-,

Class
"

Date
,-,
Name

Translation
.,
;.,

. ~:
A translation is a-type of transformation. In a translation, a geometric figure
changes
.'
position
, .
~t does not change shape or size. The original figure is called
; t. ~
j
1

the _preimr~e and lPe figure following transformation is the image,'

_ ',':: I)

The diagram at the right shows a translation in the coordinate y

plane. The pteim~e is MBC,;-The image is Ll ABC, I
r' O '. ,) ; " • '.

Each poiD:t"of M'Jic

has moved 5' units left and 2 hnits up.
Moving l:e(i)~ tpe negative g direction, and moving up is in irf
the' positive y dife~!ion. So, th,(trule for the translation is
," .' . j " . ~~. :·~.t ). ~ , ~'7''" ......"":(t.-",+-+-I--I.~.
• .. J' ~\.- +. x,.,..lj
i

(x, )I) ~ (x -:5, y + ~). \'., '::::i!- =-:_~_ 1/ i

. 11 i
'i.-
Aii ;;ans;ations~'i~ometrie;'becatlse the image and the . L,. ::: ._ _j__.J
<.
preimage are'con~ent. In thi~.'case~MBC == LlABC,
~.: .. . .~~~'",;,.' .. '~ -;

.~~~~',- ..~.
"
.l(~"~~4!I'Ui~""''n:'"· .), ~
t:Ji)$ilmpJ~
~ 1
mobl~rn
~~';'''RO¥J.P .){".. . '\
imagijs
What are:th~
(x,:y) ~
"
(;f5,
of the vertices of WXYZ for the translation
yll)? Graph the image of wxyz.
;~' " i-: A

"
~'
H-
I
v
y
+ --

W(-4, 1) :':"(-4 -+;$" I -1), qr:W (1, 0) e-- _L . ) '. Vi!

",:,::- , I', " ,:~," ,
e-- i
X(-4, 4) --,>(-4+5,4 -l),ot,X(l, 3) ,.
IV
T I-
~-
1I
". • L_ ,_ '\. '{';'

I ,
ot 'y (4,3)
I
..

Y(-1,4) --,>(-1+' 5, '4 -1), . i_

,
• e(- I..
e=: jL-t~- t- c--.
~I
i
I
I.
r.
Z( ~i,1) --,>(-1 +':5, 1 -1),
: :f~." ,:~
or:'Z (4, 0)'
~',.'. .i
e--
I
"'-,-f-.
_L 1 . .1. __ 1 ___ J l
. ... ..
Exercises ';:.
Use the rule to :O"d the images of the vertices for the translation.
. .' ,,'
"".
. ~
'.
.
I !.'~ . I ';.;

1. AMNO(x, y) '~(x + 2,y-J) 2. square,lKLM(x, y) ~ (x -l,Y)

,. M-' » ,-1 "~:, :
..
. .., y
-' i /( ;
1·-f-fJ . ··-1
,
Xi I

i -- --l--:.L:! .J
~- ,_JM.
f-
._
--t~.y~.
_l__:L-.-..-- ~_
" "'1
L-...J

P::InF' I 1
Graph the image of each figure under the given translation.

3. (x,y) ~,(~-l,~:4)
,.
4. (x"y) -*(x+3,y+3)
II Y l :j i
I
I 1 I
i '. ':'I I I I
I I': !. i 'I I
I
I, fA IX'
J I ..

rt I
..
jl> ~
l7r+ f-{
k"l'"'n
....!. I

I l'r '1 I ! I

..
..
'

Graph the image of each figure under the given translation.

r • ~
~;~.
- • - .. '_ A

~. (x,y)~(x':_~~y+4) .
6. (X,y) ~ (x-5,ytl)

Page 12
 Example Problem

Yr-+- -1'-
I
What rule describes the translation of ABCL? I -., 1--;-
j c i
to A 'B'C'D'? ., " I v

