Download Transformations as functions

• A function f(x) is defined as f(x) = −8x2. What is f(−3)?
A. −72
B. 72
C. 192
D. −576
E. 576
5.1.2: Transformations As
Functions
Do Now
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5.1.2: Transformations As
Functions
Good things!!!!
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Note Taking
1. How do you translate a pre-image by
changing the function?
2. How do you reflect a figure over an axis?
5.1.2: Transformations As
Functions
• Today’s Date: 9/7
• Essential Questions
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Intro to transformations
Define isometric transformations
Translations
Reflections
Guided practice
Exit Ticket + group check for understanding
Problem Based Task
5.1.2: Transformations As
Functions
Agenda
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Introduction
The word transform means “to
change.”
A transformation changes the
position, shape, or size of a figure on
a coordinate plane.
The preimage is the original image
It is changed or moved though
a transformation, and the resulting
figure is called an image
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5.1.2: Transformations As Functions
Introduction
Today will be focus on two transformations:
Also known as a
1. Translations - slide
“rigid
2. Reflections – mirror image
transformation”
These transformations are examples of isometry,
meaning the new image is congruent to the preimage
Figures are congruent if they both have the same
shape, size, lines, and angles. The new image is simply
moving to a new location.
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5.1.2: Transformations As Functions
Identifying Isometric Transformations
Which transformation is isometric? How do you know?
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5.1.2: Transformations As Functions
Transformations as functions
A function is a relationship between two sets of data
Each input has exactly one output
We can define points of a shape as functions because
for each x, we only have one y
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5.1.2: Transformations As Functions
Transformations as Functions, continued
In the coordinate plane we define each coordinate, or
point, in the form (x, y)
The potential inputs for a transformation function f in the
coordinate plane will be a real number coordinate pair,
(x, y)
Each output will be a real number coordinate pair, f(x, y)
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5.1.2: Transformations As Functions
Transformations as Functions, continued
Looking at point C:
Preimage: C is at (1, 3)
Image: C’ is at (5, 2)
We applied a
transformation to this
figure  same
process as putting
inputs into a function
and generating
outputs
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5.1.2: Transformations As Functions
Transformations as Functions, continued
Transformations are generally applied to a set of points on a figure
In geometry, these figures are described by points, P, rather than
coordinates (x, y)
Transformation functions are often given the letters R, S, or T.
T(x, y) or T(P) = Transformation on coordinates (x,y) or point P
P’ = “P prime”
A transformation T on a point P is a function where
T(P) is P'.
When a transformation is applied to a set of points, such as a triangle, then all
points are moved according to the transformation.
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5.1.2: Transformations As Functions
Introduction, continued
For example, if T(x, y) = (x + h, y + k), then
would be:
ALWAYS MOVE
X FIRST
We move the “x” by positive h, and the “y” by positive k
5.1.2: Transformations As Functions
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Translations
Translation = To move something
The figure does not change size, shape, or
direction – it is simply being moved from one
place to another
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5.1.2: Transformations As Functions
Translations
Represented by addition and subtraction
If we have a point (x,y), we can translate it “up
10 points” and “to the right 20 points” through
the translation
(x,y)  (x+20, y+10)
This tells us “all the x units moved by positive 20 and all the y
units moved by positive 10 units”
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5.1.2: Transformations As Functions
Translations
What is the
transformation from the
preimage to the image?
T(x,y)  ?
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5.1.2: Transformations As Functions
Translations
Let’s look at one point
A: (-5, 3)
A’: (2, -2)
• The x coordinate has
moved 7 units to the
right (positive)
• The y coordinate has
moved 5 units down
(negative)
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5.1.2: Transformations As Functions
Translations
Point A Transformation
A: (-5, 3) A’: (2, -2)
(x,y)  (x+7, y-5)
• Why don’t we move
the y first?
• Is this an isometric
transformation?
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5.1.2: Transformations As Functions
Translations – quick practice
-You can translate a figure without seeing the image
Point A: (3,5)
Point B: (5, 5)
Point C: (4, 7)
Apply the transformation T(x,y) = (x-3, y+6)
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5.1.2: Transformations As Functions
BRAIN BREAK!!!!!!!
Find a reason why each one does not belong in
the set.
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5.1.2: Transformations As Functions
Reflections
Mirror image over an axis
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5.1.2: Transformations As Functions
Reflections
The reflected image is always the same size, it is just
facing a different direction.
Is this an isometric transformation? Why or why not?
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5.1.2: Transformations As Functions
Reflections
For today, we will only focus on reflections across:
1. X-axis
2. Y-axis
3. Y = x
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5.1.2: Transformations As Functions
Reflections - Tips
When reflecting across the x-axis…
WHY DOES
THIS MAKE
SENSE??
If I want to reflect the point (9, 14) over the x-axis, what
will my new coordinates be?
5.1.2: Transformations As Functions
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Reflections – Tips
If I want to reflect the point (9, 14) over the y-axis, what
will my new coordinates be?
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5.1.2: Transformations As Functions
Reflections – Tips
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5.1.2: Transformations As Functions
Key Concepts, continued
• Isometry - a transformation where the preimage and the
image are congruent
• An isometry is also referred to as a “rigid
transformation” - the shape still has the same size,
area, angles, and line lengths
A
B
A'
C
Preimage
5.1.2: Transformations As Functions
B
'
C'
Image
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Guided Practice
Example 1
Given the point P(5, 3) and T(2, 2) = (x + 2, y + 2), what
are the coordinates of T(P) – also known as P’?
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5.1.2: Transformations As Functions
Guided Practice
Example 2
Given the transformation of a translation T(5,-3) = (x+5,
y-3), and the points P (–2, 1) and Q (4, 1), show that the
transformation is isometric by calculating the distances,
or lengths, of PQ and P ¢Q¢ .
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5.1.2: Transformations As Functions
Guided Practice: Example 3, continued
1. Plot the points of the preimage.
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5.1.2: Transformations As Functions
Guided Practice: Example 3, continued
2. Transform the points.
T5, –3(x, y) = (x + 5, y – 3)
T5, -3 (P) = (–2 + 5, 1- 3) Þ P ¢(3, - 2)
T5, -3 (Q) = (4 + 5, 1- 3) Þ P ¢(9, - 2)
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5.1.2: Transformations As Functions
Guided Practice: Example 3, continued
3. Plot the image points.
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5.1.2: Transformations As Functions
Guided Practice: Example 3, continued
4. Calculate the distance, d, of each
segment from the preimage and the
image and compare them.
Since the line segments are horizontal, count the number of units the
segment spans to determine the distance.
The distances of the segments are the same.
The translation of the segment is isometric.
✔
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5.1.2: Transformations As Functions
Exit Ticket
1. True or False: Under the translation (x,y)  (x+3, y+2),
the point (2,5) will become (5,7)
2. Reflect the point (6, 8) over the x-axis. What is your new
point?
3. You have a triangle with vertices A, B, C.
1. A is at (-5,4)
2. B is at (-2,4)
3. C is at (-2, 2)
Apply transformation: T(6,-2)
List new vertices A’, B’, C’
5.1.2: Transformations As Functions
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Problem Based Task (p.43)
“The mail room at a growing retail company stuffs,
addresses, weighs, and stamps hundreds of envelopes
to be mailed each day.”
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5.1.2: Transformations As Functions
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5.1.2: Transformations As Functions