Name: ______________________________________________

Unit 3: TransformationsThe Basics

Recommended Calendar

SectionRecommended

Completion Date

4.7 10/19

9.1 10/23

9.2 10/26

9.3 10/29

This packet is due in its entirety on Monday, 10/30.

The purpose of this packet is to familiarize you with the basic transformations (reflections, translations, and rotations.) You should complete this entire packet in order—it is designed so that one idea leads to the next, and so on. If you are confused, refer to Chapter 9 of your textbook. It is your responsibility to become familiar with these ideas, and to be able to do these problems with ease. On Monday, 10/30, this packet will be checked for completion, and the page of Exit Cards at the end of the packet will be collected for a grade.

Section 4.7: Congruence Transformations(Use pages 296 – 298 to complete this section.)

Learn:

Define the following terms.

Transformation: ________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________________________

Pre-image: ______________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________________________

Image: ___________________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________________________

Congruence Transformation: __________________________________________________________________________________________________

____________________________________________________________________________________________________________________________________

Isometry: ________________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________________________

Reflection Translation RotationDefinition

Example

Important Notation:

A A’

A is a point in the ________________ A’ is a point in the ________________

A’ is read out loud as “_______________________” A A’ is read out loud as “A is _______________________________ A’. ”

Section 9.1: Reflections(Refer to pages 623 – 626.)

Learn:

Recall: A reflection is a transformation over a line called the line of reflection. Each point of the pre-image and its image are the same distance from the line of reflection.

1. Graph the following lines. One has been done for you.

y = 0 x = 0 y = x

y = -x y = 4 x = -3

2. Triangle ABC has vertices A(3, –3), B(–5, –1), and C(3, 0). Plot ∆ABC on each graph above. Then, reflect ∆ABC in the line to obtain the image, ∆A’B’C’, by reflecting each vertex. If necessary, use a separate piece of graph paper, and fold the paper along the line of reflection to see the image of each vertex. One has been done for you.

3. Write the coordinates of the image for each reflection in the chart below.

Line of Reflection Pre-Image Image Line of Reflection Pre-Image Image

y = 0A(3, –3)B(–5, –1)C(3, 0)

A’(3, 3)B’(–5, 1)C ’(3, 0)

y = –xA(3, –3)B(–5, –1)C(3, 0)

x = 0A(3, –3)B(–5, –1)C(3, 0)

y = 4A(3, –3)B(–5, –1)C(3, 0)

y = xA(3, –3)B(–5, –1)C(3, 0)

x = –3A(3, –3)B(–5, –1)C(3, 0)

B’

A’

B

C and C’••

•

••

A

4. Observations:a. When is a pre-image point the same as its image?

b. There are special patterns that you can use for reflecting in the axes, and the lines y = x and y = –x. Fill in both columns of the chart.

Line of Reflection Mapping

y = 0 (___-axis) (x, y) ( , )

x = 0 (___-axis) (x, y) ( , )

y = x (x, y) ( , )

y = –x (x, y) ( , )

c. Suppose A A’ by a reflection. Consider the segment AA ' . How is the line of reflection related to AA ' ? Fill

in the blanks: The line of reflection is the __________________________ __________________________ of AA ' .

Practice:Answer the questions. Use graph paper, if necessary.

1. What is the image of (–2, –4) under a reflection in the line x = 3?

2. What is the image of (1, 3) under a reflection in the x-axis?

3. What is the image of (–2, 2) under a reflection in the line y = –x?

4. If A --> A' by a reflection in the y-axis, find A if A' is (7, -2).

5. What is the pre-image of (–6, 1) under a reflection in the line y = x?

6. Use the diagram to the right to name the image of ∆1 after the reflection in the given line.

a. x-axis b. y-axis c. y = x d. y = –x e. y-axis followed by x-axis

7. Multiple Choice: If R(5, –3) R’(5, 3) by a reflection, what is the line of reflection?

A. x = 0 B. x = 5 C. x-axis D. y = x E. none of these

8. Multiple Choice: If R(5, –3) R’(–3, 5) by a reflection, what is the line of reflection?

A. x = 0 B. x = 5 C. x-axis D. y = x E. none of these

9. Multiple Choice: If R(5, –3) R’(1, 1) by a reflection, what is the line of reflection?

A. x = 0 B. x = 5 C. x-axis D. y = x E. none of these

10. As you noticed in #9, it is not always easy to identify the line of reflection when given a pre-image and its image. Follow the steps below to write an equation for the line of reflection.

