1. on the Grid, Plot These Points to Create the Vertices of Quadrilateral ABCD

Translations

1. On the grid, plot these points to create the vertices of quadrilateral ABCD:

A(-3, -2), B(-2, 1), C(0,1), D(2, -2)

Connect the vertices to create quadrilateral ABCD. (Optional: Trace this quadrilateral onto a piece of patty paper.) Translate the quadrilateral 3 units to the right and 4 units up. Sketch the translated quadrilateral on the same grid.

2. Record the vertices of the original image and the vertices of the new image on the chart.

Point / x
(orig) / x'
(new) / y
(orig) / y' (new)
A
B
C
D

3. How are the values for x'of each vertex in the new image related to the values of x of each vertex in the original image?

a. Explain how these values show the transformation performed. ______

4. How are the values of y' of each vertex in the new image related to the values of y of each vertex in the original image?

a. Explain how these values show the transformation performed. ______

5. Describe an algebraic rule you use to obtain the image of any point (x,y) under this translation.

Algebraic Generalization
(x, y) → ( , )

6. Will this generalization apply for any quadrilateral that is translated 3 units to the right and 4 units

up? Justify your answer.

______

7. Now translate the quadrilateral 3 units to the left and 4 units up. Sketch the translated quadrilateral on the same grid.

Point / x
(orig) / x'
(new) / y
(orig) / y' (new)
A
B
C
D
Algebraic Generalization / x / y

8. How are the image(new) points related to the pre-image(original) points? Be sure to address both the x and y coordinates.

9. Describe an algebraic rule you use to obtain the image of any point (x,y) under this translation.

(x, y) → ( , )

10. Describe the transformation from (x, y) given the following generalizations:

a. (x+5, y)

b. (x, y + 5)

c. (x–3, y)

d. (x, y–3)

e. (x–3, y+5)

Reflections across the y-axis

1. On the grid, plot these points to create the vertices of quadrilateral ABCD in quadrant II:

A(-1, 1), B(-1, 2), C(-2, 3), D(-4, 1)

Connect the vertices to create quadrilateral ABCD. (Optional: Trace this quadrilateral onto a piece of patty paper.). Reflect the image so that the line of reflection is the y-axis. Sketch the reflected quadrilateral on the same grid.

2. Record the vertices of the original image and the vertices of the new image in the table.

Point / x
(orig) / x'
(new) / y (orig) / y' (new)
A
B
C
D
Algebraic Generalization / x / y

3. How are the values of x' for each vertex of the new image related to the values of x for each vertex of the original image?

4. How are the values of y' of each vertex in the new image related to the values of y of each vertex in the original image?

5. On the grid, plot these points to create the vertices of pentagon ABCDE:

A(0, -4), B(1, -2), C(3, -1), D(4, -2), E(4, -4)

Connect the vertices to create pentagon ABCDE. (Optional: Trace this pentagon onto a piece of patty paper.) Reflect the pentagon so that the line of reflection is the y-axis. Sketch the reflected pentagon on the same grid.

Point / x
(orig) / x'
(new) / y
(orig) / y' (new)
A
B
C
D
E
Algebraic Generalization / x / y

6. On the grid, plot these points to create the vertices of triangle ABC:

A(-2, 3), B(4, 6), C(2, 2)

Connect the vertices to create triangle ABC. (Optional: Trace this triangle onto a piece of patty paper.) Reflect the triangle so that the line of reflection is the y-axis. Sketch the reflected triangle on the same grid.

Point / x
(orig) / x'
(new) / y
(orig) / y' (new)
A
B
C
Algebraic Generalization / x / y

7. Investigate the patterns in the coordinate of the pre-image and image coordinates when reflected across the y-axis. Describe any patterns you see.

Reflections across the x-axis

1. On the grid, plot these points to create the vertices of quadrilateral ABCD in quadrant II:

A(-1, 1), B(-1, 2), C(-2, 3), D(-4, 1)

Connect the vertices to create quadrilateral ABCD. (Optional: Trace this quadrilateral onto a piece of patty paper.). Reflect the image so that the line of reflection is the x-axis. Sketch the reflected quadrilateral on the same grid.

2. Record the vertices of the original image and the vertices of the new image in the table.

Point / x
(orig) / x'
(new) / y (orig) / y' (new)
A
B
C
D
Algebraic Generalization / x / y

3. How are the values of x' for each vertex of the new image related to the values of x for each vertex of the original image?

______

a. Explain how these values show the transformation performed. ______

______

4. How are the values of y' of each vertex in the new image related to the values of y of each vertex in the original image?

