Name
Date
Integrated ll Unit 6
Transformations
J:_o95i\b\§
Wrong
RIXSAJWS
Period:
transformation is a function that changes the position, shape, and/or size of a figure. The inputs
for the function are points in the plane; the outputs are other points in the plane. A figure that is used
is
as the input of a transformation is the pre-image. The output the image.
A
For example, the transformation T moves point A to point A’.
Point A is the pre-image, and A’ is the image. You can use
function notation to write T (A) = A’. Note that a transformation
is sometimes called a mapping. Transformation T maps point
A to point A’.
V
A’
1
H
T"
.
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).
;1
’
\
\
image
preqmage
For example, the notation (x, y) —> (x + 2, y - 3) means that the transformation adds 2 to the xcoordinate of a point and subtracts 3 from its y-coordinate. Thus, this transformation maps the point
(6, 5) to the point (8, 2).
I
write T(E) = F, identify the pre-image and the image.
?re
imacqm
E
:llYkx3c:
F
1.
If
2.
Consider the transformation given by the rule (x, y) —> (x + 1, y + 1). What is the domain of this
function? What is the range? Describe the transformation.
(4
cM\<\ "V‘-C
\3;u’\*
x+\
TN Domrx';1,§g
<;m:3c
‘
as
.“’
"L
?.
.
"Um: is ON amt‘ <55)”
one cin-‘t
triangle.
right
the
transformations
of
given
on
Investigate the effects various
0
o
0
A
.
.
.
.
CW5‘
LLB
.
Use coordinate notation to help you find the image of each vertex of the triangle.
Plot the images of the vertices.
Connect the images of the veitices to draw the image of the triangle.
(x.y)—>(x-y4.y+3)
B
(x.y)~(-x.yy)
Adapted from Oncore Mathematics: Geometry
C
(x.y)~(-yiyx)
Page
1
D
3.
A transformation
(XiY)—>(Xi‘/$3/)
F
E (xiv)->(2X.yY)
(Xiy)->(2X,y2Y)
if
preserves distance the distance between any two points of the pre-image
equals the distance between the corresponding points of the image. Which of the transformations
A-F (above and on the previous page) preserve distance?
A(B‘0«\c3C/
4.
A transformation
if
preserves angle measure the measure of any angle of the pre-image equals
the measure of the corresponding angle of the image. Which of the above transformations
preserve angle measure?
Asewéb
motion (or isometry) is a transformation that changes the position of a figure without
changing the size or shape of the figure.
A rigid
The figures below show the pre-image (AABC) and image (AA'B'C' ) under a transformation.
yes
Determine whether the transformation appears to be a rigid motion.
image aepaug
(
B
+0
A:
Cv
\
fit
have (orresponéinj
L)(“s3\U~Qi'L§’ <11‘
§t£)€S
arc‘
coniji-amt
C
A
'
pie
3'
A
I
We
win
Fmag Q.
g[&
UNI
B;
C
C’
A’
6. The transformation changes the shape of the figure. Therefore, +
[0p$(i,terl5'
Adapted from OnCore Mathematics: Geometry
GM‘;
7: *‘l~€r€'S>(€
5. The transformation does not change the size or shape of the figure. Therefore
B
o.\ۤ
is
not
4'
S
'traLV\Si-o\VV\&&i0"
M
mi
(On
Nio¢.»;or\ 0
5i'(§*{_{€(§
r
at H5
isawci r5.
CA
Page2
Rigid motions have some important properties.
These are summarized below.
These properties ensure that if a
- Rigid motions
preserve distance.
o Rigid motions
preserve angle
measure.
Rigid
motions
preserve
bctwoenness.
- Rigid motions
preserve collinearity.
7.
_
;
figure is determined by certain
points, then its image after a
rigid motion is also determined
by those points.
lflnt M is the midpoint of
After a rigid motion,
can you conclude that M’
is the midpoint of A'B' ?
Why or why not?
W5,
m '»°“\
AB
For example, AABC is
determined by its vertices, points
A, B, and C. The image of AABC
after a rigid motion is the triangle
determined by A’, B’, and C’.
.
because
u_\(\b€r5o ‘W9, Sa0N,_
(i\ow,m.rt\s as ml ‘-be
0“”'.‘&r
J96 ‘A65,
2
Draw the image of the triangle under the given transformation. Tell whether the transformation
appears to be a rigid motion.
