1 . 3 Representing and Describing Transformations

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Name
Class
1.3
Date
Representing and Describing
Transformations
Essential Question: How can you describe transformations in the coordinate plane using
algebraic representations and using words?
Resource
Locker
Performing Transformations Using
Coordinate Notation
Explore
A transformation is a function that changes the position, shape, and/or size of a figure. The
inputs of the function are points in the plane; the outputs are other points in the plane. A
figure that is used as the input of a transformation is the preimage. The output is the image.
Translations, reflections, and rotations are three types of transformations. Thedecorative tiles
shown illustrate all three types of transformations.
You can use prime notation to name the image of a point. In the diagram, the transformation
T moves point A to point A′ (read “A prime”). You can use function notation to writeT( A ) = A′.
Note that a transformation is sometimes called a mapping. Transformation T maps A to A′.
A'
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©Antony McAulay/Shutterstock
A
Image
T
Preimage
Coordinate notation is one way to write a rule for a transformation on a coordinate plane. The notation
uses an arrow to show how the transformation changes the coordinates of a general point,( x, y ).
Find the unknown coordinates for each transformation and draw the image. Then
complete the description of the transformation and compare the image to its preimage.

(x, y) → (x - 4, y - 3)
Rule
Preimage
(x, y)
A(0, 4)
→
(x, y) → (x - 4, y - 3)
A′(0 − 4, 4 − 3)
5
Image
(x - 4, y - 3)
=
B(3, 0)
→
B′(3 − 4, 0 − 3)
=
C( 0, 0 )
→
C′( 0 − 4, 0 − 3 )
=
A′( −4, 1 )
⎛
⎞
B′ ⎜ -1 , -3 ⎟
⎝
⎠
⎛
⎞
C′ ⎜ -4 , -3 ⎟
⎝
⎠
y
A
A'
-5
C'
The transformation is a translation 4 units (left/right)
x
0 C
B
5
B'
-5
and 3 units (up/down).
A comparison of the image to its preimage shows that
Possible answer: the image is the same size and shape as the preimage
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B
DO NOT
Correcti
(x, y) → (-x, y)
Rule
Preimage
(x, y)
Image
(-x, y)
(x, y) → (-x, y)
R′(-(−4), 3)
=
→
S′(-(−1), 3)
=
→
T′( -(−4), 1 )
=
R(-4, 3)
→
S(-1, 3)
T(-4, 1)
⎛
⎞
R′⎜ 4 , 3 ⎟
⎝
⎠
⎛
⎞
S′⎜ 1 , 3 ⎟
⎝
⎠
⎛
⎞
T′⎜ 4 , 1 ⎟
⎝
⎠
5
R
-5
y
S S'
R'
0
T' x
5
T
-5
The transformation is a reflection across the (x-axis/y-axis).
A comparison of the image to its preimage shows that
Possible answer: the image is the same size and shape as the preimage, but it is flipped over
the y-axis
C
.
(x, y) → (2x, y)
Preimage
(x, y)
⎛
⎞
J ⎜ -1 , 2 ⎟ →
⎝
⎠
⎛
⎞
K⎜ 2 , 2 ⎟ →
⎝
⎠
⎛
⎞
L ⎜ 2 , -4 ⎟ →
⎝
⎠
Rule
(x, y) → (2x, y)
⎛
⎞
J ′ ⎜2 ⋅ -1 , 2 ⎟
⎝
⎠
⎛
⎞
K′ ⎜2 ⋅ 2 , 2 ⎟
⎝
⎠
⎛
⎞
L′ ⎜2 ⋅ 2 , -4 ⎟
⎝
⎠
Image
(2x, y)
⎛
⎞
= J ′ ⎜ -2 , 2 ⎟
⎝
⎠
⎛
⎞
= K′ ⎜ 4 , 2 ⎟
⎝
⎠
⎛
⎞
= L′ ⎜ 4 , -4 ⎟
⎝
⎠
5
J' J
y
K
K'
x
-5
The transformation is a (horizontal/vertical) stretch by a
factor of 2 .
