Geometry - Unit 3 Day 4

Unit 3: Symmetry & Transformations

Day 4: Sequences of Transformations

Essential Question: What happens when we combine multiple transformations, and does the order matter?

Today's Goals

🎯 I will be able to perform a sequence of two or more transformations on a figure.

🎯 I will be able to determine if the order of transformations affects the final image.

Key Vocabulary

  • Sequence of Transformations: Applying more than one transformation to a figure. Also called a composition of transformations.
  • Intermediate Image: The result after the first transformation in a sequence. We often label it with one prime mark (A').
  • Final Image: The result after all transformations are completed. Labeled with two prime marks (A'').

I Do: Translate, then Reflect

Let's perform two transformations on triangle ABC.
1st: Translate (x, y) → (x + 6, y - 1).
2nd: Reflect across the x-axis.

A(-8,3) A'(-2,2) A''(-2,-2)

Preimage:

A(-8, 3)

1. Translate (x+6, y-1):

A'(-2, 2)

2. Reflect over x-axis:

A''(-2, -2)

We Do: Does Order Matter?

Let's see. We'll use point P(2, 4). First, we'll reflect then rotate. Then we'll rotate then reflect. Are the final images the same?

Path 1: Reflect, then Rotate

P(2, 4)

↓ Reflect over y-axis

P'(-2, 4)

↓ Rotate 90° CCW

P''(-4, -2)

Path 2: Rotate, then Reflect

P(2, 4)

↓ Rotate 90° CCW

P'(-4, 2)

↓ Reflect over y-axis

P''(4, 2)

The final images are different! So, the order of transformations does matter.

You Do: Describe the Sequence

How can we map the blue Preimage (Triangle A) to the green Final Image (Triangle A'')? There might be more than one correct answer!

A A''

Example Answer 1:

1.

2.

Example Answer 2:

1.

2.

My Sequence:

Special Sequence: Glide Reflection

A glide reflection is a special sequence: a translation (slide) followed by a reflection over a line that is parallel to the direction of the slide.

Slide Flip

Think of footprints in the sand!

Independent Practice

Level: Green

1. Take point P(1, 2). First, reflect it over the x-axis. Then, translate it left 3 units. What are the final coordinates?

Level: Yellow

2. Triangle ABC has vertices A(1,1), B(4,1), C(1,3). First, rotate it 90° CCW. Then, dilate it by a scale factor of 2. What are the coordinates of A'', B'', and C''?

Level: Red

3. The point Z(-2, 5) is transformed to Z''(5, 2). Describe a sequence of two transformations that could have mapped Z to Z''.

✅ Exit Ticket

Answer the following questions to show what you've learned today.

1. What are the coordinates of (3, -4) after a translation (x, y) → (x+2, y-5) followed by a reflection over the y-axis?

2. True or False: A 180° rotation followed by a translation right 2 units is the same as a translation right 2 units followed by a 180° rotation. Explain or show why.