Rotation of Shapes worksheet with six exercises involving different types of rotations (90° counterclockwise, 90° clockwise, 180°) on a coordinate grid.

To solve the problem of graphing the image of each shape after rotating it about its origin, we need to understand how rotations affect the coordinates of points. Here's a step-by-step explanation for each rotation type:

1. 90° Counterclockwise Rotation

- Rule: When a point \((x, y)\) is rotated 90° counterclockwise about the origin, its new coordinates become \((-y, x)\).
- Explanation: The \(x\)-coordinate becomes the negative of the original \(y\)-coordinate, and the \(y\)-coordinate becomes the original \(x\)-coordinate.

2. 90° Clockwise Rotation

- Rule: When a point \((x, y)\) is rotated 90° clockwise about the origin, its new coordinates become \((y, -x)\).
- Explanation: The \(x\)-coordinate becomes the original \(y\)-coordinate, and the \(y\)-coordinate becomes the negative of the original \(x\)-coordinate.

3. 180° Rotation

- Rule: When a point \((x, y)\) is rotated 180° about the origin, its new coordinates become \((-x, -y)\).
- Explanation: Both the \(x\)-coordinate and the \(y\)-coordinate become their negatives.

4. Repeating the Rules

The same rules apply for all shapes. We will apply these transformations to each vertex of the given shapes and then connect the new vertices to form the rotated image.

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Solutions for Each Shape

#### Shape 1: 90° Counterclockwise Rotation
1. Identify the vertices of the shape.
2. Apply the rule \((x, y) \rightarrow (-y, x)\) to each vertex.
3. Plot the new vertices and connect them to form the rotated shape.

#### Shape 2: 90° Clockwise Rotation
1. Identify the vertices of the shape.
2. Apply the rule \((x, y) \rightarrow (y, -x)\) to each vertex.
3. Plot the new vertices and connect them to form the rotated shape.

#### Shape 3: 180° Rotation
1. Identify the vertices of the shape.
2. Apply the rule \((x, y) \rightarrow (-x, -y)\) to each vertex.
3. Plot the new vertices and connect them to form the rotated shape.

#### Shape 4: 90° Counterclockwise Rotation
1. Identify the vertices of the shape.
2. Apply the rule \((x, y) \rightarrow (-y, x)\) to each vertex.
3. Plot the new vertices and connect them to form the rotated shape.

#### Shape 5: 90° Clockwise Rotation
1. Identify the vertices of the shape.
2. Apply the rule \((x, y) \rightarrow (y, -x)\) to each vertex.
3. Plot the new vertices and connect them to form the rotated shape.

#### Shape 6: 180° Rotation
1. Identify the vertices of the shape.
2. Apply the rule \((x, y) \rightarrow (-x, -y)\) to each vertex.
3. Plot the new vertices and connect them to form the rotated shape.

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Final Answer

The final answer involves graphing the rotated shapes based on the rules above. Since I cannot draw the graphs here, the solution process is as follows:

1. For each shape, identify the vertices.
2. Apply the appropriate rotation rule to each vertex.
3. Plot the new vertices on the grid.
4. Connect the new vertices to form the rotated shape.

The exact coordinates and graphs would depend on the specific vertices of each shape in the image. If you provide the coordinates of the vertices, I can give more detailed steps or results.

Final Answer (Conceptual):
\[
\boxed{\text{Apply the rotation rules to each vertex and plot the new positions.}}
\]