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Write a rule to describe each transformation

The problem appears to involve a transformation that maps one set of points to another. The points provided are:

- Original Points:  
  \( E(3, 1) \), \( F(3, 4) \), \( G(5, 1) \)

- Transformed Points:  
  \( F(-4, -3) \), \( G(-1, -5) \), \( E(-1, -3) \)

Let's analyze how each point is transformed.

1. **Point \( E(3, 1) \) to \( E(-1, -3) \)**:
   - The \( x \)-coordinate changes from \( 3 \) to \( -1 \) (a decrease of 4).
   - The \( y \)-coordinate changes from \( 1 \) to \( -3 \) (a decrease of 4).

2. **Point \( F(3, 4) \) to \( F(-4, -3) \)**:
   - The \( x \)-coordinate changes from \( 3 \) to \( -4 \) (a decrease of 7).
   - The \( y \)-coordinate changes from \( 4 \) to \( -3 \) (a decrease of 7).

3. **Point \( G(5, 1) \) to \( G(-1, -5) \)**:
   - The \( x \)-coordinate changes from \( 5 \) to \( -1 \) (a decrease of 6).
   - The \( y \)-coordinate changes from \( 1 \) to \( -5 \) (a decrease of 6).

### Rule for Transformation

The transformation rule for this mapping is a **translation**. Specifically, the translation can be described as:

\[
(x, y) \rightarrow (x - k, y - k)
\]

where \( k \) differs for each point. Here's how the transformation is applied:

- For \( E(3, 1) \) to \( E(-1, -3) \): \( k = 4 \)
- For \( F(3, 4) \) to \( F(-4, -3) \): \( k = 7 \)
- For \( G(5, 1) \) to \( G(-1, -5) \): \( k = 6 \)

Since each point is translated by a different amount, this is not a uniform translation but rather a transformation that applies a different shift for each point.

Would you like further details on the transformation process or additional questions to consider?

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Here are five related questions to deepen your understanding:

1. What would the rule be if all points underwent the same translation?
2. How can we describe the transformation if we consider it in the form of vectors?
3. What would the transformation look like if reflected over the origin?
4. How would you determine if a transformation is a rotation instead of a translation?
5. How does this translation affect the orientation of the figure formed by these points?

**Tip:** When dealing with transformations, always check if there is consistency in how each point is mapped to identify whether it is a translation, rotation, reflection, or dilation.

Explore the concept of transformation rules in geometry with detailed examples and explanations. Learn how to describe transformations such as translation using specific points and rules. Suitable for Grades 9-12.

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