______

a. Explain how these values show the transformation performed. ______

______

5. Complete the chart using the vertices of the original and the new image.

Original image / New image
x / y / x' / y'
Algebraic Generalization
(x, y) → ( , )

6. On the grid, plot these points to create the vertices of pentagon ABCDE:

A(0, -4), B(1, -2), C(3, -1), D(4, -2), E(4, -4)

Connect the vertices to create pentagon ABCDE. (Optional: Trace this pentagon onto a piece of patty paper.) Reflect the pentagon so that the line of reflection is the x-axis. Sketch the reflected pentagon on the same grid.

7. Record the vertices of the original image and the vertices of the new image on the chart.

Point / x
(orig) / x'
(new) / y
(orig) / y' (new)
A
B
C
D
E
Algebraic Generalization
(x, y) → ( , )

8. On the grid, plot these points to create the vertices of triangle ABC:

A(0, 0), B(3, 2), C(5, -1)

Connect the vertices to create triangle ABC. (Optional: Trace this triangle onto a piece of patty paper.) Reflect the triangle so that the line of reflection is the x-axis. Sketch the reflected triangle on the same grid.

9. Record the vertices of the original image and the vertices of the new image on the chart.

Point / x
(orig) / x'
(new) / y
(orig) / y' (new)
A
B
C
Algebraic Generalization
(x, y) → ( , )

Reflections across other lines

1. What did you notice about reflections across the x-axis and y-axis?

______

2. What did they have in common?

______

3. What was different?

______

4. In the previous activities, you explored reflecting an image so that the line of reflection was either the x-axis or y-axis. Another way to describe this is to say “an image is reflected across the axis.” Do you think it is possible to reflect an image across a line other than one of the axes? Explain your reasoning.

Reflection across x = 3

5. On the grid, plot these points to create the vertices of quadrilateral ABCD in quadrant II:

A(-1, 1), B(-1, 2), C(-2, 3), D(-4, 1)

Connect the vertices to create quadrilateral ABCD. (Optional: Trace this quadrilateral onto a piece of patty paper.).

x / y

Fill in the table of values for the line x = 3.
Using a colored pencil, graph the line x = 3.
Reflect the image so that the line of reflection is x = 3.
Sketch the reflected quadrilateral on the same grid.

6. Record the vertices of the original image and the vertices of the new image in the table.

Point / x
(orig) / x'
(new) / y (orig) / y' (new)
A
B
C
D
Algebraic Generalization / X / y

7. How are the values of x' for each vertex of the new image related to the values of x for each vertex of the original image?

______

8. How are the values of y' of each vertex in the new image related to the values of y of each vertex in the original image?

______

9. Complete the chart using the vertices of the original and the new image.

Original image / New image
x / y / x' / y'
Algebraic Generalization
(x, y) → ( , )

Reflection across y = -2

10. On the grid, plot these points to create the vertices of quadrilateral ABCD in quadrant II:

A(-1, 1), B(-1, 2), C(-2, 3), D(-4, 1)

Connect the vertices to create quadrilateral ABCD. (Optional: Trace this quadrilateral onto a piece of patty paper.).

x / y

Fill in the table of values for the line y = -2.
Using a colored pencil, graph the line y = -2.
Reflect the image so that the line of reflection is y = -2.
Sketch the reflected quadrilateral on the same grid.

11. Record the vertices of the original image and the vertices of the new image in the table.

Point / x
(orig) / x'
(new) / y (orig) / y' (new)
A
B
C
D
Algebraic Generalization / x / y

12. How are the values of x' for each vertex of the new image related to the values of x for each vertex of the original image?

______

13. How are the values of y' of each vertex in the new image related to the values of y of each vertex in the original image?

____________

______

14. Complete the chart using the vertices of the original and the new image.

Original image / New image
x / y / x' / y'
Algebraic Generalization
(x, y) → ( , )

16. On the grid, plot these points to create the vertices of quadrilateral in quadrant II:

A(-1, 1), B(-1, 2), C(-2, 3), D(-4, 1)

Connect the vertices to create quadrilateral ABCD. (Optional: Trace this quadrilateral onto a piece of patty paper.).

x / y

Fill in the table of values for the line y = x.
Using a colored pencil, graph the line y = x.
Reflect the image so that the line of reflection is y = x.
Sketch the reflected quadrilateral on the same grid.

17. Record the vertices of the original image and the vertices of the new image in the table.

Point / x
(orig) / x'
(new) / y (orig) / y' (new)
A
B
C
D
Algebraic Generalization / x / y

18. How are the values of x' for each vertex of the new image related to the coordinates of the original image?

______

19. How are the values of y' of each vertex in the new image related to coordinates of the original image?

20. Complete the chart using the vertices of the original and the new image.

Original image / New image
x / y / x' / y'
Algebraic Generalization
(x, y) → ( , )

Reflection across the line y = -x

21. On the grid, plot these points to create the vertices of quadrilateral ABCD in quadrant II:

A(-1, 1), B(-1, 2), C(-2, 3), D(-4, 1)

Connect the vertices to create quadrilateral ABCD. (Optional: Trace this quadrilateral onto a piece of patty paper.).

x / y

Fill in the table of values for the line y = -x.
Using a colored pencil, graph the line y = -x.
Reflect the image so that the line of reflection is y = -x.
Sketch the reflected quadrilateral on the same grid.

