14-1 Mappings and Functions
Transformational Geometry
One branch of geometry, known as
transformational geometry, investigates
how one geometric figure can be transformed
into another. In transformational geometry
we are required to reflect, rotate, and change
the size of the figures.
Mapping
Image and Preimage
Mappings and Functions
• Mapping  Geometry: Correspondence
between a set of points.
• Function  Algebra: Correspondence
between sets of numbers.
One-to-one
• A mapping (or a function) from set A to set
B is called a one-to-one mapping (or
function) if every member of B has exactly
one preimage in A.
y = x2 is not a one-to-one function
9 has two preimages, 3 and -3
Example 1
• Function k maps every number to a number
that is two less than one third of the number.
– Express this fact using function notation
– Find the image of 9
– Find the preimage of 16
Example 2
• Mapping T maps each point (x,y) to the
point (x+2, 3y)
– Express this fact using mapping notation
– Find P’ and Q’ the images of P(2,4) and Q(-2,6)
– Decide whether T maps M, the midpoint of PQ
to M’ the midpoint of P’Q’.
– Decide whether PQ = P’Q’
Transformation
• A one-to-one mapping from the whole plane
to the whole plane.
–
–
–
–
–
Reflection
Translation
Glide Reflection
Rotation
Dilation
Isometry
• If a transformation maps every segment to a
congruent segment
• “Preserves distance”
Theorem
• An isometry maps a triangle to a congruent
triangle
Corollary
• An isometry maps an angle to a congruent
angle
Corollary
• An isometry maps a polygon to a polygon
with the same area.
Example 3
• Mapping S maps each point (x,y) to and
image point (x,-2y). Given A(-3,1) B(-1,3)
C(4,1) and D(2,-1)
– Decide whether S is an isometry
14-2 Reflections
Reflection
A reflection is another type of geometric
transformation.
A reflection is a mirror image that is
created when a figure is flipped over a
line.
Example: Reflection Image About Line
m
m
m
m
Reflections
Line m is called the line of reflection
We call A’ the reflection image of the point A
A
m
A’
The dashed line shows that the points are images of
each other under this transformation.
Line m is perpendicular to the line segment AA’ and
also bisects it.
A
m
A’
• We say A is reflected in line m to A’
• To abbreviate this “reflection in line m” we write
Rm:AA’ or
Rm:(A) = A’
A
m
A’
Theorem 14-2
• A reflection in a line is an isometry
Isometry
• Preserves distance
• Preserves angle measure
• Preserves area of a polygon
Invariant
• Another way to say that the distance, angle
measure and area are preserved when doing
a reflection, is to say
– Distance, angle measure and area are invariant
under a reflection.
Triangle ABC has vertices A(2,4), B(0,6), and C(-2,2). Graph the
figure and its reflected image over the x-axis. Then find the
coordinates of the reflected image.
B
A
C
Triangle ABC has vertices A(2,4), B(0,6), and C(-2,2). Graph the
figure and its reflected image over the x-axis. Then find the
coordinates of the reflected image.
B
A
C
C’
A’
B’
Quadrilateral RSTV has vertices R(2,3), S(-1,5), T(-3,0), V(3,-4).
Graph the figure and its reflected image over the y-axis. Then find
coordinates of the reflected image.
Triangle ABC has the vertices A(-6,-1) B(-2,-1) C(-5,-6). Graph the
figure and its reflected image over the line y=x. Then find
coordinates of the reflected image.
Triangle ABC has the vertices A(-6,-1) B(-2,-1) C(-5,-6). Graph the
figure and its reflected image over the line y=x. Then find
coordinates of the reflected image.
