Unit 3: Symmetry & Transformations

Day 2: Reflections & Rotations

Essential Question: How do reflections and rotations change a figure's position and orientation on a coordinate plane?

Today's Goals

🎯 I will be able to reflect figures across the x-axis, y-axis, and the lines y = x and y = -x.

🎯 I will be able to rotate figures 90°, 180°, and 270° counterclockwise around the origin.

Key Vocabulary

Preimage: The original figure before a transformation.
Image: The new figure after a transformation. (We use prime notation, like A')
Reflection: A transformation that flips a figure across a line, creating a mirror image.
Rotation: A transformation that turns a figure around a fixed point.

I Do: Reflecting across the y-axis

A reflection flips a figure. When we reflect across the y-axis, every point moves horizontally to the opposite side. The distance from the y-axis remains the same.

Preimage A: (2, 1)

Image A': (-2, 1)

Notice: The x-coordinate becomes its opposite, but the y-coordinate stays the same!

Rule: (x, y) → (-x, y)

My Notes on Reflections:

Reflecting over the y-axis changes the sign of the x-coordinate.

We Do: Reflecting across the x-axis

Let's reflect this triangle across the x-axis together. What do you predict will happen to the coordinates?

Preimage P: (-6, 2) → P':(-6, -2)

Preimage Q: (-2, 8) → Q':(-2, -8)

Preimage R: (-8, 5) → R':(-8, -5)

Rule: (x, y) → (x, -y)

You Do: Reflecting across y = x

Now it's your turn! Reflect this trapezoid across the diagonal line y = x. Write down the new coordinates.

A(2,5) → A':(, )

B(6,5) → B':(, )

C(8,2) → C':(, )

D(0,2) → D':(, )

Rule: (x, y) → (y, x)

I Do: Rotating 90° CCW

A rotation turns a figure around a center point (today, the origin). A 90° counterclockwise (CCW) rotation is one turn to the left.

Preimage A: (2, 3)

Image A': (-3, 2)

Notice: The coordinates swap, and the new x-coordinate becomes its opposite.

Rule: (x, y) → (-y, x)

We Do / You Do: 180° & 270° Rotations

Let's discover the rules for 180° and 270° CCW rotations together. Use the point P(5, 2).

180° Rotation (2 turns):

P(5, 2) → P'(-5, -2)

Rule: (x, y) → (-x, -y)

270° Rotation (3 turns):

P(5, 2) → P''(2, -5)

Rule: (x, y) → (y, -x)

Transformation Rules Summary

This is a great page for your notes! These are the most common rules for reflections and rotations about the origin.

Reflections

Over x-axis: (x, y) → (x, -y)

Over y-axis: (x, y) → (-x, y)

Over y = x: (x, y) → (y, x)

Over y = -x: (x, y) → (-y, -x)

Rotations (CCW)

90°: (x, y) → (-y, x)

180°: (x, y) → (-x, -y)

270°: (x, y) → (y, -x)

360°: (x, y) → (x, y)

Important Note:

A clockwise rotation of 90° is the same as a counterclockwise rotation of 270 degrees.

Independent Practice

Level: Green

1. Triangle ABC has vertices A(1,2), B(5,5), and C(5,2). Find the coordinates of the image after a reflection over the x-axis.

2. Point Z(4, -3) is rotated 180° about the origin. What are the coordinates of Z'?

Level: Yellow

3. A square has vertices at (1,1), (4,1), (4,4), and (1,4). Find the coordinates of the image after a 90° CCW rotation about the origin.

4. A line segment has endpoints J(2, -5) and K(6, -1). It is reflected over the line y = x. What are the endpoints of the image J'K'?

Level: Red

5. Point P(-3, 7) is transformed to P'(-7, -3). Describe the reflection that maps P to P'.

6. A figure is in Quadrant II. After a rotation of 270° CCW about the origin, which quadrant will the image be in? Explain your reasoning.

✅ Exit Ticket

Answer the following questions to show what you've learned about reflections and rotations.

1. What are the coordinates of point M(5, -1) after a reflection over the y-axis?

2. What are the coordinates of point T(-4, -6) after a 90° counterclockwise rotation about the origin?

3. The point G(2, 3) is transformed to G'(-2, -3). Describe the rotation that occurred.

Geometry - Unit 3 Day 2