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Name________________________ Date _____ Per. _____ Teacher
Chapter 12
Transformations:
Shapes in Motion
2
Table of Contents
Reflections – Day 1 ……………………………………………..…………….. Page 3
Translations – Day 2 ……………….……….…………….……..…………….. Page 10
Rotations – Day 3…………………………………………..………………….. Page 17
Dilations – Day 4 ……………………………………………………….….….. Page 24
Symmetry – Day 5 ………………………………………..……..…………….. Page 33
Glide Reflections – Day 6……………………………………………….…….. Page 38
Compositions – Day 7………………………………………………..………….. Page 44
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Reflections – Day 1
An Introduction to Transformations
Image - _______________________________________________________
Preimage - ____________________________________________________
Isometry - ____________________________________________________
_____________________________________________________________
4
Reflections
Reflection-
Reflection Over the X-Axis a. Graph triangle PQR with vertices P(-3, 4), Q(4, 2), and R(-1, 1). Then graph the image of
triangle PQR after a reflection over the x-axis. Write the coordinates of its vertices.
P(-3, 4) Pl________
Q(4, 2) Ql_______
R(-1, 1) Rl________
Rule
5
Reflection Over the Y-Axis b. Graph quadrilateral ABCD with vertices A(-4, 1), B(-2, 3), C(0, -3) and D(-3, -2). Then graph the
image of ABCD after a reflection over the y-axis. Write the coordinates of its vertices.
A(-4, 1) Al________
B(-2, 3) Bl________
C(0, -3) Cl________
D(-3, -2) Dl________
Rule
Reflection Over the line y = x.
c. Graph trapezoid ABCD with vertices A(-4, 7), B(-1, 7), C(0, 5), and D(-5, 5). Then graph the
image of ABCD after a reflection over the line y = x. Write the coordinates of the new vertices
Rule
A(-4, 7) Al________
B(-1, 7) Bl________
C(0, 5) Cl_______
D(-5, 5) Dl________
6
Reflection over a Horizontal or Vertical Line d. Graph triangle ABC with vertices A(-4, 1), B(-1, 1), and C(-3, 4). Then graph the image of
ABC after a reflection over the line y = -3. Write the coordinates of the new vertices.
Identifying a Reflection Write a rule to describe each transformation.
a. b.
A(-4, 1) Al________
B(-1, 1) Bl________
C(-3, 4) Cl________
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Summary
Find the coordinates of the vertices for the reflection images of the triangle in the x- axis,
y-axis, and the line y = x.
8
Homework
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10
Translations – Day 2
Warm - Up
1.
2.
Translation- ___________________________________________________
_____________________________________________________________
_____________________________________________________________
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1) Triangle DEF is a translation of triangle ABC.
Use the diagram to write a rule for the translation of triangle ABC to Triangle DEF.
2) Pentagon ABCDE is drawn on the grid below.
On the grid, draw a translation of pentagon ABCDE using the rule T3,-5.
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3) Pentagon ABCDE is plotted on the grid below.
On the grid, draw the translation of pentagon ABCDE using the rule T-4,3. Label the
translated figure A’B’C’D’E’.
4.
5.
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Summary
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Homework
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16
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Rotations – Day 3
Warm - Up
1.
2.
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Example 1: Graph triangle XYZ with vertices X(2, 2), Y(4, 3), and Z(3, 0).
Then graph the image of triangle XYZ after a rotation 90o counterclockwise about the
origin, and write the coordinates of its vertices.
X(2, 2) Xl( , )
Y(4, 3) Yl( , )
Z(3, 0) Zl( , )
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Example 2: Graph trapezoid ABCD with vertices A(1, 3), B(4, 4), C(4, 0), and
D(1, 1). Then graph the image of trapezoid ABCD after a 180o rotation about
the origin, and write the coordinates of its vertices.
Example 3: Graph polygon TUVW with vertices T(2, -4), U(3, -1), V(-1, 0), and
W(-2, -3). Then graph the image of polygon TUVW after a 270o rotation, and
write the coordinates of its vertices.
A(1, 3) Al ( , )
B(4, 4) Bl ( , )
C(4, 0) Cl ( , )
D(1, 1) Dl ( , )
T(2, -4) Tl ( , )
U(3, -1) Ul ( , )
V(-1, 0) Vl ( , )
W(-2, -3) Wl ( , )
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Identifying a Rotation Write a rule to describe each transformation.
a.
b.