White Board Practice
1. Rm : stands for ?
White Board Practice
2. Rk :A  ____
A
j
S
B
W
T
D
C
X
U
Y
k
White Board Practice
3. Rk (B) = ____
A
j
S
B
W
T
D
C
X
U
Y
k
White Board Practice
4. Rk AB ____
A
j
S
B
W
T
D
C
X
U
Y
k
White Board Practice
5. Rk (C) = ____
A
j
S
B
W
T
D
C
X
U
Y
k
White Board Practice
6. Rk :T = ____
A
j
S
B
W
T
D
C
X
U
Y
k
White Board Practice
7. Rk :BC = ____
A
j
S
B
W
T
D
C
X
U
Y
k
White Board Practice
8. Rk :STU  ____
A
j
S
B
W
T
D
C
X
U
Y
k
White Board Practice
9. Rj :(S) = ____
A
j
S
B
W
T
D
C
X
U
Y
k
White Board Practice
10. Rj :ST = ____
A
j
S
B
W
T
D
C
X
U
Y
k
White Board Practice
11. Rj : (
A
) =XY
j
S
B
W
T
D
C
X
U
Y
k
White Board Practice
12. Rj : line k  ______
A
j
S
B
W
T
D
C
X
U
Y
k
14-3 Translations and Glide
Reflections
Translation
Translation
• A transformation glides all points of the
plane the same distance in the same
direction.
• A translation is a transformation that
corresponds to physical sliding without
turning.
Vectors
A’
B’
A
B
C’
C
Coordinates
• You don’t need to know the coordinates,
you just need to know that if one point
slides up 5 and to the right 3, then all points
slide up 5 and to the right 3
If a transformation is a translation
then all arrows
• Must be parallel and the same length
Example 1
•
The translation T: (x,y)(x+3, y-1) maps
triangle ABC to triangle A’B’C’. A(3,-1),
B(0,2), C(2,-3)
(a) Graph triangle ABC and its image
(b) Draw arrows connecting A to A’, B to B’, and
C to C’
(c) Are the arrows the same length and parallel?
Example 2
•
If T: (2,2)(-2,-2), then
T: (4,4)( ? , ? )
Glide Reflection
• Glide reflection is a transformation
where a translation is followed by a
reflection in a line parallel to the
direction of translation.
•The order of the two transformations
(translation and reflection) is not important.
•You will get the same result by first
reflecting and then translating the image.
Example 3
• A glide reflection moves all points down 3
units and reflects all points in the x-axis.
Find the image of A(2,-1), B(1,1) and
C(3,3)
14.4 Rotations
To avoid confusion
• R (Reflection)
• RP,45° (Rotation)
A ROTATION of a geometric figure
is the turn of the figure around a
fixed point.
5
A
4
3
2
C
-5
-4
B
-3
-2
1
-1
1
-1
-2
Rotate the figure
90 around the
origin.
-3
-4
-5
2
3
4
5
5
A
C’
A’
4
3
2
C
-5
-4
B
-3
-2
1
-1
1
-1
-2
Rotate the figure
clockwise 90
around the origin.
B’
-3
-4
-5
2
3
4
5
Rotate the figure -90
around the origin.
5
4
3
2
1
-5
-4
-3
-2
-1
1
-1
A
2
3
4
5
B
-2
-3
D
-4
-5
C
Rotate the figure -90
around the origin.
5
B’
C’
4
3
2
A
1
-5
-4
-3
-2
-1
D’
1
-1
A
2
3
4
5
B
-2
-3
D
-4
-5
C
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
-1
Rotate the figure
180 around the
origin.
-2
A
-3
-4
-5
B
C
5
B’
C’
5
4
3
2
1
A’
-5
-4
-3
-2
-1
1
2
3
4
-1
Rotate the figure
180 counterclockwise around
the origin.
-2
A
-3
-4
-5
B
C
5
Theorem
• A rotation is an isomety
Special Rotations
• 360
• 180
• 390
360 rotation
• Rotates any point P around to itself.
180
• A rotation about point O of 180 is called a
half turn.
• A Halfturn about the origin can be written
Ho: (x,y)(-x,-y)
Rotation of 390 ??
• 360 + 30
Example 1
•
(a)
(b)
(c)
(d)
State another name for each rotation
Ro,-270°
Ro,180°
Ro,450°
Ro,135°
Example 2
•
(a)
(b)
(c)
(d)
The diagonals of square ABCD intersect
at O. Complete each statement.
