understand reflection€¦ · understand a reflection is a transformation that flips a figure...
TRANSCRIPT
![Page 1: UNDERSTAND reflection€¦ · 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](https://reader035.vdocuments.us/reader035/viewer/2022070105/605b50bdf1f7d37be47f0b92/html5/thumbnails/1.jpg)
Dup
licat
ing
this
pag
e is
pro
hibi
ted
by la
w. ©
201
6 Tr
ium
ph L
earn
ing,
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:
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
572GASE_GEO_SE_U1_PDF.indd 18 10/9/15 4:21 PM
![Page 2: UNDERSTAND reflection€¦ · 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](https://reader035.vdocuments.us/reader035/viewer/2022070105/605b50bdf1f7d37be47f0b92/html5/thumbnails/2.jpg)
ConnectD
uplic
atin
g th
is p
age
is p
rohi
bite
d by
law
. © 2
016
Triu
mph
Lea
rnin
g, L
LC
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
572GASE_GEO_SE_U1_PDF.indd 19 10/9/15 4:21 PM
![Page 3: UNDERSTAND reflection€¦ · 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](https://reader035.vdocuments.us/reader035/viewer/2022070105/605b50bdf1f7d37be47f0b92/html5/thumbnails/3.jpg)
Dup
licat
ing
this
pag
e is
pro
hibi
ted
by la
w. ©
201
6 Tr
ium
ph L
earn
ing,
LLC
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
572GASE_GEO_SE_U1_PDF.indd 20 10/9/15 4:21 PM
![Page 4: UNDERSTAND reflection€¦ · 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](https://reader035.vdocuments.us/reader035/viewer/2022070105/605b50bdf1f7d37be47f0b92/html5/thumbnails/4.jpg)
Lesson 3: Reflections 21
Dup
licat
ing
this
pag
e is
pro
hibi
ted
by la
w. ©
201
6 Tr
ium
ph L
earn
ing,
LLC
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
572GASE_GEO_SE_U1_PDF.indd 21 10/9/15 4:21 PM
![Page 5: UNDERSTAND reflection€¦ · 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](https://reader035.vdocuments.us/reader035/viewer/2022070105/605b50bdf1f7d37be47f0b92/html5/thumbnails/5.jpg)
Practice
Dup
licat
ing
this
pag
e is
pro
hibi
ted
by la
w. ©
201
6 Tr
ium
ph L
earn
ing,
LLC
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
572GASE_GEO_SE_U1_PDF.indd 22 10/9/15 4:21 PM
![Page 6: UNDERSTAND reflection€¦ · 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](https://reader035.vdocuments.us/reader035/viewer/2022070105/605b50bdf1f7d37be47f0b92/html5/thumbnails/6.jpg)
Lesson 3: Reflections 23
Dup
licat
ing
this
pag
e is
pro
hibi
ted
by la
w. ©
201
6 Tr
ium
ph L
earn
ing,
LLC
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
572GASE_GEO_SE_U1_PDF.indd 23 10/9/15 4:21 PM