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G.CO.S STUDENT NOTES
&
PRACTICE WS #8
- geometrycommoncore.com
7
COMPOSITE FUNCTIONS
Composite function is the term for when a sequence of transformation take place. So if we reflection point A
over the x axis we get A' and then if we rotate A' 90o about the origin we get A". The first thing to notice is
that as we perform transformation in a succession we continue to label with the primes to show who is the
pre-image of who. A is the pre-image of A', A' is the pre-image of A", A" is the pre-image of A"', and so on.
DOES THE ORDER WE DO THEM IN MATTER?
A translation of <3,-5> followed by
a reflection over the y axis.
A reflection over the y axis followed
by a transfation of <3,-5>.
il
td
\<
1
:l,l
Y
"I
=l
l-
+
_.1
:l
Notice that the two composite transformations result in different locations,
NYTS
(Now You Trv Somel
1. A rotation of 90" about O followed by
a
2. A translation of <3,-5) followed by
translation of <3,-5>
a rotation.gf 90" about O.
'lo'
3. Why do you
think order alters the resultant location of AA"B"C"?
Tne DE(ftr.rCE. €Ron -THe Ltxti- oF
PeP*crtor.r ,s ALTFPEO BY IuL
o{LDW-IuxT -fuPY Aar- DontE lN '
G.co.s sruDENT NorEs & PRAC-fiQE ws #g - geometrycommoncore.com
2
lnterpreting the notation (Work from the inside out)
Trr,r, o R1, *ir(LABC)
(LA' B'C')
4-,,,,
:
The reflection over the y axis happens first to create AA'B'C'.
-
The translation of <-3,5> happens second to create AA"B',C,'
The result.
AAU BU CU
o
Ro,ro'
(MACI -
Ry
*i,
Ry
o,i,(aA' B 'c') =
AAtr
.
The rotation of 90o about the origin happens first to create AA,B,C,.
The reflection over the y axis happens second to create AA,B,C,.
gtt gn
The result.
English: A rotation of 270" about the origin followed by a translation of <9,-3>.
Math Notation: Trn._, o Ro,zt0"(LABC)
- A,An Bn Cn
English: A reflection of over the x axis followed by a rotation of 180o about the origin.
Math Notation: Ro.,ro"
o
R*
*i,(LABC)
- A,An Bn Ctt
NYTS (Now You Trv Some)
4. Which transformation would you perform FIRST?
a)
R,r,,ro.
oT...o,(MBC)
Ro)8..(aABC)
b)
o*@
Rr*i,o Ry=,(MBC)
&=,(
OR
5. Create the composite transformation statement for the given description.
a) Given AABC, translate it by <-5,3> followed by a reflection over the x axis.
P,
*,, o T/-',,1>(^ABC)
b) Given aABC, rotate it73" about o followed by a translation of <11,9>.
Tztfi-o
Po,-..
(^ABC)
R,
",i_(LABC)
#8- geometrycommoncore.com
G.CO.S STUDENT NOTES & PRAC{ICE WS
3
DOUBLE REFLECTIONS OVER PARALLEL LINES
Rr=,
o
Rr=-,
(LABC) = LAU Bu Cu
R*=2
Analysis of AABC & AA"B"C"
o
o
o
o
&=_, (LABC) = LAn Bn Cn
Analysis of AABC & AA"B"C"
o Orientation is the same.
o All points moved a fixed distance.
o All points moved a fixed direction.
Orientation is the same.
All points moved a fixed distance.
All points moved a fixed direction.
A double reflection is a TRANSLATION.
The translation is double the distance between the parallel lines.
NIIIS (Now You Trv Somel
&=-o o R.=IMBC)
.
6a. What do you expect
t-
x =t
Te.At rSuy?lcr.r
s
)1
\
A
,g
.l
n
tc)
c
tt
F
l
l-
6d. What is the distance between the
parallel lines?
S
Given the parallel lines we can determine the resultant translation distance and direction.
o R==_r(LABC)= LAU
O
6c. What was the translation distance?
I
&:_,
I
Yes
I
\ /
(,
L'FT
6b. Did the double reflection over
parallel lines result in a translation?
B
\
to happen?
BnCu y:-2
is ahorizontal line andy: -5 is ahorizontal line.
Going from y: -2 downward towards y = -5 tells us that the
translation will go down.
The distance between the two parallel lines is 7 units.
So this
will
be a TRANSLATION OF 14
tiNITS DOWN.
G.CO.S STUDENT NOTES & PRACTICE WS #8- geometrycommoncore.com
R,=_z o &=,
x : 1 is a vertical line and x: -2 is a vertical line.
Going from x: 1 left towards x: -2 tells us that the translation will
go left.
(LABC) = A,Au Bu Cu
The distance between the two parallel lines is 3 units.
So this
will
be a TRANSLATION OF 6 LINITS LEFT.
N[/IS (Now You Trv Somel
7
. R,=+ o R,='(A,ABC)
-
Translation distance
A4" Bu C u
uP
Translation direction
8.
Ry=a o
Translation direction
-
Translation direction
Lo. R,=-z " R"=r(A'ABC) = AAU Bu Cu
Translation distance
Translation direction
t
TRANSLATTON TO THE
UP
DOWN
=
I
I
B"
t
--1." c"
RIGHT 8
@
LEFT
LEFT
UP DOWN RIGHT @
m
m
.-t c
oOWu RIGHT
R* o R,(LABC) = LAU Bu Cu
R, o R*(LABC) = LAU Bu Cu
B
@
Translation distance =
L'An Bu Ctt
(m
DOWN RIGHr
q
Translation distance =
Rr-z(LABC) = LAU Bu Cu
9' R,=rr o R,=t(LABC)
IO
=
n
BI
.J
B'
o
.-1 C
oo
c' .1'
\.
TRANSLATION TO THE
LEFT 8
Some students think that the location of pre-image determines the direction that it will go during the
reflections. lT HAS NOTHING TO DO WITH lT. Double distance between the parallel lines determines the
distance and the ORDER determines the DIRECTION.