Ro,-90° :B
Ro,-270°:C
Ro,180° :A
RD,-90°:A
Page 589
• Classroom Exercises 1-11
14.5 Dilations
Isometries
•
•
•
•
Reflection
Translation
Glide reflection
Rotation
Dilations
A dilation is a
transformation that
changes the size but
not the shape of an
object or figure.
Every dilation has a fixed
point that is called the
center of dilation.
So a dilations is related to….
Do,k
• O is the center of dilation
• k is the scale factor
• If k>1, the dilation is called an
expansion.
– The shape will get bigger
•If k<1, the dilation is called an
contraction.
–The shape will get smaller
Dilations
To dilate an object with a center of dilation of
the origin only:
1) Graph object if necessary.
2) Multiply the coordinates of the object by
the scale factor.
3) Graph new coordinates.
Example 1
Do,2
Example 2
Your turn:
D0,-1
-1
A negative scale factor
• Changes the direction of the dilation
• It will create opposite rays
To do a dilation with a center of
dilation not at the origin
• Measure from the center of dilation to a
point.
• Multiply that distance by the absolute value
of the scale factor.
• Measure from the center of dilation to a
new point with your new distance.
Remember….
• If the scale factor is negative you would
measure in the opposite direction.
Example 3
• Find the image of WXYZ under D0,1/2
X
Y
O
W
Z
Example 4
• Find the image of RST under D0,3
S
R
T
O
Theorem
• A dilation maps a triangle to a
similar triangle
Corollary
• A dilation maps an angle to a
Congruent angle
Corollary
• A dilation D0,k maps any segment to a
parallel segment k  times as long.
Corollary
• A dilation D0,k maps any polygon to a
similar polygon whose area is k2times
as large
14.6 Composites of Mappings
Theorem
• The composite of two isometries is an
isometry.
Theorem
• A composite of reflections in two parallel
lines is a translation. The translation glides
all points through twice the distance from
the first line of reflection to the second.
Theorem
• A composite of reflections in two
intersecting lines is a rotation about the
point of intersection of the two lines. The
measure of the angle of rotation is twice the
measure of the angle from the first line of
reflection to the second.
Corollary
• A composite of reflections in perpendicular
lines is a half turn about the point where the
lines intersect.
White Board Practice
Page 602 # 3
14-7 Inverses and the Identity
T: glides every runner one place
to the right
T2: glides every runner two
places to the right
The inverse of T
Written T-1
T-1: glides every runner one place
to the left
T-1 ° T:PP
• Keeps all points fixed
Identity
• The mapping that maps every point
to itself is called the identity
transformation.
• I is the identity
• T ° I = T and I ° T = T
Inverse
• The inverse of a
transformation T is defined
as the transformation such
-1
-1
that T ° T = I or T ° T = I
Example 1
• The symbol 2-1 stands for the inverse
of 2 or ½ . They multiple to be 1.
Give the value of the following.
a) 3-1
b) 7-1
c) (4/5)-1
d) (2 -1)-1
Example 2
• Find the inverses of the following
transformations.
a) Reflection Rx
b) Translation T: (x,y)(x-2, y+3)
c) Rotation R o,a
d) Dilation D o,3
Example 3
• Which pairs of transformations
are inverses?
a) R o,180 and R o,-180
b) R o,270 and R o,-90
c) T: (x,y) (x+1, y-2) and U:
(x,y) (x-2, y-1)
d) Rx ° Ry and Ry ° Rx
14.8 Symmetry in the Plane and
in Space
Symmetry
• A figure in the plane has symmetry if there
is an isometry, other than the identity that
maps the figure to itself.
Line Symmetry
What is a line of
symmetry?
• A line on which a figure can be folded so
that both sides match
Here are some examples of common
geometric figures and their lines of
symmetry.
Line symmetry
• is really reflecting
Point Symmetry
Point Symmetry
Point Symmetry
Point Symmetry
• is really half turns
Rotational Symmetry
Rotational Symmetry
Translational Symmetry
Glide reflection Symmetry