0
-5
5
L
L'
Possible answer: the image and the preimage are both right triangles, but they do not
have the same size or shape
.
Reflect
1.
Discussion How are the transformations in Steps A and B different from the transformation in Step C?
The transformations in Steps A and B preserve the size and shape of the right triangle. The
transformation in Step C changes the shape of the right triangle.
2.
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A comparison of the image to its preimage shows that
For each transformation, what rule could you use to map the image back to the preimage?
A. (x, y) → (x + 4, y + 3); B. (x, y) → (-x, y); C. (x, y) → (0.5x, y)
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Describing Rigid Motions Using
Coordinate Notation
Explain 1
Some transformations preserve length and angle measure, and some do not. A rigid motion
(or isometry) is a transformation that changes the position of a figure without changing
the size or shape of the figure. Translations, reflections, and rotations are rigid motions.
Properties of Rigid Motions
• Rigid motions preserve distance.
• Rigid motions preserve collinearity.
• Rigid motions preserve angle measure.
• Rigid motions preserve parallelism.
• Rigid motions preserve betweenness.
If a figure is determined by certain points, then its image after a rigid motion is
determined by the images of those points. This is true because of the betweenness and
collinearity properties of rigid motions. For example, suppose △ABC is determined
by its vertices, points A, B, and C. You can find the image of △ABC by finding
the images of points A, B, and C and connecting them with segments.
Example 1 Use coordinate notation to write the rule that maps each preimage to its
image. Then identify the transformation and confirm
that it preserves length and angle measure.
B'

Preimage
A(1, 2)
B(4, 2)
C(3, −2)
→
→
→
Image
A′(−2, 1)
B′(−2, 4)
C′(2, 3)
-5
© Houghton Mifflin Harcourt Publishing Company
The
x-coordinate of each image point is the opposite of the
y-coordinate of its preimage.
The
y-coordinate of each image point equals the x-coordinate
of its preimage.
√(3 − 4) 2 + (-2 − 2) 2
2
Since
A'
B
0
x
5
C
-5
△ ABC and △A′B′C′. Use the Distance Formula as needed.
――――――――
= √―
17
――――――――
AC = √(3 − 1) + (-2 − 2)
= √―
20
BC =
C'
° counterclockwise around the origin given by
The transformation is a rotation of 90
the rule (x, y) → (−y, x).
AB = 3
y
A
Look for a pattern in the coordinates.
Find the length of each side of
5
2
A′ B′ = 3
B′ C′ =
――――――――
+ (3 − 4)
√(2 − (-2))
2
2
_
= √17
____
A′ C′ = √ (2 − (-2)) + (3 − 1)
2
2
_
= √20
AB = A′ B′ , BC = B′ C′ , and AC = A′ C′ , the transformation preserves length.
Find the measure of each angle of
△ABC and △A′B′C′. Use a protractor.
m ∠A = 63°, m∠B = 76°, m∠C = 41° m
∠A′ = 63°, m∠B′ = 76°, m∠C′ = 41°
Since m
∠A = m∠A′ , m∠B = m∠B′ , and m∠C = m∠C′ , the transformation
preserves angle measure.
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B
Preimage
P(-3, -1) →
Q(3, -1)
→
R(1, −4)
→
DO NOT
Correcti
Image
P′(−3, 1)
Q′(3, 1)
R′(1, 4)
5
Look for a pattern in the coordinates.
-5
The x-coordinate of each image point
equals
the x-coordinate of its preimage.
P'
P
y
R'
Q'
Q
0
The y-coordinate of each image point
is the opposite of the y-coordinate of its preimage.
-5
x
5
R
reflection across the x-axis
The transformation is a
x, y) → (x, -y) .
given by the rule (
Find the length of each side of △PQR and △P′Q′R′.