Step 1: Graph R(5, –3) and R’(1, 1). Draw a segment connecting the two points.

Step 2: Find the midpoint of RR 'and name it Q.

Step 3: Find the slope of RR ' . Then write the slope of a line perpendicular to RR ' .

Step 4: Write an equation of the line that is perpendicular to RR ' and passes through Q.

11. Use the steps above to find an equation of the line of reflection that maps R(5, 4) onto R’(1, –2).

Assess:Complete the 9.1 Exit Card at the end of the packet. This will be graded.

Section 9.2: Translations(Refer to pages 632 – 634.)

Learn:Computer and video game programmers use transformations to create patterns and animations. For instance, the car to the right was translated to obtain the second car.

Recall: A translation is a transformation that moves all points on the original figure the same distance in the same direction.

1. In the figure, what point is the pre-image? What point is the image?

2. Describe, in words, how to slide the pre-image to the image.

3. Let’s write that algebraically. If the x-coordinate of the pre-image is x, then the x-coordinate of the image is x + _____. Similarly, if the y-coordinate of the pre-image is y, then the y-coordinate of the image is y + _____. In other words, (x, y) (_________, _________). This is called “coordinate notation.”

*Note: Your textbook teaches about “translation vectors”, which you will NOT be expected to learn. You will only be expected to know coordinate notation.*

Describe the translation using coordinate notation. Use the example to help you.Example: A figure is moved 5 units to the right and 4 units down.Answer: (x, y) (x + 5, y – 4)

5. A figure is moved 7 units to the left and 2 units up.

6. Quadrilateral JKLM has vertices J(–5, 3), K(–4, 5), L(–3, 3), and M(–4, 1). Find the coordinates of the vertices of the image after the translation (x, y) → (x + 6, y – 2).

Practice:Answer the questions.

1. Use coordinate notation to describe the translation on the graph.

C ’

C

2. Use coordinate notation to describe the translation of PQ to P 'Q' if P(-3, 3), Q(5, 4), P'(7, -1), and Q'(15, 0).

3. The vertices of quadrilateral ABCD are A(3, 4), B(2, 4), C(3, 2), and D(4, 1). After a translation, the coordinates of A’ are (5, 1). Describe the translation using coordinate notation. Then find the coordinates of B’, C’, and D’.

For #4-7, consider the translation that is defined by (x, y) → (x + 5, y – 2).

4. What is the image of (2, 4)? 5. What is the image of (–5, 7)?

6. What is the pre-image of (2, –3)? 7. What is the pre-image of (–6, 8)?

8. A translation of AB is described by (x, y) → (x – 2, y + 5). Given the ordered pairs below, find the values of k, m, n, and p.

A(7, k) A'(4m – 1, 0) B(n + 3, 1) B'(–6, 2p)

9. Draw the triangle with the given vertices: F(–2, 3), G(3, 3), H(3, –1). Then find the coordinates of the vertices of the image after a translation described by (x, y) → (x – 3, y + 3). Draw the image.

Assess:Complete the 9.2 Exit Card at the end of the packet. This will be graded.

Section 9.3: Rotations(Refer to pages 640 – 642.)

Learn:

Recall: A rotation is a transformation around a fixed point called the center of rotation, through a specific angle, and in a specific direction. Each point of the original figure and its image are the same distance away from the center.

1. Use the graph to the right. P P’ by a rotation. Identify the center of rotation, the angle of rotation, and the direction of rotation (clockwise or counterclockwise.)

Center: _________________ Angle: _________________ Direction: _________________

2. For each graph, you will rotate the given pre-image point 90°, 180°, and 270° clockwise about the origin. Plot those 3 images in 3 different colors all on the same graph. Use red for 90°, green for 180°, and blue for 270°. Then, write the coordinates of the images in the chart. To help you do the rotation correctly, hold the paper in your hands, turn it by the given angle, and look to see where the point ends up. (Hint: The point will end up in a different quadrant.)

a. b.

c.

