Composition of Transformations

Composition of Transformations
Objectives:
1. To perform
compositions of
transformations
2. To find the inverse
of a transformation
Assignment:
• 10-1 SB Worksheet
• Challenge Problems
Objective 1
You will be able to perform
compositions of transformations
Transformations
A transformation is an
operation that changes
some aspect of the
geometric figure to
produce a new figure.
A'
The original figure is
called the pre-image
A
The transformed figure
is called the image
Composition of Transformations
A composition of transformations is a sequence
of transformations performed on the same figure.
Function Composition
Function composition happens when we
take a whole function and substitute it in
for x in another function.
h( x)  g  f ( x) 
Substitute f(x) in for x in g(x)
– The “interior” function gets substituted in for x
in the “exterior” function
Exercise 1
Let f(x) = 2x − 4 and g(x) = 1/2x + 2.
Evaluate f(g(8)).
Transformation Notation
Don’t let this stuff confuse you; it’s just notation.
Notation
Transformation
𝑟𝑦=0
Reflection across the 𝑥axis
𝑅
0,0 ,90°
𝑇 5,−2
Rotation of 90° counterclockwise around the origin
Translation under the
vector 5, −2
Transformation Notation
The notation for the composition of transformations
is similar to function composition in that you work
from the interior to the exterior.
Notation
𝑟𝑦=0 𝑅 0,0
,90°
𝑇 5,−2
1. Translation under the vector
5, −2
Transformations 2. Rotation 90°CC around the
origin
3. Reflection across 𝑥-axis
Transformation Notation
𝑟𝑦=0 𝑅
0,0 ,90°
𝑇 5,−2
1. Translation under the
vector 5, −2
2. Rotation 90°CC around
the origin
3. Reflection across 𝑥axis
Transformation Notation
𝑟𝑦=0 𝑅
0,0 ,90°
𝑇 5,−2
1. Translation under the
vector 5, −2
2. Rotation 90°CC around
the origin
3. Reflection across 𝑥axis
Exercise 2a
Does the order matter when you perform multiple
transformations in a row?
1. 𝑇 −4,−1 𝑇 2,−3
2. 𝑇 2,−3 𝑇 −4,−1
Exercise 2b
Does the order matter when you perform multiple
transformations in a row?
1. 𝑟𝑦=0 𝑟𝑥=0
2. 𝑟𝑥=0 𝑟𝑦=0
Exercise 2c
Does the order matter when you perform multiple
transformations in a row?
1. 𝑟𝑥=0 𝑇 2,−3
2. 𝑇 2,−3 𝑟𝑥=0
Exercise 3
Write notation for the following compositions of
transformations. Use points 𝐴(0,0), 𝐵(1,1), and
𝐶(0, −1).
1. Clockwise rotation of 60° about the origin
followed by a translation by directed line
segment 𝐴𝐵
2. Reflection about the line 𝑥 = 1, followed by a
reflection about the line 𝑥 = 2
3. Three translations, each of directed line
segment 𝐴𝐶
You will be able to find
the inverse of a
transformation
Objective 2
Exercise 4a
Identify a composition of transformations that
could map the arrow on the left to the image
of the arrow on the right.
Exercise 4b
Identify a composition that undoes the
mapping, meaning it maps the image of the
arrow on the right to the pre-image on the left.
Inverse Relations
An inverse relation is a relation that switches the
inputs and output of another relation.
Inverse
relations
“undo”
each
other
Inverse Functions
If a relation and its inverse are both
functions, then they are called inverse
functions.
f  g ( x)   x and g  f ( x)   x
f -1 = “f inverse” or “inverse of f ”
Exercise 5
Let 𝑓(𝑥) = 2𝑥 − 4 and 𝑔(𝑥) =
each of the following. and.
1. 𝑓(5)
2. 𝑔(6)
1
𝑥
2
+ 2. Find
3. 𝑓(𝑔 4 )
Inverse Transformations
Two transformations are inverse
transformations if they reverse each other.
For all points 𝐴
on the preimage
𝑇 𝑇 −1 𝐴 = 𝐴
𝑇 4,2
−1
= 𝑇 −4,−2
Inverse Transformations
Exercise 6
1. Identify a
composition of
transformations
that maps the
purple T onto the
green T.
2. What is the inverse
transformation
from Q1?
Composition of Transformations
Objectives:
1. To perform
compositions of
transformations
2. To find the inverse
of a transformation
Assignment:
• 10-1 SB Worksheet
• Challenge Problems