P′ Q′ = 6
PQ = 6
QR =
――――――――――
√(
1− 3
) (
2
+ −4 − -1
)
2
Q′ R′ =
――
13
=√
――――――――――
√(1 − -3 ) + (−4 − -1 )
――
= √ 25 = 5
PR =
2
――――――――――
√(1 − 3 ) + (4 − 1 )
――
2
2
√ 13
――――――――――
P′ R′ = √( 1 − -3 ) + ( 4 − 1 )
――
= √ 25 = 5
=
2
2
2
Since PQ = P' Q' , QR = Q' R' , and PR = P' R' , the transformation preserves length.
Find the measure of each angle of △PQR and △P′Q′R′. Use a protractor.
m∠P ' = 37° , m∠Q' = 56° , m∠R' = 87°
Since m∠P = m∠P' , m∠Q = m∠Q' , and m∠R = m∠R' , the transformation
preserves angle measure.
Reflect
3.
How could you use a compass to test whether corresponding lengths in a preimage
and image are the same?
Place the point of the compass on one endpoint of the segment in the preimage and open it
to the length of the segment. Without adjusting the compass, move the point of the compass
to an endpoint of the corresponding segment in the image and make an arc. If the lengths
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m∠P = 37° , m∠Q = 56° , m∠R = 87°
are the same, the arc will pass through the other endpoint of the segment in the image.
4.
Look back at the transformations in the Explore. Classify each transformation as
a rigid motion or not a rigid motion.
A. rigid motion; B. rigid motion; C. not a rigid motion
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Your Turn
Use coordinate notation to write the rule that maps each preimage to its image. Then
identify the transformation and confirm that it preserves length and angle measure.
5.
Preimage
D(-4, 4)
E(2, 4)
F(−4, 1)
Image
D′(4, -4)
E′(-2, -4)
F′(4, -1)
→
→
→
Each coordinate maps to its opposite.
The transformation is a rotation of 180° around -5
the origin given by the rule (x, y) → (−x, -y).
m∠D = m∠D′ = 90°
DE = D′ E′ = 6
―
EF = E′ F′ = √45
DF = D'F' = 3
5
D
y
E
x
F
0
m∠E = m∠E′ = 27°
5
F'
E' -5
m∠F = m∠F′ = 63°
D'
The transformation preserves length and angle measure.
6.
Preimage
S(-3, 4)
T(2, 4)
U(−2, 0)
Image
S′(-2, 2)
T′(3, 2)
U′(-1, -2)
→
→
→
5
S
S'
y
T
T'
x-coordinates: image is 1 more than preimage
y-coordinates: image is 2 less than preimage
The transformation is a translation given by the
rule (x, y) → (x + 1, y - 2).
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©Mary Hockenbery/Flickr/Getty Images
ST = S′ T′ = 5
TU = T′U′ = √32
―
SU = S′ U′ = √―
17
x
-5
m∠S = m∠S′ = 76°
m∠T = m∠T′ = 45°
U 0
5
U'
-5
m∠U = m∠U′ = 59°
The transformation preserves length and angle measure.
Explain 2
Describing Nonrigid Motions
Using Coordinate Notation
Transformations that stretch or compress figures are not rigid motions
because they do not preserve distance.
The view in the fun house mirror is an example of a vertical stretch.
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DO NOT
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Example 2 Use coordinate notation to write the rule that maps each preimage to
its image. Then confirm that the transformation is not a rigid motion.

△JKL maps to triangle △J′K′L′.
Preimage
Image
J(4, 1)
→
L(0, -3)
→
K(-2, -1) →
J′(4, 3)
K′(-2, -3)
L′(0, -9)
Look for a pattern in the coordinates.
x-coordinate of each image point equals the x-coordinate of its preimage.
The y-coordinate of each image point is 3 times the y-coordinate of its preimage.
The transformation is given by the rule (x, y) → (x, 3y).
The
Compare the length of a segment of the preimage to the length of the
corresponding segment of the image.
