1. Graphing & Describing “Reflections”
Chapter 6
Day 4
Graphing & Describing “Reflections”

2. We have learned that there are 4 types of transformations:
1) Translations
2) Reflections
3) Rotations
4) Dilations
The first 3 transformations preserve the size and shape of the figure.
In other words…
If your pre-image (the before) is a trapezoid,
your image (the after) is a congruent trapezoid.
If your pre-image contains parallel lines, your image contains congruent parallel lines .
If your pre-image is an angle, your image is an angle with the same measure.

3. Yesterday… our lesson was ALL about translations
Yesterday… our lesson was ALL about translations. Today… our lesson will focus on reflections.

4. Let’s Get Started Section 1: Comparing a translation to a reflection.
Section 2: Performing a reflection over the x- or y-axis.
Section 3: Performing a reflection over a line.
Section 4: Describing a reflection.

5. Turn and Discuss
Which of the below is the translation and which is the reflection. Be ready to explain how you know to the class.
# #2
Transformation #1 is a reflection. I know because the figures are mirror images (or a flip). Some students may give more detail by indicating that it is a flip over the y-axis. Some may have determined it was the reflection by noticing that the figures are pointing in opposite directions.
Transformation #2 is a translation. I know because the figures represent a slide. Some students may give more detail by indicating that the figure slid down and left. Some may have determined it was the translation by noticing that the figures are pointing in the same direction.

6. Turn and Discuss
Which of the below is the translation and which is the reflection. Be ready to explain how you know to the class.
# #2
Transformation #1 is a translation. I know because I looked at the positioning of the letters. The letters are in the same positions on the pre-image and image. A slide will map the rectangles and letters on top of each other.
Transformation #1 is a reflection. I know because I looked at the positioning of the letters. On the pre-image the A and B are at the top of the rectangle. On the image, A’ and B’ are at the bottom of the rectangle. A flip will map the rectangles and letters on top of each other.

7. If You Remember…
A translation is a slide that moves a figure to a different position (left, right, up or down) without changing its size or shape and without flipping or turning it.
You can use a slide arrow to show the direction and distance of the movement.
Have a student come to the board and label the pre-image and image on each of these pictures.
Ask… Compare each image to its pre-image. Are they congruent? How can you tell?
Answer: The figures to the left appear congruent. The figures on the right are congruent because you can compare the lengths of the segments making up the rectangles. They are the same size.

8. A reflection (flip) creates a mirror image of a figure.
Have a student come to the board and label the pre-image and image on each of these pictures.
Ask… Compare each image to its pre-image. Are they congruent? How can you tell?
Answer: The figures to the left appear congruent. The figures on the right are congruent because you can compare the lengths of the segments making up the rectangles. They are the same size.

9. Moving On Section 1: Comparing a translation to a reflection.
Section 2: Performing a reflection over the x- or y-axis.
Section 3: Performing a reflection over a line.
Section 4: Describing a reflection.

10. A Reflection is a Flip over a Line
A reflection is a flip because the figure is “flipped” over a line. Each point in the image is the same distance from the line as the original point. A B C A' B' C' t
A and A' are both 6 units from line t.
B and B' are both 6 units from line t.
C and C' are both 3 units from line t.
Each vertex in ∆ABC is the same distance from line t as the vertices in
∆A'B'C'.
Check to see if the pre-image and image are congruent.

11. STEPS and EXAMPLE
Step 1: Start with any vertex and count the number of units to the specified axis (or line).
Step 2: Measure the same distance on the other side of the axis (or line) and place a dot. Label using prime notation.
Step 3: Repeat for the other vertices.
Reflect the figure across the y-axis.
Demonstrate and explain how to reflect while students watch. Make sure they know the x-axis from the y-axis.
Check to see if the pre-image and image are congruent.

12. Reflection Across the Y-AXIS
Let’s name the coordinates of each figure.
A ( , ) A' ( , )
B ( , ) B' ( , )
C ( , ) C' ( , )
D ( , ) D' ( , )
Pre-image Image
reflection
across y-axis
----- When you reflect across the y-axis -----
The y-coordinate will always stay the same.
The x-coordinate will always flip signs.
𝑥, 𝑦 → −𝑥, 𝑦
Compare the coordinates of the pre-image to the image.
What do you notice?
Pre-Image Image
A(-6, 5) A’(6, 5)
B(-3, 5) B’(3, 5)
C(-3, 1) C’(3, 1)
D(-6, 3) D’(6, 3)

