Transformations: Mappings and
Functions
Lesson 14.1
Pre-AP Geometry
Lesson Focus
The purpose of this lesson is to introduce and define the
concept of a mapping.
The lesson also develops two other basic ideas, namely
transformations of the plane and distance preserving
mappings called isometries.
Basic Terms
Transformation
The operation that maps, or moves, a preimage onto an
image. Three basic transformations are reflections, rotations,
and translations.
Isometry
A transformation that preserves lengths. Also called a rigid
transformation or congruence mapping. Geometric figures
which can be related by an isometry are called congruent.
The idea of isometry has many uses in art, architecture, and
mechanical engineering.
Basic Terms
Image
The new figure that results from the transformation of a
figure in a plane.
Preimage
The original figure in the transformation of a figure in a plane.
Basic Terms
Mapping
A correspondence between points. Each point P in a given set
is mapped to exactly one point P’ in the same or a different
set. P’ is called the image of P, and P is called the preimage of
P’.
Function
A correspondence between sets of numbers in which each
number in the first set corresponds to exactly one number in
the second set.
Basic Terms
One-to-one mapping (or function)
A mapping (or function) from set A to set B in which every
member of B has exactly one preimage in A.
Mapping Notation
In algebra, the function f that shows the relationship of a
value x in set A to a value y in set B is expressed as
f(x) = y or f:x→y
In geometry, the mapping of a point P in the preimage to the
point P’ in the image is expressed as
M(P)=P’ or M:P→P’
Theorem 14-1
An isometry maps a triangle to a congruent triangle.
Practice #1
Given: Function k maps every number to a number that is two
less than one-third of the number.
1. Express this fact using function notation.
2. Find the image of 9.
3. Find the preimage of 16.
Corollary 1
An isometry maps an angle to a congruent angle.
Corollary 2
An isometry maps a polygon to a polygon with the same area.
Practice #2
Given: Mapping T maps each point (x, y) to the point
(x + 2, 3y).
1. Express this fact using mapping notation.
2. Find P’ and Q’, the images of P(2, 4) and Q(-2, 6).
3. Decide whether T maps M, the midpoint of PQ to M’, the
midpoint of P’Q’.
4. Decide whether PQ = P’Q’.
Notes
• By definition, an isometry preserves distance. You can think
of an isometry as keeping a figure rigid. Because the figure is
kept rigid, its image will be a congruent figure.
• Until it is proven that a given transformation maps every
segment to a congruent segment, you may not claim that a
transformation is an isometry. You may say that it appears to
be an isometry based on your experiments with particular
segments.
Practice #3
Given: Mapping S maps each point (x, y) to an image point (x,
-2y). Also, A(-3, 1), B(-1, 3), C(4, 1), and
D(2, -1).
1. Decide whether:
AB = A’B’
BC = B’C’
AC = A’C’
CD = C’D’
2. Is S an isometry? Explain.
Written Exercises
Problem Set 14.1, p.574: # 2 - 10 (even);
Handout 14-1