To determine the vector representing the translation in this problem, we can examine the movement of each vertex of the shape from its original position (marked by points A, B, C, and D) to its translated position (marked by points $A'$, $B'$, $C'$, and $D'$).

1. Choose a corresponding point, such as $A$ to $A'$, and calculate the shift in both the x and y directions:

   • $A$ is at $(2, 4)$
   • $A'$ is at $(-2, 0)$

2. Find the differences in the x and y coordinates:

   • The change in the x-coordinate is $-2 - 2 = -4$.
   • The change in the y-coordinate is $0 - 4 = -4$.

Thus, the translation vector is: $\begin{pmatrix} -4 \\ -4 \end{pmatrix}$

This vector means that every point on the shape was moved 4 units to the left and 4 units down.

Would you like further details, or have any questions?

Here are some related questions:

1. What would the translation vector be if we moved the shape up by 3 units and right by 2 units?
2. How do you apply a translation vector to coordinates in a Cartesian plane?
3. What is the effect of a positive vs. negative sign in a translation vector on the direction of movement?
4. Can you describe a scenario where a shape’s translation results in a vector of $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$?
5. How can translation vectors be used in coordinate geometry problems?

Tip: When dealing with translation vectors, remember that the x-component affects horizontal movement (left or right), and the y-component affects vertical movement (up or down).