12-All Transformations - Kuta Software - Free Printable

Problem Analysis:

The task involves performing various geometric transformations on given figures and then describing the rules for these transformations. Let's solve each part step by step.

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Part 1: Rotation 90° Counterclockwise About the Origin

#### Given Figure:
The figure is a triangle with vertices \( A \), \( J \), and \( L \).

#### Transformation:
Rotate the triangle 90° counterclockwise about the origin.

#### Solution:
To rotate a point \((x, y)\) 90° counterclockwise about the origin, the new coordinates become \((-y, x)\).

1. Identify the coordinates of the vertices:
- \( A = (x_1, y_1) \)
- \( J = (x_2, y_2) \)
- \( L = (x_3, y_3) \)

2. Apply the rotation formula \((-y, x)\) to each vertex:
- New coordinates of \( A \): \((-y_1, x_1)\)
- New coordinates of \( J \): \((-y_2, x_2)\)
- New coordinates of \( L \): \((-y_3, x_3)\)

3. Plot the new points and connect them to form the rotated triangle.

#### Final Answer:
The image of the triangle after a 90° counterclockwise rotation about the origin is plotted using the transformed coordinates.

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Part 2: Translation: 4 Units Right and 1 Unit Down

#### Given Figure:
The figure is a triangle with vertices \( P \), \( Q \), and \( R \).

#### Transformation:
Translate the triangle 4 units right and 1 unit down.

#### Solution:
To translate a point \((x, y)\) 4 units right and 1 unit down, the new coordinates become \((x + 4, y - 1)\).

1. Identify the coordinates of the vertices:
- \( P = (x_1, y_1) \)
- \( Q = (x_2, y_2) \)
- \( R = (x_3, y_3) \)

2. Apply the translation formula \((x + 4, y - 1)\) to each vertex:
- New coordinates of \( P \): \((x_1 + 4, y_1 - 1)\)
- New coordinates of \( Q \): \((x_2 + 4, y_2 - 1)\)
- New coordinates of \( R \): \((x_3 + 4, y_3 - 1)\)

3. Plot the new points and connect them to form the translated triangle.

#### Final Answer:
The image of the triangle after translating 4 units right and 1 unit down is plotted using the transformed coordinates.

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Part 3: Translation: 1 Unit Right and 1 Unit Up

#### Given Figure:
The figure is a quadrilateral with vertices \( K \), \( N \), \( M \), and \( E \).

#### Transformation:
Translate the quadrilateral 1 unit right and 1 unit up.

#### Solution:
To translate a point \((x, y)\) 1 unit right and 1 unit up, the new coordinates become \((x + 1, y + 1)\).

1. Identify the coordinates of the vertices:
- \( K = (x_1, y_1) \)
- \( N = (x_2, y_2) \)
- \( M = (x_3, y_3) \)
- \( E = (x_4, y_4) \)

2. Apply the translation formula \((x + 1, y + 1)\) to each vertex:
- New coordinates of \( K \): \((x_1 + 1, y_1 + 1)\)
- New coordinates of \( N \): \((x_2 + 1, y_2 + 1)\)
- New coordinates of \( M \): \((x_3 + 1, y_3 + 1)\)
- New coordinates of \( E \): \((x_4 + 1, y_4 + 1)\)

3. Plot the new points and connect them to form the translated quadrilateral.

#### Final Answer:
The image of the quadrilateral after translating 1 unit right and 1 unit up is plotted using the transformed coordinates.

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Part 4: Reflection Across the x-axis

#### Given Figure:
The figure is a quadrilateral with vertices \( C \), \( D \), \( W \), and \( H \).

#### Transformation:
Reflect the quadrilateral across the x-axis.

#### Solution:
To reflect a point \((x, y)\) across the x-axis, the new coordinates become \((x, -y)\).

1. Identify the coordinates of the vertices:
- \( C = (x_1, y_1) \)
- \( D = (x_2, y_2) \)
- \( W = (x_3, y_3) \)
- \( H = (x_4, y_4) \)

2. Apply the reflection formula \((x, -y)\) to each vertex:
- New coordinates of \( C \): \((x_1, -y_1)\)
- New coordinates of \( D \): \((x_2, -y_2)\)
- New coordinates of \( W \): \((x_3, -y_3)\)
- New coordinates of \( H \): \((x_4, -y_4)\)

3. Plot the new points and connect them to form the reflected quadrilateral.

#### Final Answer:
The image of the quadrilateral after reflecting across the x-axis is plotted using the transformed coordinates.

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Part 5: Write a Rule to Describe Each Transformation

#### Given Figure:
The figure is a triangle with vertices \( B \), \( C \), and \( F \).

#### Transformation:
The triangle has been transformed, and we need to describe the rule.

#### Solution:
By observing the transformation:
- The triangle appears to have been translated.
- The new coordinates of the vertices suggest a translation of 3 units right and 2 units up.

#### Rule:
The rule for this transformation is:
\[
(x, y) \rightarrow (x + 3, y + 2)
\]

#### Final Answer:
\[
\boxed{(x, y) \rightarrow (x + 3, y + 2)}
\]

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Part 6: Write a Rule to Describe Each Transformation

#### Given Figure:
The figure is a quadrilateral with vertices \( O \), \( P \), \( Q \), and \( R \).

#### Transformation:
The quadrilateral has been transformed, and we need to describe the rule.

#### Solution:
By observing the transformation:
- The quadrilateral appears to have been rotated 90° clockwise about the origin.
- To confirm, check the coordinates of the vertices before and after the transformation. The pattern matches the rotation formula for 90° clockwise: \((x, y) \rightarrow (y, -x)\).

#### Rule:
The rule for this transformation is:
\[
(x, y) \rightarrow (y, -x)
\]

#### Final Answer:
\[
\boxed{(x, y) \rightarrow (y, -x)}
\]

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Final Summary of Answers:

1. Rotate 90° counterclockwise about the origin: Plot using \((-y, x)\).
2. Translate 4 units right and 1 unit down: Plot using \((x + 4, y - 1)\).
3. Translate 1 unit right and 1 unit up: Plot using \((x + 1, y + 1)\).
4. Reflect across the x-axis: Plot using \((x, -y)\).
5. Rule for Part 5: \(\boxed{(x, y) \rightarrow (x + 3, y + 2)}\).
6. Rule for Part 6: \(\boxed{(x, y) \rightarrow (y, -x)}\).