I ! I "l.. !

rf t±
I
To J5et from A to ~,(or from {1 to B or C -1-- t- ~ ,,q-
! : l\J__
to '(: or p to D' ), 'you move' 8 unitsleft I--- 1,-t--
I--- \-' t--- 1-' --i- . __ 1 :t. I \1
A-
arid 7 units' down. ; r:-t-r- j
J I I " I,
IU
The rule thatdescribes this translation i I
is(~,y) ~ (x-8,Y-7). )', . i- r=-~ - !-2 Q i 4- Ix
I 1 I
I "'" -.....r-, ,c11"'2-
~ • • ~ I ,

ct-J _, --+
~ ~ 1 "

:.1 --
.__
I

'";'
II\'_'
'r--:r-1--
~
I
t\F'
0'
-
J
1 -rI
--+--
I

The dashed-Iine-figure is a translation image ofthe solid-line figure. Write a rule to

des'cribe' ca~c~"translation. ". , : '. ::'
.. . ~;. . -. -'.~ .", ,

7. To start, identify the coordinates of the vertices of both figures. ID, Y

........
.'
'The ve~c~~ of the preim~~e are:'
A(':'3,' D)"BC-3, [j), and CCI, D).
,

A
, ,

" -,""
f--'"
....... r

I'" x
~f
The ye~i~es the ima~e '~e: . - ~
A'cD,+1), C'(3,D)·
~~
......
- ,...
B'CD,-l),and
. -, ',';' !:
'~
'''!,'
.j.,.
r-
,.'
The tra~slatioq
., rule is _1_;
8. 9. Y
; Y c
v
tl
IA ...J..--' I\.
., ,.; Iv' Li' )I 1\
'\ ., .., .a..
.. ..
.., , I ,~ R \ f--'" r-
,"
~, 'W' lA' ... ~ '\
I
'X " X
l. -
'\

0 , ..
_Y.

I
'J\.
, '.
Lfl'
'\
-
II ~ ~ ~ • rK
r 'T

,~ r-
'V, U

, ,
- ;., '

. "
" "

The dashed ...line figure is a-translation l . I .'

image of the solid-line figure. Write a rule 1.
to describe each translation," , i :,

10. 11.
y
"
,~
~ A Y
":>
_, _V l.{ ........
r--.,.

r:
\ "
....... w
, 'J "

~ « '
.,'
_!. 8' x /
... V'
-
, ,...,N'
- Y',
.., ,
_l I,
c ~

Page I3
 -,

Name - ~_,__---------Class Date _

Translations
Practice
......... ~~~ ~~ ~ ,............•....•.•............ , ................•.•
Use the gr~pn'atl~e right f~r Exerciaas ' '
1-3. " ":",r, i;'
Give the coordinates of point A after
it has b~eh,tranelated down 3 units.
2. Gi~~th~'66o~~!~'atesof ~bint B after
it has been tr~h$lated leff3 units,
,t, ~ , ...: ·f~.~~ "
3. Wrat ~p:l~he.~qprdinates of point N
after it is' translated right Er units
~ I ,-• " ;. f -~ . "
. ·;\",i.,

Grapheach tran~'~tion of Ap.CD~.

Use arrow potatiijn;to showthe
translation. ;"" \ fl-' .;'~
• .:. _ ' ." \ .; .~ J

S.A (2, 1), B (4,5), C (7,4),0 (5, -1);

4. A (211),:8'(4, 5}."C (7, 4):,,0 (5, '-1);
l~' '/,'
.:

rig~t 2 yhlts ~'_:

.' ',_'.

','
'f
down 1 unit, left 2 units ':-.::
.r , " \:;1 ~' • • • •

... ". " ,.~ • '1.

.. _
..: ~,'1~~'
: 1:: . ,: ', !«_~...'. :~,: ~~
" ~ - -. ~ ~ 'f"',' :.. ..... ~;

, ;"" ~.
t .... ~~~- ... "'1-- ...... ~•• ;;..~ ..,.... ....... 'j.,_,. ....'