13. Let’s take a look at the same pre-image and see what it looks like after being reflected across the x-axis …

14. Reflection Across the X-AXIS
Here is the same pre-image but this time it is reflected across the x-axis.
Pre-image Image
𝐀 (−𝟔, 𝟓) 𝐀′ (−𝟔, −𝟓)
𝐁 (−𝟑, 𝟓) 𝐁′ (−𝟑, −𝟓)
𝐂 (−𝟑, 𝟏) 𝐂′ (−𝟑, −𝟏)
𝐃 (−𝟔, 𝟑) 𝐃′ (−𝟔, −𝟑)
reflection across x-axis
When you reflect across the x-axis
The x-coordinate will always stay the same.
The y-coordinate will always flip signs.
𝒙, 𝒚 → 𝒙,− 𝒚
Use this slide to explain how to do a reflection over the x-axis. Go ahead and perform the reflection so that they can see how we came up with the reflection that we did. Then review what happens to the coordinates when a reflection over the x-axis is performed as directed on the slide.
Compare the coordinates of the pre-image to the image.
What do you notice?

15. You Try #1 Name the coordinates of your reflection:
Reflect ∆𝑨𝑩𝑪 across the y-axis.
Name the coordinates
of your reflection:
Image
A’(2,3)
B’(8,6)
C’(4,7)
You Try #1

16. You Try #2 Name the coordinates of your reflection:
Reflect figure QRST across the x-axis
Name the coordinates
of your reflection:
Image
Q’(-8, -2)
R’(-4, 2)
S’(-10, 5)
T’(-6, 5)
You Try #2

17. Turn and Discuss Which could be a point reflected across the x-axis?
Which could be a point reflected across the y-axis?
Be prepared to explain how you know to the class.
A. 𝟓, 𝟖 →(−𝟓, −𝟖)
B. 𝟓, 𝟖 →(𝟓, −𝟖)
C. 𝟓, 𝟖 →(−𝟓, 𝟖)
Answer:
“B” could be a point reflected across the x-axis because the x-coordinate stayed the same and the y-coordinate flipped signs.
“C” could be a point reflected across the y-axis because the y-coordinate stayed the same and the x-coordinate flipped signs.

18. Moving On Section 1: Comparing a translation to a reflection.
Section 2: Performing a reflection over the x or y-axis.
Section 3: Performing a reflection over a line.
Section 4: Describing a reflection.

19. Reflections Across a Vertical or Horizontal Line
You may be asked to reflect a figure across a line that is not the x-axis or the y-axis.
Let’s see how to do that…

20. Example over a Vertical Line
First, draw the line of reflection.
Then follow the normal steps for reflecting over a line.
line of reflection
Explain how this reflection was done across the line x = 3.

21. Example over a Horizontal Line
First, draw the line of reflection.
Then follow the normal steps for reflecting over a line.
line of reflection
Explain how this reflection was done across the line y = -2.

22. You Try #3 Name the coordinates of your reflection:
Reflect ∆𝑬𝑭𝑮 across the line 𝒙=𝟐
Name the coordinates
of your reflection:
Image
E’(-1, -4)
F’(-4, -1)
G’(-4, -4)
You Try #3

23. You Try #4 Name the coordinates of your reflection:
Reflect figure ABCD across the line 𝐲=𝟒
Name the coordinates
of your reflection:
Image
A’(-9,1)
B’(-5, 1)
C’(-6,3)
D’(-9,3)
You Try #4

24. Moving On Section 1: Comparing a translation to a reflection.
Section 2: Performing a reflection over the x or y-axis.
Section 3: Performing a reflection over a line.
Section 4: Describing a reflection.

25. Describing a Reflection
In this Section, you will be given a reflection that has already been performed, and you will describe what occurred.
Example:
Description
reflection across 𝒙=−𝟏
Some students may mistakenly think this is a reflection across the y-axis.
Explain to students how this is a reflection across the vertical line x = -1.

26. You Try Describe the below reflections. 5) 6) Reflection across x-axis
Reflection across y-axis

27. You Try
7) Which of the following represents a single reflection of Figure 1?
Figure 1 A C
Answer: B B D

28. You Try 8) Describe the reflection below. across the y-axis
across the x-axis
across the line y=-3
across the line x=4
Answer: D

29. You Try
9) Describe the below reflection of point (3,-7). 𝟑,−𝟕 → (−𝟑,−𝟕) A. reflection across the x-axis B. reflection across the y-axis
Answer: B

30. Questions
What is the difference between a translation and a reflection?
How do you perform a reflection?
Will a reflection ALWAYS result in a congruent figure?
How would you complete this if it was a reflection over the x-axis? (9, 3) → ( , )
How would you complete this if it was a reflection over the y-axis? (9, 3) → ( , )
How do “describe” a reflection?
A translation is a slide. A reflection is a flip (mirror image)
You take a vertex and count how many units away it is from the line of reflection. Then, measure same distance on the other side of the line of reflection. Label using prime notation.
Yes. Translations, Reflections, and Rotations always produce congruent figures.
(9, -3)
(-9, 3)
You describe across which line the figure was reflected.

31. END OF POWERPOINT