Page 2: Advanced Reflection Examples

This page expands on reflection concepts with multiple examples across different lines of reflection.

Example: When reflecting across x=4:

• Point J(1,-1) becomes J'(7,-1)
• Point K(2,3) becomes K'(6,3)
• Point L(3,-2) becomes L'(5,-2)

Highlight: Key rules for different reflection lines:

• For x=a line: y-coordinate stays the same
• For y=b line: x-coordinate stays the same
• For y=x line: x and y coordinates switch places

Page 3: Complex Reflections

This page covers more advanced reflection examples, particularly focusing on diagonal reflections and their rules.

Example: For reflection across y=-x:

• Point S(-1,-6) becomes S'(6,1)
• Point T(0,-3) becomes T'(3,0)
• Point U(3,-4) becomes U'(4,-3)

Highlight: When reflecting across y=-x, coordinates switch places and change signs.

Page 4: Introduction to Translations

This page introduces translations as another type of rigid motion.

Definition: A translation is a transformation that slides a figure vertically and/or horizontally without changing its size or shape.

Vocabulary: Translation notation: (x,y) → $x+h, y+k$ or <h,k>

• h represents horizontal shift
• k represents vertical shift

Example: Translation T<5,7>:

• Point D(-3,3) becomes D'(4,7)
• Point E(0,2) becomes E'(7,6)
• Point F(-1,-3) becomes F'(6,7)

Page 5: Composition of Transformations

This page explores how multiple transformations can be combined.

Definition: A composition of rigid motions involves applying two or more transformations sequentially.

Example: Reflecting across x-axis followed by translation T<9,-1>:

• Point X(-3,1) becomes X'(-3,-1) then X"(6,-8)
• Point Y(-2,1) becomes Y'(-2,-1) then Y"(7,-2)

Highlight: When combining transformations, the order of operations matters and affects the final result.

Page 6: Introduction to Rotations

This page introduces rotation as a rigid motion, focusing on rotations around the origin.

Definition: Rotation rules around the origin:

• 90° CCW: (x,y) → $-y,x$
• 180°: (x,y) → $-x,-y$
• 270° CCW: (x,y) → $y,-x$

Example: 90° rotation about the origin:

• Point A(3,5) becomes A'(-5,3)
• Point B(1,7) becomes B'(-7,1)

Page 7: Advanced Rotations

This page explores more complex rotation examples and combinations with other transformations.

Example: 270° rotation about the origin:

• Point A(2,7) becomes A'(-7,2)
• Point B(6,5) becomes B'(-5,6)

Highlight: Multiple transformations can be combined, such as translation followed by rotation or reflection followed by rotation.

Page 8: Rotations Around Points

This page explains how to perform rotations around points other than the origin.

Definition: Four-step process for rotating around a point:

1. Write original points
2. Subtract point of rotation
3. Apply rotation rules
4. Add back point of rotation

Example: 180° rotation around point (1,1):

• Original point F(1,2)
• Subtract (1,1): (0,1)
• Apply rotation: (0,-1)
• Add (1,1): F'(1,0)

Page 9: Final Rotation Examples

This page provides additional examples of rotations around specific points.

Example: 90° CCW rotation around point (0,1):

• Point G(-3,-2) becomes G'(3,-2)
• Point F(-3,-5) becomes F'(6,-2)
• Point H(0,-1) becomes H'(7,1)

Highlight: The process demonstrates how complex rotations can be broken down into manageable steps using coordinate geometry.

Page 1: Introduction to Reflections

This page introduces the fundamental concepts of rigid motion in geometry and reflections. A reflection involves flipping a figure across a line while maintaining equal distances from the line of reflection.

Definition: A rigid motion is a transformation that preserves both length and angle measurements.

Vocabulary: A reflection is a flip over a line called the line of reflection, where each point and its image are equidistant from the line.

Example: When reflecting point A(-4,2) across the x-axis, its image becomes A'(-4,-2), demonstrating how the y-coordinate changes sign while the x-coordinate remains the same.

Highlight: Common lines of reflection include:

• x-axis and y-axis
• Vertical lines $x=a$ and horizontal lines $y=b$
• Diagonal lines $y=x or y=-x$