Composition and Symmetry

Lesson 14.6Pre-AP Geometry

Lesson Focus

Two transformations can be combined to produce a new transformation called a composite. The focus of this lesson is

to study composites of transformations.

Basic Terms

Composite of mappingsA transformation that combines two mappings.

The composite of mappings S and T maps P to P’’ where T:P→ P’ and S:P’→ P’’ such that

S o T:P →P’’

Also called a product of mappings.

Theorem 14-6

The composite of two isometries is an isometry.

Theorem 14-7

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 14-8

A composite of reflections in tow 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 to Theorem 14-8

A composite of reflections in perpendicular lines is a half-turn about the point where the lines intersect.

Notes

• In algebra, the parentheses in (f o g)(x) are retained because writing g o f(x) would be a nonsensical composition of the function g and the number f(x). When using mapping notation, these parentheses are no longer needed because the colon serves as a grouping symbol, as in g o f: x → 2x2.

• The composition of a half-turn and a reflection is not commutative.

• A rotation is a transformation, so the image of any point can be found by using the center, the magnitude, and the direction of rotation (clockwise or counterclockwise).

Practice1. If f(x) = 3x + 1 and g(x) = x2, find

(a) (g o f)(x)(b) (f o g)(x)

2. Given: T:(x, y)→(x + 2, y – 1), find(a) O, 30 o O, 60

(b) HO o Rx

(c) Rx o Ry

(d) Rx o T(e) Ry o Ry

Written Exercises

Problem Set 14.6, p. 603: # 2 – 12 (even)Handout: 14-6