90° cw 180° cw 270° cw

A(0, 4)

B(2, 5)

C(-7, –3)

3. Observations:a. There are special patterns that you can use for rotating clockwise about the origin by 90°, 180°, and 270°. Fill

in the chart.Rotation Mapping

90° cw (x, y) ( , )

180° cw (x, y) ( , )

270° cw (x, y) ( , )

AB

C

b. Some rotations yield the same image. Fill in the blanks:Rotating 90° clockwise about the origin is the same as rotating ___________ counterclockwise about the origin.

Rotating 180° clockwise about the origin is the same as rotating ___________ counterclockwise about the origin.

Rotating 270° clockwise about the origin is the same as rotating ___________ counterclockwise about the origin.

Practice:Answer the questions. Use graph paper, if necessary.

1. Suppose a point lies in the given quadrant. Determine in which quadrant its image lies, after a rotation by the given angle about the origin.

a. quadrant II, 270° cw: ___________________ b. quadrant IV, 180° cw: ___________________

c. quadrant III, 270° ccw: ___________________ d. quadrant II, 90° ccw: ___________________

2. What is the image of (2, 4) under a 90° ccw rotation about the origin?

3. What is the image of (–5, 7) under a 90° cw rotation about the origin?

4. What is the pre-image of (2, –3) under a 270° ccw rotation about the origin?

5. What is the pre-image of (–6, 8) under a 180° rotation about the origin?

6. Circle all that apply: P(–2, 5) P’(2, –5) under a rotation about the origin. Which of the following is the angle of rotation?

A. 90° cw B. 270° cw C. 180° ccw D. 270° ccw E. 90° ccw

7. Circle all that apply: P(–8, –4) P’(4, –8) under a rotation about the origin. Which of the following is the angle of rotation?

A. 90° cw B. 270° cw C. 180° ccw D. 270° ccw E. 90° ccw

8. Quadrilateral LOVE Quadrilateral L’O’V’E’ under a rotation about the origin so that L(1, –3) L’(–3, –1). Find the coordinates of O’, V’, and E’ if O(-2, 4), V(0, 1) and E(5, 3).

Assess:Complete the 9.3 Exit Card at the end of the packet. This will be graded.

Sections 9.1-9.3: Review

Practice:Answer the questions.

Perform the following transformations with the point (–2, 7).1. translation described by (x, y) → (x + 1, y – 1)

2. translation described by (x, y) → (x, y – 8)

3. reflection in the x-axis

4. reflection in the line x = 1

5. reflection in the line y = x

6. rotation 180 clockwise about (0, 0)

7. rotation 90 counterclockwise about (0, 0)

8. For each graph, draw ΔFGH with vertices F(2, 3), G(3, 3), and H(2, 1). Then draw and label the image after the specified transformation.

a) Translation: (x, y) → (x + 2, y) b) Reflection: in the line y = 1 c) Rotation: 90° cw about (0, 0)

9. Segment AB is reflected in the line y = x. Given the ordered pairs below of the pre-image and image points, find the values of c, d, e, and f.

A(c + 1, –8) A'(d, 2) B(5, 2e) B'(–1, 3f – 7)

9.1 Exit Card [6 pts]

Complete the table by determining either the pre-image or image under a reflection in the given line.

Pre-Image Line of Reflection Image

1. (–9, 1) y-axis

2. (3, 5) x = –1

3. (–2, –4) y = x

4. x- axis (3, –8)

5. (0, –1) y = –1

6. y = 2 (6, –4)

9.2 Exit Card [5 pts]

1. Triangle ABC has vertices A(4, 0), B(0, 4), and C(0, 0). Find the coordinates of the vertices of the image after the translation described by (x, y) → (x – 3, y – 2).

2. Use coordinate notation to describe the translation that maps KLMN onto K’L’M’N’.

9.3 Exit Card [7 pts]

Use the diagram to the right to identify the image after the given rotation.1. A clockwise rotation of 45° about P maps A onto…

2. A clockwise rotation of 90° about P maps C onto…

3. A clockwise rotation of 180° about P maps onto…

4. A counterclockwise rotation of 90° about P maps H onto…

5. A counterclockwise rotation of 45° about P maps onto…

18

For #6-7, identify the image after the given rotation.

6. Z(3, –2), 270° clockwise about the origin 7. W(–1, –4), 270° counterclockwise about the origin