Transformations Practice Worksheet featuring eight exercises on geometric transformations including reflections, rotations, and translations on a coordinate plane.

The worksheet involves performing various geometric transformations on given shapes and identifying the coordinates of the transformed images. Below, I will explain each transformation step-by-step.

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1. Reflection across \( x = 1 \)

#### Steps:
1. Identify the vertices of the original shape.
2. For each vertex \((x, y)\), reflect it across the line \( x = 1 \). The formula for reflection across \( x = 1 \) is:
\[
(x', y') = (2 - x, y)
\]
3. Plot the new vertices and connect them to form the reflected shape.

#### Example:
- Suppose a vertex is at \((3, 4)\).
- Reflecting across \( x = 1 \):
\[
x' = 2 - 3 = -1, \quad y' = 4
\]
So, the new vertex is \((-1, 4)\).

#### Final Answer:
Draw the reflected shape and label the coordinates of the new vertices.

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2. Reflection across \( y = x \)

#### Steps:
1. Identify the vertices of the original shape.
2. For each vertex \((x, y)\), reflect it across the line \( y = x \). The formula for reflection across \( y = x \) is:
\[
(x', y') = (y, x)
\]
3. Plot the new vertices and connect them to form the reflected shape.

#### Example:
- Suppose a vertex is at \((3, 4)\).
- Reflecting across \( y = x \):
\[
x' = 4, \quad y' = 3
\]
So, the new vertex is \((4, 3)\).

#### Final Answer:
Draw the reflected shape and label the coordinates of the new vertices.

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3. Reflection across the \( y \)-axis

#### Steps:
1. Identify the vertices of the original shape.
2. For each vertex \((x, y)\), reflect it across the \( y \)-axis. The formula for reflection across the \( y \)-axis is:
\[
(x', y') = (-x, y)
\]
3. Plot the new vertices and connect them to form the reflected shape.

#### Example:
- Suppose a vertex is at \((3, 4)\).
- Reflecting across the \( y \)-axis:
\[
x' = -3, \quad y' = 4
\]
So, the new vertex is \((-3, 4)\).

#### Final Answer:
Draw the reflected shape and label the coordinates of the new vertices.

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4. Rotation 180° about the origin

#### Steps:
1. Identify the vertices of the original shape.
2. For each vertex \((x, y)\), rotate it 180° about the origin. The formula for a 180° rotation is:
\[
(x', y') = (-x, -y)
\]
3. Plot the new vertices and connect them to form the rotated shape.

#### Example:
- Suppose a vertex is at \((3, 4)\).
- Rotating 180° about the origin:
\[
x' = -3, \quad y' = -4
\]
So, the new vertex is \((-3, -4)\).

#### Final Answer:
Draw the rotated shape and label the coordinates of the new vertices.

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5. Rotation 90° counterclockwise about the origin

#### Steps:
1. Identify the vertices of the original shape.
2. For each vertex \((x, y)\), rotate it 90° counterclockwise about the origin. The formula for a 90° counterclockwise rotation is:
\[
(x', y') = (-y, x)
\]
3. Plot the new vertices and connect them to form the rotated shape.

#### Example:
- Suppose a vertex is at \((3, 4)\).
- Rotating 90° counterclockwise:
\[
x' = -4, \quad y' = 3
\]
So, the new vertex is \((-4, 3)\).

#### Final Answer:
Draw the rotated shape and label the coordinates of the new vertices.

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6. Rotation 180° about the origin

#### Steps:
This is the same as problem 4. Use the formula:
\[
(x', y') = (-x, -y)
\]

#### Final Answer:
Draw the rotated shape and label the coordinates of the new vertices.

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7. Translation: \((x, y) \to (x - 2, y + 6)\)

#### Steps:
1. Identify the vertices of the original shape.
2. For each vertex \((x, y)\), apply the translation:
\[
(x', y') = (x - 2, y + 6)
\]
3. Plot the new vertices and connect them to form the translated shape.

#### Example:
- Suppose a vertex is at \((3, 4)\).
- Translating:
\[
x' = 3 - 2 = 1, \quad y' = 4 + 6 = 10
\]
So, the new vertex is \((1, 10)\).

#### Final Answer:
Draw the translated shape and label the coordinates of the new vertices.

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8. Translation: \((x, y) \to (x + 2, y + 5)\)

#### Steps:
1. Identify the vertices of the original shape.
2. For each vertex \((x, y)\), apply the translation:
\[
(x', y') = (x + 2, y + 5)
\]
3. Plot the new vertices and connect them to form the translated shape.

#### Example:
- Suppose a vertex is at \((3, 4)\).
- Translating:
\[
x' = 3 + 2 = 5, \quad y' = 4 + 5 = 9
\]
So, the new vertex is \((5, 9)\).

#### Final Answer:
Draw the translated shape and label the coordinates of the new vertices.

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9. Translation: \((x, y) \to (x + 5, y - 5)\)

#### Steps:
1. Identify the vertices of the original shape.
2. For each vertex \((x, y)\), apply the translation:
\[
(x', y') = (x + 5, y - 5)
\]
3. Plot the new vertices and connect them to form the translated shape.

#### Example:
- Suppose a vertex is at \((3, 4)\).
- Translating:
\[
x' = 3 + 5 = 8, \quad y' = 4 - 5 = -1
\]
So, the new vertex is \((8, -1)\).

#### Final Answer:
Draw the translated shape and label the coordinates of the new vertices.

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Summary of Final Answers:

For each problem, draw the transformed shape and label the coordinates of the new vertices. The specific coordinates will depend on the original vertices of the shapes provided in the worksheet.

\boxed{\text{Follow the steps above to complete each transformation.}}