B (X. Y)
A (X,Y)—’(X+3.Y)
Yeedws
5-.’
R ‘igx
..
I
D (X. Y)
H
(—X.
l'Y\o'¥‘;at.\
-<'
"
C
(3X. 33')
thfé
5! )9 No.
f\0‘\'O.f‘\3» ,5
\_.
r\\0Jf\Gf‘\
(X, Y)
(X. -Y)
"
VQSI‘-h/‘lg
‘S<~y
._
.
1
:
A
rbi
motion
V
Y€5,+W5 is
5»
ia
at ri3.'c\
<—_—F—+—-——r--+--II?
V.
I
.,5_.
Adapted from OnCore Mathematics: Geometry
YQ5,“W\l’S
._
..
a
c.'3.‘c\
l-§
motion
\
Page 3
Translations
translation slides all points of a figure the same distance in
the same direction. The figure shows a translation of a triangle.
A
:7
E
is convenient to describe
It
p
A
of
kT‘*""“""" translations using the language
3
A
is quantity
C
point
Mia,
130"“
You can use vectors to give a formal definition of translation.
P
‘
is a transformation along a vector such that the segment
joining a point and its image has the same length as the vector and is
A
{.__.
vectors. vector a
that has both direction and magnitude. The initial point of a vector is
the starting point. The terminal poir1t_c>f a vector is the ending point.
The vector at right may be named EF or v.
translation
~
-
.“‘»~--‘_.P’
parallel to the vector.
The notation Tv(P)
= P’
,7
says that the image of point P after a translation along vector
:2
is P’.
Trace ABCDEFGH on
The figure below shows pre-image ABCDEFGH and translation vector
patty paper and use the translation vector to draw the image of ABCDEFGH. Once you see the
pattern, draw the translation on the grid below.
8. Write the
coordinates of each
pre-image point and
its corresponding
image on the point.
Also write the
coordinates of the
endpoints of
I
I
.
.
.
,,
,
,
V
I
I
I
I
.
‘I
‘
QX_PX=
7
Qy_Py:
'2
Another way to
describe PQ is to
describe the change
x and y coordinates.
For this vector, you
in
would write <7, -2>
9. Describe how you can use the coordinates of
Adapted from OnCore Mathematics: Geometry
of the image.
E to find the(ti:coordinatesChalai
in X 0'“ J
I
9
K.
Ik
+9
vW£
eaét/u (Q0/[\i
‘RI
Page 4
(x, y) —>(x + 7, y — 2).
-2‘> and coo_rdinate notation (x, y) —>(x + 7, y — 2) give you the same
,,—,-.4.‘.8 21 ..,«,;u ‘to 1-.) {I
Q,
1 U,-,r.-‘
Still another way to describe the translation is to
Explain how vector notation
‘D9-Wt
information. “; Q 4
use coordinate notation
-
J
V
V
‘L
10. You are going to translate this
translation vector <-3, 2>.
I
_
What does <-3, 2> tell you?
Xeacl 3.
up 7-
X
b.
Write this in coordinate notation
c.
-7
Now draw the translated triangle on the grid to
5
‘fl
#5
fl
7
11. Is translation a rigid motion? Explain.
‘Q34 A» preserueswo. 517.2 amt‘ ’J\M.9€
Draw the image of
the figure under the
given translation.
Write the coordinate
notation I vector
notation for each.
5
-
d.
ét-s,g+.7_)
the left.
Give a translation that would move the original
triangle completely into Quadrant IV.
(X,j)'7
«’C""‘v'~
OH}
(X,
f_.V
*\’3‘L\\)
Z 3,
P‘<‘W<{j’—-
Qgj) ,7 (x+5ltj.2).
a. <3, -2>
12.
the
a.
_
_
+fq
I
L’
A
1 veelterflguslng
la
‘-
*
y) —>(x —4,yH+ 4)
5,
“'7
or
A -L]‘L’7
5’
«
13. Use both vector and coordinate notation to name the
[A
translation that maps ABC to W A'B'C'
(x.~D—* (><~I ,3—5\
4-\_-57
What distance does each point move under this translation?
\[-Eb
Adapted from Oncore Mathematics: Geometry
Page 5