22. Record the vertices of the original image and the vertices of the new image in the table.

Point / x
(orig) / x'
(new) / y (orig) / y' (new)
A
B
C
D
Algebraic Generalization / x / y

23. How are the values of x' for each vertex of the new image related to the coordinates of the original image?

______

24. How are the values of y' of each vertex in the new image related to the coordinates of the original image?

______

25. Complete the chart using the vertices of the original and the new image.

Original image / New image
x / y / x' / y'
ALGEBRAIC GENERALIZATION
(x, y) → ( , )

What happened here?

Use your experience from the previous activities about reflections to answer the following questions.

List the types of reflection you have performed in the previous activities.

1. ______

2. ______

3. ______

4. ______

5. ______

6. ______

26. Describe a reflection process that could explain the following:

(2, -7) reflected across is (2, 7)
(2, -7) reflected across is (-7, 2)
(2, -7) reflected across is (-2, -7)
(2, -7) reflected across is (7, -2)

Translate and Translate Again

1. On the grid, plot these points to create the vertices of a triangle:

A (-2, 1), B (-2, 6), C (-8, 1)

2. On the same grid, translate the triangle 4 units to the right and 3 units up. Record the vertices of the 1st translation in the chart. Then translate the new image 2 units to the left and 5 units down. Record the vertices of the final images in the chart.

Original Image / 1st Translated Image / Final Translated Image
A / (-2, 1)
B / (-2, 6)
C / (-8, 1)
Algebraic Generalization / (x, y)

3. How is writing the algebraic generalization for the final translated image different than writing the algebraic generalization for the 1st translated image?

4. On the grid, plot these points to create the vertices of a triangle:

A (-2, 1), B (-2, 6), C (-8, 1)

5. On the same grid, translate the original triangle 2 units to the left and 5 units down. Record the vertices of the 1st translation in the chart. Then translate the new image 4 units to the right and 3 units up. Record the vertices of the final images in the chart.

Original Image / 1st Translated Image / Final Translated Image
A / (-2, 1)
B / (-2, 6)
C / (-8, 1)
Algebraic Generalization / (x, y)

a. Do the ordered pairs of the second translated image in #1 correspond to the ordered pairs

of the second translated image in #4?

b. What can you conclude about the order in which translations are completed? Be specific and use data from the tables.

Reflect and Reflect Again

1. On the grid, plot these points to create the vertices of a quadrilateral:

A (3, 2), B (9, 2), C (9, 6), D (3, 6)

2. On the same grid, reflect the quadrilateral across the y-axis and then across the x-axis. Record the vertices of the reflected image in the chart.

Original Image / Reflected across the y-axis / Reflected across the x-axis
A / (3, 2)
B / (9, 2)
C / (9, 6)
D / (3, 6)
Algebraic Generalization / (x, y)

3. On the grid, plot these points to create the vertices of a quadrilateral:

A (3, 2), B (9, 2), C (9, 6), D (3, 6)

4. On the same grid, reflect the original quadrilateral across the x-axis and then the y-axis. Record the vertices of the reflected image in the chart.

Original Image / Reflected across the x-axis / Reflected across the y-axis
A / (3, 2)
B / (9, 2)
C / (9, 6)
D / (3, 6)
Algebraic Generalization / (x, y)

a. Do the ordered pairs of the reflected image in #1 correspond to the ordered pairs of the

reflected image in #3?

b. What can you conclude about the order in which reflections are completed? Be specific and use data from the tables.

Sequences of Translations and Reflections

1. On the grid, plot these points to create the vertices of quadrilateral ABCD:

A (1, 2), B (7, 2), C (5, 6), D (1, 6)

2. On the same grid, translate the quadrilateral 3 units to the right and record the vertices. Then reflect the new image across the x-axis and record the vertices.

Original Image / Translated Image / Reflected Image
A / (1, 2)
B / (7, 2)
C / (5, 6)
D / (1, 6)
Algebraic Generalization / (x, y)

3. On the grid, plot these points to create the vertices of quadrilateral ABCD:

A (1, 2), B (7, 2), C (5, 6), D (1, 6)

4. On the same grid, reflect the original quadrilateral across the x-axis and record the vertices. Then translate the new image 3 units to the right and record the vertices.

