Duplicating this page is prohibited by law. © 2016 Triumph Learning, LLC 18 Unit 1: Transformations in the Coordinate Plane UNDERSTAND A reflection is a transformation that flips a figure across a line called a line of reflection Each reflected point is the same distance from the line of reflection as its corresponding point on the preimage, but it is on the opposite side of the line The resulting image and the preimage are mirror images of one another The line of reflection can be the x-axis, the y-axis, or any other line in the coordinate plane You can think of a reflection of a figure as a function in which the input is not a single value, x, but rather a point on the coordinate plane, (x, y) When you apply the function to a point on a figure, the output will be the coordinates of the reflected image of that point When a point is reflected across the y-axis, the sign of its x-coordinate changes The function for a reflection across the y-axis is: R y-axis (x, y) 5 (2x, y) When a point is reflected across the x-axis, the sign of its y-coordinate changes The function for a reflection across the x-axis is: R x-axis (x, y) 5 (x, 2y) Another common line of reflection is the diagonal line y 5 x To reflect over this line, swap the x- and y-coordinates The function for a reflection across line y 5 x is: R y 5 x (x, y) 5 (y, x) The path that a point takes across the line of reflection is always perpendicular to the line of reflection Perpendicular lines form right angles when they cross one another As shown in the diagram on the right, the path from point P to point P9 forms right angles with the line of reflection, y 5 x. y –2 –4 –6 –6 2 4 6 0 2 4 6 –2 –4 x M N P M' N' P' 3 units 3 units y –2 –4 –6 –6 2 4 6 0 2 4 6 –2 –4 x M N P M' N' P' y = x path is perpendicular to y = x y –2 –4 –6 –6 2 4 0 2 4 6 –2 –4 x M N P 2 units 2 units M' N' P' 6 LESSON 3 Reflections
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201
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18 Unit 1: Transformations in the Coordinate Plane
UNDERSTAND A reflection is a transformation that flips a figure across a line called a line of reflection . Each reflected point is the same distance from the line of reflection as its corresponding point on the preimage, but it is on the opposite side of the line . The resulting image and the preimage are mirror images of one another . The line of reflection can be the x-axis, the y-axis, or any other line in the coordinate plane .
You can think of a reflection of a figure as a function in which the input is not a single value, x, but rather a point on the coordinate plane, (x, y) . When you apply the function to a point on a figure, the output will be the coordinates of the reflected image of that point .
When a point is reflected across the y-axis, the sign of its x-coordinate changes . The function for a reflection across the y-axis is:
Ry-axis(x, y) 5 (2x, y)
When a point is reflected across the x-axis, the sign of its y-coordinate changes . The function for a reflection across the x-axis is:
Rx-axis(x, y) 5 (x, 2y)
Another common line of reflection is the diagonal line y 5 x . To reflect over this line, swap the x- and y-coordinates . The function for a reflection across line y 5 x is:
Ry 5 x(x, y) 5 (y, x)
The path that a point takes across the line of reflection is always perpendicular to the line of reflection . Perpendicular lines form right angles when they cross one another . As shown in the diagram on the right, the path from point P to point P9 forms right angles with the line of reflection, y 5 x.
y
–2
–4
–6
–6 2 4 60
2
4
6
–2–4x
M
N
P
M�
N�
P�
3 un
its
3 un
its
y
–2
–4
–6
–6 2 4 60
2
4
6
–2–4x
M
NP
M�N�
P�
y � x
path is perpendicular
to y = x
y
–2
–4
–6
–6 2 40
2
4
6
–2–4x
M
N
P2 units
2 unitsM�
N�
P�6
LESSON
3 Reflections
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Lesson 3: Reflections 19
Reflect nABC across the x-axis . Then reflect nABC across the y-axis .
y
–2
–4
–6
–6 2 4 60
2
4
6
–4x
AB
C
–2
Identify the coordinates of the vertices of nABC .
The vertices of the triangle are A(26, 21), B (22, 21), and C (22, 24) .
1
To reflect the vertices of nABC across the x-axis, change the signs of the y-coordinates . Then draw the image .
A(26, 21) A9(26, 1)
B (22, 21) B9(22, 1)
C (22, 24) C9(22, 4)
y
–2
–4
–6
–6 2 4 60
2
4
6
–4x
AB
C
A�B�
C�
–2
2
To reflect the vertices of nABC across the y-axis, change the signs of the x-coordinates . Then draw the image .
A(26, 21) A0(6, 21)
B (22, 21) B 0(2, 21)
C (22, 24) C 0(2, 24)
y
–2
–4
–6
–6 2 4 60
2
4
6
–2–4x
AB
C
A�B�
C�
C �
B �A �
3
Use function notation to describe how nABC is transformed to nA9B9C9 and how nABC is transformed to nA0B 0C 0 .
TRY
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20 Unit 1: Transformations in the Coordinate Plane
EXAMPLE A Graph the image of quadrilateral JKLM after the reflection described below .
R(x, y) 5 (y, x) .
Then describe the reflection in words .
