1. Do Now A function f(x) is defined as f(x) = −8x2. What is f(−3)?
B. 72
C. 192
D. −576
E. 576
5.1.2: Transformations As Functions A

2. Good things!!!!
5.1.2: Transformations As Functions

3. Note Taking 2. How do you reflect a figure over an axis?
Today’s Date: 9/7
Essential Questions
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

4. Agenda 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

5. 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
5.1.2: Transformations As Functions

6. Introduction Today will be focus on two transformations:
1. Translations - slide
2. Reflections – mirror image
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.
Also known as a “rigid transformation”
5.1.2: Transformations As Functions

7. Identifying Isometric Transformations
Which transformation is isometric? How do you know?
5.1.2: Transformations As Functions

8. 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
In this lesson, we will learn to describe transformations as functions on points in the coordinate plane. Let’s first review functions and how they are written.
5.1.2: Transformations As Functions

9. 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)
5.1.2: Transformations As Functions

10. 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
5.1.2: Transformations As Functions

11. Transformations are generally applied to a set of points on a figure
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.
5.1.2: Transformations As Functions

12. 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

13. Translation = To move something
Translations
Translation = To move something
The figure does not change size, shape, or direction – it is simply being moved from one place to another
5.1.2: Transformations As Functions

14. Represented by addition and subtraction
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”
5.1.2: Transformations As Functions

15. What is the transformation from the preimage to the image?
Translations
What is the transformation from the preimage to the image?
T(x,y)  ?
5.1.2: Transformations As Functions

16. The x coordinate has moved 7 units to the right (positive)
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)
5.1.2: Transformations As Functions

17. Point A Transformation
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?
5.1.2: Transformations As Functions

18. 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)
A: (0, 11)
B: (2, 11)
C: (1, 13)
5.1.2: Transformations As Functions

19. Find a reason why each one does not belong in the set.
BRAIN BREAK!!!!!!!
Find a reason why each one does not belong in the set.
5.1.2: Transformations As Functions

20. Reflections Mirror image over an axis
Every point on the mountain and the reflected image of the mountain is the same distance from the line of reflection
5.1.2: Transformations As Functions

21. Is this an isometric transformation? Why or why not?
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?
5.1.2: Transformations As Functions

22. For today, we will only focus on reflections across: X-axis Y-axis
Y = x
5.1.2: Transformations As Functions

23. 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

24. Reflections – Tips
If I want to reflect the point (9, 14) over the y-axis, what will my new coordinates be?
5.1.2: Transformations As Functions

25. Reflections – Tips
5.1.2: Transformations As Functions

26. 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 A' B C B' C'
Preimage
Image
5.1.2: Transformations As Functions

27. 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’?
5.1.2: Transformations As Functions

28. 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 and
5.1.2: Transformations As Functions

29. Guided Practice: Example 3, continued Plot the points of the preimage.
5.1.2: Transformations As Functions

30. Guided Practice: Example 3, continued Transform the points.
T5, –3(x, y) = (x + 5, y – 3)
5.1.2: Transformations As Functions

31. Guided Practice: Example 3, continued Plot the image points.
5.1.2: Transformations As Functions

32. ✔ Guided Practice: Example 3, continued
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. ✔
5.1.2: Transformations As Functions

33. Apply transformation: T(6,-2) List new vertices A’, B’, C’
Exit Ticket
True or False: Under the translation (x,y)  (x+3, y+2), the point (2,5) will become (5,7)
Reflect the point (6, 8) over the x-axis. What is your new point?
You have a triangle with vertices A, B, C.
A is at (-5,4)
B is at (-2,4)
C is at (-2, 2)
Apply transformation: T(6,-2)
List new vertices A’, B’, C’
5.1.2: Transformations As Functions

34. 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.”
5.1.2: Transformations As Functions

35. 5.1.2: Transformations As Functions