THE ALGEBRA / GEOMETRY CONNECTION
Algebra
FUNCTION: A correspondence between
two sets of numbers
Geometry
MAPPING: A correspondence between the
sets of points.
Pre-image (A)  Image (A’)
Input  Output
Pre-image
Input
f(x) = x2 – 3
f(-2) = (-2) 2 – 3
f(-2) = 4-3
f(-2) = 1
Output
T (x,y)  (x + 3, y – 6)
T(-1,5)  (-1 + 3, 5 - 6)
T(-1, 5)  (2, -1)
Image
T (x,y)  (x + 3, y – 6) is read “Coordinate
Rule T maps all points (x,y) to (x + 3, y – 6)”
ONE TO ONE FUNCTION: when there is
exactly the same number of elements in the
domain (x or input) as there is in the range (y or
output)
FUNCTION BUT NOT A ONE
TO ONE FUNCTION
TRANSFORMATION: when you have the
same number of points in the pre-image as in the
image
MAPPING BUT NOT A
TRANSFORMATION
L
M
N
K
T
U
S
FUNCTION AND A
ONE TO ONE FUNCTION
MAPPING AND A
TRANSFORMATION
L
M
K
N
L'
M'
K'
N'
An ISOMETRIC TRANSFORMATION (RIGID MOTION) is a transformation that preserves the
distances and/or angles between the pre-image and image.
Example #1
Example #2
Example #3
D
I
C'
B'
C
I'
J
L'
L
J'
E
E'
F
D'
F'
B
M
H
Rotate (Turn) – Example #1
H'
K'
K
Translate (Slide) – Example #2
T (x,y)  (y, -x)
M'
T (x,y)  (x + 3 , y + 0)
Reflection (Flip) - Example #3
T (x,y)  (-x, y)
A NON-ISOMETRIC TRANSFORMATION (NON-RIGID MOTION) is a transformation that does not
preserve the distances and angles between the pre-image and image.
Example #1
Example #2
F
C'
C
D'
D
B
B'
Dilation- Example #1
Example #3
E
G
E'
Stretch – Example #2
F'
G'
L
M
K
N
L'
M'
K'
N'
Stretch – Example #3
Dilation – Where both dimension’s scale factors is the same. The shape is proportional, not identical.
Dilation changes the size of the shape making it a NON-ISOMETRIC transformation.
Example: T (x,y)  (2x, 2y)
Stretch – Where one dimension’s scale factor is different than the other dimension’s scale factor.
Examples #2 and #3 represent stretches. A stretch definitely distorts the shape making it a NONISOMETRIC transformation. Example: T (x,y)  (2x, 3y)
------------------------------------------------------------------------------------------------------------------SYMMETRY: To carry a shape onto itself is another way of saying that a shape has symmetry.
There are three types of symmetries that a shape can have:
1. line symmetry
2. rotation symmetry
3. point symmetry
Line symmetry: A figure in the plane has a line symmetry if the figure can be mapped onto itself by a
reflection in the plane.
*The maximum lines of symmetry that a polygon can have are equal to its number of
sides. The maximum is always found in the regular polygon, because all sides and all
angles are congruent.
Rotational symmetry: A figure has rotational symmetry if the figure is the image of itself under a
rotation about a point through any angle whose measure is strictly between 0° and
360°. 0° and 360°are excluded from counting as having rotational symmetry
because it represents the starting position.
*ANGLE OF ROTATION - the SMALLEST angle through which the figure can be
rotated to coincide with itself. This number will always be a factor of 360°.
Angle of rotation = 360 ÷ order
*ORDER OF A ROTATION SYMMETRY -- The number of positions in which the
object looks exactly the same is called the order of the symmetry. When
determining order, the last rotation returns the object to its original
position. Order 1 implies no true rotational symmetry since a full 360 degree
rotation was needed.
Point Symmetry: Point Symmetry exists when a figure is built around a point such that every point in
the figure has a matching point that is:
1. the same distance from the central point
2. but in the opposite direction.
*A simple test to determine whether a figure has point symmetry is to turn it
upside-down and see if it looks the same. A figure that has point symmetry is
unchanged in appearance by a 180 degree rotation – all shapes that have point
symmetry have rotational symmetry with an even order.
*The point of rotation is a midpoint between every point and its image
Name
Parallelogram
Line Symmetry
Diagram
None
Rectangle
Line Symmetry
Count
0
2
Rotation Symmetry
Diagram
1
2
1
2
Rotation
Symmetry
Order
2
2
Trapezoid
0/1
Regular Polygon
1
1
2
3
4
5
6
None
Equal to the
number of sides of
the regular
polygon
2
3
6
5
4
0
Equal to the
number of sides
of the regular
polygon
THE ISOMETRIC TRANSFORMATIONS
Isometric Properties - the following properties are preserved between the pre-image and its image:

Distance (lengths of segments are the same)

Angle measure (angles stay the same)

Parallelism (things that were parallel are still parallel)