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Summary
22
Homework
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Dilations – Day 4 SWBAT: Identify and Draw Dilations
Warm – Up
1. 2.
4. What are the coordinates of the image
(2,-5) after a counterclockwise rotation
of 270˚ about the origin?
a. (-2, 5)
b. (2, 5)
c. (-5, -2)
d. (5, -2)
Dilation - _____________________________________________________
_____________________________________________________________
Scale factor- ___________________________________________________
_____________________________________________________________
Enlargement - _________________________________________________
_____________________________________________________________
Reduction - ___________________________________________________
_____________________________________________________________
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To find the image of a point under a dilation, you multiply by the scale factor.
a) D4 (x,y) (4x,4y) b) D6
1(x,y) ( Yx
6
1,
6
1)
Example- Find a) D2(3,4)
b)
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D (6,9)
Example 1-Enlargement
Graph triangle PQR with vertices P(-1, 1), Q(4, 1), and R(-1, 3). Then Graph its image
triangle P’Q’R’ after a dilation with a scale factor of 2.
P(-1, 1) P’________
Q(4, 1) Q’_______
R(-1, 3) R’________
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Example 2-Reduciton
Graph quadrilateral EFGH with vertices E(-6, 4), F(-4, 8), G(4, 2), and H(-2, -4). Then
graph the image of EFGH after a dilation with a scale factor of ½ .
Example 3-Determining the Scale Factor
Graph triangle ABC with vertices A(-4, 6), B(-2, 4), and C(-8, 4). Then graph the image
of ABC after a dilation triangle A’B’C’ with vertices Al(-2, 3), Bl(-1, 2), and Cl(-4, 2).
Find the scale factor of the dilation and classify as an enlargement or a reduction.
E(-6, 4) E’________
F(-4, 8) F’_______
G(4, 2) G’_______
H(-2, -4) H’________
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You Try!!!
1. Graph triangle ABC with vertices A(-2, 1), B(3, 1), and C(-3, 3). Then graph the
image of ABC after a after a dilation with a scale factor of 3.
2. Graph quadrilateral EFGH with vertices E(-6, 4), F(-4, 8), G(4, 2), and H(-2, -4).
Then graph the image of quadrilateral EFGH after a dilation with a scale factor of2
3.
3. Triangle ABC with vertices A(-3, 6), B(-6, 9), and C(-9, -3). The graph of the image of
ABC after a dilation triangle A’B’C’ with vertices Al(-2, 4), Bl(-4, 6), and Cl(-6, -2).
Find the scale factor of the dilation and classify as an enlargement or a reduction.
A(-2, 1) Al________
B(3, 1) Bl________
C(-3, 3) Cl________
E(-6, 4) E’________
F(-4, 8) F’_______
G(4, 2) G’_______
H(-2, -4) H’________
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Name: _________________________ Per.: ______ Date: _________ Transformation Review – Homework
1) Which transformation is shown in the accompanying diagram?
a) Reflection
b) Rotation
c) Translation
d) Dilation
2) A translation maps (x,y) to (x – 5, y + 3). In which quadrant does the point (-3, -2) lie under the
same translation?
a) I
b) II
c) III
d) IV
3) A translation moves point B (-5, 3) to point B’ (2, 1). What is the image of (x,y) under this
translation?
a) (x + 7, y – 2)
b) (x + 7, y + 2)
c) (x – 3, y – 2)
d) (x – 3, y + 2)
4) Which transformation for ∆RST is shown in the diagram below?
a) Line reflection
b) Translation
c) Rotation
d) Dilation
5) What is the image of (1, -3) when reflected about the y-axis?
a) (-1, 3)
b) (3, -1)
c) (-3, -1)
d) (-1, -3)
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6) If the point (5,1) is reflected in the x-axis, the image is:
a) (-5, 1)
b) (-5,-1)
c) (5, 1)
d) (5, -1)
7) What are the coordinates of P’, the image of P(-1,4) after a reflection in the line x = 2?
a) (4,1)
b) (-1,1)
c) (0,4)
d) (5,4)
8) In the accompanying diagram of right triangle ABC with the right angle C, line l is drawn
through C and is parallel to AB .
If ∆ABC is reflected in line l, forming the image ∆A’B’C’, which statement is not true?
a) C and C’ are the same point
b) m<ABC = m<A’B’C’
c) The area of ∆A’B’C’ is twice the area of ∆ABC.
d) Line l is equidistant from A and A’.