Original Image / Reflected Image / Translated Image
A / (1, 2)
B / (7, 2)
C / (5, 6)
D / (1, 6)
Algebraic Generalization / (x, y)

a. Do the ordered pairs of the final image in #1 correspond to the ordered pairs of the final

image in #3?

b. What can you conclude about the order in which translations and reflections are completed? Be specific and use data from the tables.

5. On the grid, plot these points to create the vertices of quadrilateral ABCD:

A (1, 2), B (7, 2), C (5, 6), D (1, 6)

6. On the same grid, translate the quadrilateral 1 unit to the right and record the vertices. Then reflect the new image across the y-axis and record the vertices.

Original Image / Translated Image / Reflected Image
A / (1, 2)
B / (7, 2)
C / (5, 6)
D / (1, 6)
Algebraic Generalization / (x, y)

7. On the grid, plot these points to create the vertices of quadrilateral ABCD:

A (1, 2), B (7, 2), C (5, 6), D (1, 6)

8. On the same grid, reflect the quadrilateral across the y-axis and record the vertices. Then translate the new image 1 unit to the right and record the vertices.

Original Image / Reflected Image / Translated Image
A / (1, 2)
B / (7, 2)
C / (5, 6)
D / (1, 6)
Algebraic Generalization / (x, y)

a. Do the ordered pairs of the final image in #5 correspond to the ordered pairs of the final

image in #7?

b. What can you conclude about the order in which translations and reflections are completed? Be specific and use data from the tables.

Sequence of Translations and Reflections – Optional Activities

9. On the grid, plot these points to create the vertices of quadrilateral ABCD:

A (1, 2), B (7, 2), C (5, 6), D (1, 6)

10. On the same grid, translate the quadrilateral 3 units up and record the vertices. Then reflect the new image across the x-axis and record the vertices.

Original Image / Translated Image / Reflected Image
A / (1, 2)
B / (7, 2)
C / (5, 6)
D / (1, 6)
Algebraic Generalization / (x, y)

11. On the grid, plot these points to create the vertices of quadrilateral ABCD:

A (1, 2), B (7, 2), C (5, 6), D (1, 6)

12. On the same grid, reflect the original quadrilateral across the x-axis and record the vertices. Then translate the new image 3 units up and record the vertices.

Original Image / Reflected Image / Translated Image
A / (1, 2)
B / (7, 2)
C / (5, 6)
D / (1, 6)
Algebraic Generalization / (x, y)

a. Do the ordered pairs of the final image in #9 correspond to the ordered pairs of the final image in #11?

b. What can you conclude about the order in which translations and reflections are completed?

Be specific and use data from the tables.

13. On the grid, plot these points to create the vertices of quadrilateral ABCD:

A (1, 2), B (7, 2), C (5, 6), D (1, 6)

14. On the same grid, translate the quadrilateral 3 units up and record the vertices. Then reflect the new image across the y-axis and record the vertices.

Original Image / Translated Image / Reflected Image
A / (1, 2)
B / (7, 2)
C / (5, 6)
D / (1, 6)
Algebraic Generalization / (x, y)

15. On the grid, plot these points to create the vertices of quadrilateral ABCD:

A (1, 2), B (7, 2), C (5, 6), D (1, 6)

16. On the same grid, reflect the quadrilateral across the y-axis and record the vertices. Then translate the new image 3 units up and record the vertices.

Original Image / Reflected Image / Translated Image
A / (1, 2)
B / (7, 2)
C / (5, 6)
D / (1, 6)
Algebraic Generalization / (x, y)

a. Do the ordered pairs of the final image in #13 correspond to the ordered pairs of the final

image in #15?

b. What can you conclude about the order in which translations and reflections are completed?

Be specific and use data from the tables.

Reflect and Apply

1. Using these coordinates as the vertices for an image, answer the following questions.

Original Image / New Image
x / y / x' / y'
1 / 1 / -5 / 1
2 / 3 / -4 / 3
4 / 3 / -2 / 3
5 / 1 / -1 / 1

a. Is the image a reflection or a translation?

b. How do you know?

c. Plot an original image and a new image so that the new image could be either a reflection or a translation of the original image.

e. Describe the reflection and translation that you used. ______

______

2. Given a polygon with the following vertices A(-3, 1), B(-2, 4), C(0, 5), D(1, 1) write a translation rule that would produce a new image that is entirely in Quadrant IV. Use the grid below to confirm your solution. (x, y)

______

3. Given a polygon with the following vertices A(-6, -4), B(-6, -2), C(-5, -2), D(-5, -4) write a translation that would produce a new image that has a vertical line of reflection. Use the grid below to confirm your solution. (x, y)

Turn Me 90 Rotation

1. On the grid, plot these points to create the vertices of triangle ABC.

A (-1, 1), B (-8, 1), C (-3, 5)

2. On the same grid, rotate the triangle about the origin 90° counterclockwise. Record the vertices of the rotated image in the table.