~
...-: r ;-/ l ~~~~.':}
.....-! __
-
,_...- J ~ <9_ ~ .J.., T _..;:

1 ....·;, -c
f
, > , '\

, . _'fJ --f i-; ":?,1 :' HJ-'~

>

L~;E:,.'~H:±~::~f'
r_:;~:;.:~; r

o
'"

.,,'

Page I·
 ____________________________ --------Class
Name Date

Reflections

A reflection is a type oftransfonnation in which a geometric figure is flipped over

a line of reflection.
In a reflection, a preimage and an image have opposite orientations, but are the
same shape and size. Because the preimage and image are congruent, a reflection
is an isometry.

To graph a refl~ction image on a coordinate plane, graph the images of each

vertex. Each vertex in the image must be the same distance from the line of
reflection as the corresponding vertex in the preimage.

Reflection across the x-axis: Reflection across the j-axis:

(x, y) ~ (z, ~y) (x, y) -+ (~x, y)
The x-coordinate does not change. The j-coordinate does not
change.
The j-coordinate tells the distance The x-coordinate tells the
from the x-axis. distance from the y-axis

~xampie Problem:
MBC.has vertices atA(2, 4), B(6, 4); and C(3, 1). What is the
I
image of lUBe reflected over the x-axis? I -I
I Xi
Step 1: Graph AI, the image of A, It is the same distance from
the x-axis as A. The distance from the y-axis has not
changed. The coordinates for A' are (2, ~4).

Step 2: Graph 11 . It is the same distance from the x-axis as B.

The distance from the j-axis has not changed. The
coordinates for 11 are (6, -4).

Step 3: Graph C. It is the same distance from the x-axis as C.

the coordinates for Care (3, -1).
 9, Quadrilateral :WXYZwithvertices :W(~3, 4),X(-4, 6), Y(-2, 6), and Z(-1, 4) reflected across j--axis.

..LLli i
illi-tl
rr""t-r--i- - i I I 1 rr
II
it---j
I

: I I j I! r! 1 I . 1.1 I !
! i I I i [i i I I I
i \ t Ii! I ! i I I .1
1\" ..H_
I., , '_."' .• ,

Each point is reflected across the line indicated. Find the coordinates of each
image.
y
1. A across the x-axis
.r·
.,_.
2. B across tile y-axis "')
I
k

3. C acrossy == 1 X
.- - 0
4. D across x = ~1 -> I'
L
t 0 I
5. E across y = ~3 D

6. F across x = -2

Given points S(I, -2); r(4, -1), and V(4, -4), graph ASTVand its reflection
image across each line.

7. the x-axis
s.x =-1

.11 Y I Y

I
I-
"'}
-
"')

X
x
- - - 0 - - - 0
- ..,
-L
~

I /I

"t, .
Each figure is reflected across the line indicated. Find the coordinates of the vertices for each image.

6. MGR with vertices F(-1, 3), G(-5, 1), and H(- 3, 5) reflected across x-axis

t1 Y
"}
~
x
- - - 0
~-.
~
"- ~

7. !'!.CDE with vertices CC2, 4), D(5, 2), and E(6, 3) reflected across x-axis.

t l ill ! I !! I I I r

I I [ r I.

" ~---./

8. MKL with vertices J(-l, -5), K(-2, -3), and

L( -4, -6) reflected across y-axis.

I I j ! I
! j J i I
,
I I

Ll.L
! I rr -rr-'-,
r r t : i i
i"+_Jr-r-f- --t-1 t-·t·- -" -j-·+·-i-4C-I·--+--+---f--l
I -i

;-.ji -+-1 --!---+-4--i-t--I··-I--+·-I---I

t- _ ,._;_ !--.
\ i!. r f 1 ~