Collinearity (points on a line, remain on the line)
(1) THE REFLECTION
DEFINITION
A reflection in a line m is a isometric transformation that maps every point P in the plane to a point P’, so
that the following properties are true;
1. If point P is NOT on line m, then line m is the
2. If point P is ON line m, then P = P’
perpendicular bisector of PP ' .
m
m
P = P'
P
P'
EXPLANATION
A reflection over a line m (notation Rm) is an isometric transformation
m
in which each point of the original figure (pre-image) has an image that
A'
A
is the same distance from the line of reflection as the original point
but is on the opposite side of the line.. The line of reflection is the
perpendicular bisector of the segment joining every point and its
image.
B
Rm  ABC   A ' B ' C '
B'
C
C'
After a reflection, the pre-image and image are identical but

DISTANCES ARE DIFFERENT -- Points in the plane move different distances, depending on their
distance from the line of reflection. Points farther away from the line of reflection move a greater
distance than those closer to the line of reflection. Notice how AA’ is greater than BB’. Notice that
because line m is the perpendicular to AA ' , BB ' and

CC ' they are all parallel to each other.
ORIENTATION IS REVERSED – The pre-image has a reversed orientation than its image. Orientation
order of A – B – C but in the image the points in a clockwise direction come in the order of A’ – C’ – B’.
This occurs because a reflection creates the mirror image.

SPECIAL POINTS – Points on the line of reflection do not move at all under the reflection. The preimage (D) = image (D’) when the point is on the line of reflection.
(2) THE ROTATION
DEFINITION
A rotation about a Point O through Ɵ degrees is an isometric transformation that maps every point P in the
plane to a point P’, so that the following properties are true;
1. If point P is NOT point O, then OP = OP’ and
mPOP’ = Ɵ
2. If point P IS point O, then P = P’. The center of
rotation is the ONLY point in the plane that is
unaffected by a rotation.
P'
A'
P
θ
O
A
θ
O = P = P'
EXPLANATION
A rotation is an isometric transformation that turns a figure
about a fixed point called the center of rotation. Rays drawn
from the center of rotation to a point and its image form an
angle called the angle of rotation (notation Rcenter, degree).
An object and its rotation are the same shape and size, but the
figures may be turned in different directions.
RO,  ABC   A ' B ' C '
A'
A
B'
C
θ
C'
B
O
ROTATION DIRECTION
EQUIVALENT ROTATIONS
Because angles are formed along an arc of a circle there are two ways to get to the same location, a positive
direction and a negative direction. A 60° rotation is the same as a -300° rotation. Co-terminal angles can be
calculated using the formula, a coterminal angle = initial angle + 360n, where n is an integer.
Because a rotation is a transformation that maps all points along an arc the following properties are present.

DISTANCES ARE DIFFERENT -- Points in the plane move
different distances, depending on their distance from the center
A'
A
B'
of rotation. A point farther away from the center of rotation
maps a greater distance than those points closer to the center of
B
rotation. Notice that different from a reflection, AA ' , BB ' and
C'
CC ' are NOT parallel to each other.

C
ORIENTATION IS THE SAME – The pre-image has the same
O
orientation as its image. Orientation is the order of the points
about the shape. In ∆ABC the points in a clockwise direction come in the order of A – B – C and in the
image the points in a clockwise direction come in the order of A’ – B’ – C’. They are the same!!

SPECIAL POINTS – The center of rotation is the only point in the plane that is unchanged, O = O’.
SPECIAL ROTATION – ROTATION OF 180°
A rotation of 180° maps A to A’ such that:
a) m
b) OA = OA’ (from definition of rotation)
c) Ray
A
O
OA and Ray OA ' are opposite rays. (They form a line.)
AO is the same line as AA '
A'
(3) THE TRANSLATION
DEFINITION
A translation is an isometric transformation that maps every two points A and B in the plane to points A’
and B’, so that the following properties are true;
1. AA’ = BB’ (a fixed distance).
2. AA '|| BB ' (a fixed direction).
A'
A
B'
B
EXPLANATION
A translation slides an object a fixed distance in a given
direction. When working in the plane this is usually represented by an
arrow, the arrow provides both distance and direction of the
A'
A
translation. When working on the coordinate plane, a vector is used to
B'
B
describe the fixed distance and the given direction often denoted by
<x,y>. The x value describes the effect on the x coordinates (right
or left) and the y value describes the effect on the y coordinates (up
or down).
T x, y   ABC   A ' B ' C '
C
C'
TRANSFORMATION PROPERTIES – Because a translation is a transformation that maps all a fixed distance and
in a fixed direction the following properties are present.

DISTANCES ARE THE SAME -- Points in the plane all map the
exact same distance. Notice how AA’ is EQUAL TO BB’.

A'
A
B'
ORIENTATION IS THE SAME – The pre-image has the same
B
C'
direction come in the order of A – B – C and in the image the
C
points in a clockwise direction come in the order of A’ – B’ – C’.

SPECIAL POINTS – There are NO special points, ALL POINTS
IN THE PLANE MOVE!!!
SPECIAL TRANSLATION PROPERTY – TRANSLATING AN ANGLE ALONG ONE OF ITS RAYS
A translation of ABC by vector BA maps all points such that
B
C
B
C
1. ABC  A’B’C’ (Isometry)
A
2. B, A, B’ and A’ are collinear (translation on angle ray)
Because the two angles are equal and formed on the same ray, then:
BC || B ' C '
A = B'
This is a key property to translations – All segments that are
translated are parallel to each other.
A'
C'