9) Point P’ (-6,-4) is the image of point P (-2, 3) under translation T. What is the image of (5,-1)
under the same translation?
a) (9,6)
b) (1,-8)
c) (-1,-5)
d) (3,2)
10) What are the coordinates of the image (2,-5) after a counterclockwise rotation of 90˚ about the
origin?
a) (-2, 5)
b) (2, 5)
c) (-5, -2)
d) (5, 2)
11) The enlargement of a photograph is an example of a:
a) Line reflection
b) Translation
c) Dilation
d) Rotation
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12) Which transformation is represented by the illustration?
a) Reflection
b) Dilation
c) Translation
d) Rotation
13) What is the image of point (4,-2) after a dilation of 3?
a) (4/3, -2/3)
b) (12, -6)
c) (7, 1)
d) (1, -5)
14) The image of P(6, -9) after a dilation with respect to the origin is (4, -6). What is the constant of
dilation?
a) 1
3
b) 2
3
c) 3
2
d) -2
15) In which quadrant will the image of A (4, -2) lie after dilation D-2?
a) I
b) II
c) III
d) IV
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Part Two:
16.
32
33
Symmetry – Day 5
Warm - up
1.
2.
34
35
36
Homework
37
38
Glide Reflections – Day 6
Warm – Up
1.
2.
Which rotation about point O maps B to D? _______
39
The symbol for a composition of transformations is an open circle.
The notation is read as a reflection in the x-axis following a translation of
(x+3, y+4). Be careful!!! The process is done in reverse!!
You may see various notations which represent a composition of transformations:
could also be indicated by
Identifying Glide Reflections
1.
40
2)
3) Is T0, 5 ry-axis a glide reflection?
4) Is T7, 0 rx-axis a glide reflection?
5) Is T3,3 ry=x a glide reflection?
6) Is T4,-1 ry-axis a glide reflection?
7) Is T-2, 8 rx-axis a glide reflection?
Practice Problem #8
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Practice Problem #9
Given: ABC: A (-4, 5), B (-6, 7), C (-8, 3)
Practice Problem #10
Given: ABC: A (5, 1), B (7, 4), C (9, 1)
T-5,-4 rx-axis
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Homework
43
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Compositions (More Practice) – Day 7
When two or more transformations are combined to form a new transformation, the result
is called a composition of transformations. In a composition, the first transformation
produces an image upon which the second transformation is then performed.
The symbol for a composition of transformations is an open circle.
The notation is read as a reflection in the x-axis following a translation of
(x+3, y+4). Be careful!!! The process is done in reverse!!
45
46
Try it Problems
1. The transformation (x, y) → (2x + 4, 2y + 1) is the composition of what two transformations?
2.
3.
4.
5.
47
Homework: Transformations Review
1) In the accompanying diagram, K is the image of A after a translation. Under the same translation,
which point is the image of J?
2) A translation moves A (-1, 3) to A’ (-3, 7). What are the coordinates of B’, the image of B(5,-3) under
the same translation?
3) Find the coordinates of P’, the image of P (-3, 4) under the translation T4,1.
4) Translation T maps point (2, 6) to point (4, -1). What is the image of point (-1, 3) under translation T?
5) Which graph shows a reflection ABC in the x-axis?
6)
48
7) Find the coordinates of P’, the image of P(3, -1) under the transformations (x,y) (-y, -x).
8) What is the image of (6, 5) under a counterclockwise rotation of 180˚?
9) If point P (3, -2) is rotated 90˚ about the origin, what is the image of P?
10) Which transformation does not preserve size?
11) The image of (-2, 6) after a dilation with respect to the origin is (-10, 30). What is the constant of
dilation?
12) Which transformation for letter M is shown in the accompanying diagram?
13) If a dilation maps (-3, 2) to (x, 8), what is the value of x?
14) If the coordinates of point A are (–2,3), what is the image of A under r Dy axis 3 ?
49
15) On the accompanying grid, graph and label AB , where A is (0,5) and B is (2,0). Under
the transformation ),(ABrr axisyaxisx A maps to A , and B maps to B . Graph and
label A B .
What single transformation would map AB to A B ?
50
22.
In 16 - 21: a. Find the image of P(5, -3) under each of the given transformations.
16. rx-axis 17. R180 18. R90
19. T0,3 20. rx-axis T2,2 21. rx-axis ry=x
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23.