https://file.aiquickdraw.com/mathbot/file/1726285521439kww3sbmm.jpg

In the image, the problem asks to determine the rule for the transformation that maps the triangle \( \Delta AVF \) to its image \( \Delta A'V'F' \). Here’s how we can approach it:

1. **Original Points:**
   - \( A(-4, -4) \)
   - \( V(2, 0) \)
   - \( F(3, 4) \)

2. **Transformed Points:**
   - \( A'(0, -2) \)
   - \( V'(6, 2) \)
   - \( F'(7, 6) \)

### Step 1: Examine the transformation
To find the transformation rule, compare the changes between the original and transformed points.

For point \( A(-4, -4) \) to \( A'(0, -2) \):
- \( x \) changes from \( -4 \) to \( 0 \) (a shift of \( +4 \)).
- \( y \) changes from \( -4 \) to \( -2 \) (a shift of \( +2 \)).

For point \( V(2, 0) \) to \( V'(6, 2) \):
- \( x \) changes from \( 2 \) to \( 6 \) (a shift of \( +4 \)).
- \( y \) changes from \( 0 \) to \( 2 \) (a shift of \( +2 \)).

For point \( F(3, 4) \) to \( F'(7, 6) \):
- \( x \) changes from \( 3 \) to \( 7 \) (a shift of \( +4 \)).
- \( y \) changes from \( 4 \) to \( 6 \) (a shift of \( +2 \)).

### Step 2: Write the transformation rule
The transformation involves shifting the \( x \)-coordinate by \( +4 \) and the \( y \)-coordinate by \( +2 \). Therefore, the rule for the transformation is:

\[
(x, y) \rightarrow (x + 4, y + 2)
\]

Would you like further details or clarifications on this solution?

Here are 5 related questions for further exploration:
1. How do you determine if a transformation is a translation, rotation, reflection, or dilation?
2. Can you represent this transformation as a matrix operation?
3. What happens to the distances between points under a translation?
4. If you reverse the transformation, what would the rule look like?
5. How would this transformation look on a three-dimensional coordinate plane?

**Tip:** To verify a transformation, always check multiple points to ensure consistency!

Below is a graph of triangle ΔAVF and its image ΔA'V'F'. Write the rule for the given transformation.

Learn how to determine the rule for transforming triangle ΔAVF to its image ΔA'V'F' by applying a translation of (x + 4, y + 2). This problem covers concepts of translation in coordinate geometry. Suitable for students in Grades 8-10.

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