K
y
–2
–4
–6
–6 2 4 60
2
4
6
–2–4x
LM
J
Identify the coordinates of the vertices of quadrilateral JKLM .
The quadrilateral has vertices J (24, 3), K (0, 4), L(2, 2), and M (21, 1) .
1
Apply the function to the vertices .
R (24, 3) 5 (3, 24)
R (0, 4) 5 (4, 0)
R (2, 2) 5 (2, 2)
R (21, 1) 5 (1, 21)
2
Graph the image .
▸ y
–2
–4
–6
–6 2 4 60
2
4
6
–2–4x
KL
M
J
J�
M�
L�K�
3
Describe the reflection in words .
Find the line that lies halfway between corresponding points of the figure .
y
–2
–4
–6
–6 2 4 6
2
4
6
–2–4x
K
LJ
J�
L�K�
M
M�
y � x
Each of these halfway marks lies on the line y 5 x .
▸The function performs a reflection across the line y 5 x.
4
Apply the same function, R(x, y) 5 (y, x), to J9K9L9M9 . What image results?
TRY
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Lesson 3: Reflections 21
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Figures can be reflected over horizontal or vertical lines that are not the x- or y-axis as well .
EXAMPLE B Trapezoid STUV is graphed on the right . Reflect this trapezoid over the line x 5 4 .
y
–2
–4
–6
2 4 6 80
2
4
6
–2x
T
S V
U
Reflect vertices U and V .
Point U, at (2, 1), is 2 units to the left of x 5 4 . So, its reflection will be 2 units to the right of x 5 4 . So, plot a point at (6, 1) and name it U9 .
Use the same strategy to plot point V9 .
y
–2
–4
–6
2 6 80
2
4
6
–2x
U�
V �
2 units2 units
2 units
2 units
T
S V
U
1
Find and plot the other two points of the image .
Point T at (21, 1) is 5 units to the left of x 5 4 . So, plot point T9 5 units to the right of x 5 4 at (9, 1) .
Point S is 6 units to the left of x 5 4 . So, plot point S9 at (10, 24), which is 6 units to the right of x 5 4.
▸
4
y
–2
–4
–6
2 6 8 100
2
4
6
–2x
U�
V � S �
T �T
S V
U
2
How could you describe the reflection of trapezoid STUV over the line x 5 4 using function notation?
DISCUSS
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Practice
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22 Unit 1: Transformations in the Coordinate Plane
Draw each reflected image as described and name its vertices. Identify the coordinates of the vertices of the image.
1. Reflect nABC across the x-axis .
y
–2
–4
–6
–6 2 4 60
2
4
6
–2–4x
A
B
C
A9( , ) B9( , ) C9( , )
REMEMBER When a point is reflected across the x-axis, the sign of its y-coordinate changes.
2. Reflect pentagon GHJKL across the line y 5 3 .
y
–2
–6 2 4 6
2
4
6
8
–2–4x
H J
K
LG
G9( , ) H9( , ) J9( , )
K9( , ) L9( , )
Fill in each blank with an appropriate word or phrase.
3. A reflection results in two figures that look like of each other .
4. Lines that meet and form right angles are called lines .
5. A point and its reflection are each the same distance from .
6. The path that a point takes across the line of reflection is to the line of reflection .
Use the given function to transform DEF. Then describe the transformation in words.
7. R(x, y) 5 (2x, y)
y
–4
–6
–6 2 4 6
2
4
6
–2–4x
D
E
F–2
8. R(x, y) 5 (y, x)
y
–4
–6
–6 2 4 6
2
4
6
–2–4x
D
E
F–2
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Lesson 3: Reflections 23
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Identify the coordinates of the image for each reflection as described.
9. Reflect M (3, 4) across the x-axis .
M9( , )
10. Reflect N (22, 28) across the y-axis .
N9( , )
11. Reflect P (22, 0) across the line y 5 x .
P9( , )
12. Reflect Q (5, 10) across the line y 5 x .
Q9( , )
Describe how quadrilateral ABCD was reflected to form quadrilateral A9B9C9D9, using both words and function notation.
13. y
–2
–4
–6
–6 2 4 60
2
4
6
–2–4x
A�
B�
D�A
BC
D
C�
Words:
Function:
14. y
–2
–4
–6
–6 2 4 60
2
4
6
–2–4x
A�
B�
D�
A
B
D
C�
C
Words:
Function:
Solve.
15. JUSTIFY Camille drew the square below on a coordinate plane . She says that if she reflects the square over the x-axis it will look exactly the same as if she reflects it over the y-axis . Is she correct or incorrect? Use words, numbers and/or drawings to justify your answer .
y
–2
–4
–6
–6 2 4 60
2
4
6
–2–4x
16. DRAW Patrick reflected a figure in two steps . The result was that each point (x, y) was transformed to the point (2y, x) . Draw a triangle (any triangle) on the plane below and transform it as described . Then describe what two reflections Patrick performed .
x
y
–2
–4
–6
–2–4–6 2 4 60
2
4
6
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