The transformation shown involves a movement of the original figure (labeled as $A$, $B$, $C$, $D$) to a new figure (labeled as $A'$, $B'$, $C'$, $D'$). Let's analyze this transformation by comparing the coordinates of each corresponding point before and after the transformation.

Step 1: Identify Coordinates

• Original coordinates:

  • $A(2, 4)$, $B(4, 4)$, $C(4, 2)$, $D(2, 2)$

• Transformed coordinates:

  • $A'(-4, -2)$, $B'(-2, -2)$, $C'(-2, -4)$, $D'(-4, -4)$

Step 2: Determine the Rule of Transformation

From the original to the transformed figure, the points have undergone a shift in both the $x$- and $y$-coordinates:

• Comparing $A$ to $A'$:
  • $x$ shifted from $2$ to $-4$: $2 - 6 = -4$ (6 units to the left)
  • $y$ shifted from $4$ to $-2$: $4 - 6 = -2$ (6 units down)

Step 3: Verify the Same Pattern for All Points

We observe the same change consistently for all the points:

• $B(4, 4)$ to $B'(-2, -2)$ follows $(x - 6, y - 6)$
• $C(4, 2)$ to $C'(-2, -4)$ follows $(x - 6, y - 6)$
• $D(2, 2)$ to $D'(-4, -4)$ follows $(x - 6, y - 6)$

Conclusion

The transformation involves a translation by 6 units down and 6 units left, which corresponds to the rule $T_{(-6, -6)}(x, y) = (x - 6, y - 6)$.

Correct Answer(s) Selection:

The rules that apply are:

• A: $T_{(-6, -6)}(x, y) = (x - 6, y - 6)$
• B: 6 units down; 6 units left

Do you have any questions or need further details on this?

Here are 5 related questions to consider:

1. What is a translation in the context of geometric transformations?
2. How do we describe other types of transformations like rotations or reflections?
3. Can you find the translation rule for a movement of 4 units right and 3 units up?
4. How do we differentiate between a translation and a rotation?
5. How would you write a translation rule for a shift of 5 units left and 2 units down?

Tip: Always compare the coordinates of corresponding points to understand the transformation applied to a geometric figure.