____
―――――――
2
2
2
2
JK = √(-2 − 4) + (-1 − 1)
J′ K′ = (-2 − 4) + (-3 − 3)
―
= √40
=
_
√ 72
Since JK ≠ J′K′ , the transformation is not a rigid motion.

△MNP maps to triangle △M′ N′ P′.
Preimage
M(-2, 2)
→
P(-2, -2)
→
N(4, 0)
→
Image
M′(-4, 1)
N′(8, 0)
P′(-4, -1)
The x-coordinate of each image point is twice the x-coordinate of its preimage.
The y-coordinate of each image point is half the y-coordinate of its preimage.
(
)
1
(x, y) → 2x, __2 y .
Compare the length of a segment of the preimage to the length of the corresponding
segment of the image.
――――――――
――――――――――
= √( 4 − -2 ) + ( 0 − 2 )
―――――
= √ 6 + -2
――
MN =
√(x 2 − x 1) 2 + (y 2 − y 1) 2
2
2
=
Since
――――――――
――――――――――――
= √( 8 − -4 ) + ( 0 − 1 )
―――――
= √ 12 + -1
――
M′ N′ =
2
2
2
2
√ 40
=
MN ≠ M′ N′ , the transformation is not a rigid motion.
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√(x 2 − x 1) 2 + (y 2 − y 1) 2
36
2
2
√145
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The transformation is given by the rule
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Reflect
7.
How could you confirm that a transformation is not a rigid motion by using a protractor?
If any angle measure in the preimage is different from the corresponding angle measure in
the image, then the transformation is not a rigid motion. Therefore, use the protractor to
check corresponding angles. If all angle measures are preserved, then check lengths.
Your Turn
Use coordinate notation to write the rule that maps each preimage to its image. Then
confirm that the transformation is not a rigid motion.
8.
△ ABC maps to triangle △ A′ B′ C′.
Preimage
A(2, 2)
→
C(2, -4)
→
B(4, 2)
→
9.
△ RST maps to triangle △ R′S′ T ′ .
Image
Preimage
B′(6, 3)
S(4, 2)
A′(3, 3)
C′(3, -6)
x, y) → (1.5x, 1.5y)
(_
Image
R(-2, 1)
→
R′(-1, 3)
T(2, -2)
→
T ′(1, -6)
(
→
1 x, 3y
(x, y) → _
2
)
S′(2, 6)
――――――――
37
√(4 − (-2)) +( 2 − 1) = √―
――――――――
18
R′ S′ = √(2 − (-1)) + (6 − 3) = √―
AB is horizontal and AB = 2.
_
A′ B′ is horizontal and A′B′ = 3.
RS =
Since AB ≠ A′ B′, the transformation
is not a rigid motion.
2
2
2
2
Since RS ≠ R′ S′, the transformation
is not a rigid motion.
Elaborate
10. Critical Thinking To confirm that a transformation is not a rigid motion, do you have to check the
length of every segment of the preimage and the length of every segment of the image? Why or why not?
No; once you find a segment of the preimage whose length is not equal to the length of
© Houghton Mifflin Harcourt Publishing Company
the corresponding segment of the image, you can stop checking lengths. You only need
to find one pair whose lengths are not equal in order to confirm that the transformation is
not a rigid motion.
11. Make a Conjecture A polygon is transformed by a rigid motion. How are the perimeters of the preimage
polygon and the image polygon related? Explain.
The perimeters are equal. Each side of the preimage polygon is transformed to a side of
the image polygon with the same length. The sum of the side lengths of the preimage is
equal to the sum of the side lengths of the image.
12. Essential Question Check-In How is coordinate notation for a transformation, such as
(x, y) → (x + 1, y - 1), similar to and different from algebraic function notation, such as ƒ(x) = 2x + 1?
In both cases, the notation shows how an input is changed by the transformation or
function. In coordinate notation, the input is a point of the coordinate plane. In algebraic
function notation, the input is a real number.
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