABC with coordinates A(-1, -2), B(0, -4), and C(3, -1). a) On the graph below, draw and label ΔABC.

b) Graph and label ΔA`B`C`, the image of ΔABC after translation T4, -3. c) Graph and label ΔA``B``C``, the image of ΔA`B`C` after a reflection in the origin. d) Graph and label ΔA```B```C```, the image of ΔA``B``C`` after a reflection in the line y = -x.

113. Triangle ABC has coordinates A(-3, -7), B(-3, -3), and C(0, -3).

a) On the graph below, graph and label ΔABC. b) Graph and state the coordinates of ΔA`B`C`, the image of ΔABC after a point reflection in the origin. c) Graph and state the coordinates of ΔA``B``C``, the image of ΔA`B`C` reflected in the line y = 2. d) Graph and state the coordinates of ΔA```B```C```, the image of ΔA``B``C`` after translation T-8, 2.

x


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114. a) On the accompanying grid, graph the equation 2y = 2x2 - 4 in the interval -3 ≤ x ≤ -3 and label it a.

b) On the same grid, sketch the image of a under T5,-2 ! rx-axis and label it b.

115. Triangle ABC has coordinates A(1, 1), B(5, 1), and C(4, 3). Given the transformations T, U, and W described below: T: (x, y) → (x, -y) U: (x, y) → (x – 6, y + 6) W: (x, y) → (-2x, -2y) a) Graph ΔABC and graph and state the coordinates of its image ΔA`B`C`, after transformation T. b) Graph and state the coordinates of ΔA``B``C```, the image of ΔABC after transformation U. c) Graph and state the coordinates of ΔA```B```C```, the image of ΔABC after transformation W. d) Which transformation, T, U, or W, is not an isometry? e) Which transformation, T, U, or W, does not preserve orientation?


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116. Triangle ABC has vertices A(2, -2), B(5, -2), and C(3, -4). a) On the set of axes below, graph and label ΔABC and its image under each of the following

transformations. State the coordinates of the vertices for each image of ΔABC. T: (x, y) → (-x, y) U: (x, y) → (x – 4, y + 4) W: (x, y) → (-2x, -2y) b) Which transformation, T, U, or W, is not an isometry? c) Which transformation, T, U, or W, does not preserve orientation?


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ConstructionsofTransformationsTranslations

' ( )AAT ABCΔ

' ( )CCT ABCD

A

B

C

A'

B

A D

C

C'


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Reflections

' ( )AAT ABCΔ andthen ( ' ' ')mR A B CΔ

Rm

m

A

B

C

m

A

B

C

A'


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LinesofReflection

A'

B'

C'

C B

A

A'

B'

C'

C B

A


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Dilations

DA,3(BC)

DA,2.5(BC)

A

B

C

A

B

C


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Rotations

RA,180(BC)

A

B

C

C'

B'

A'

A

B

C

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