To determine the translation vector between the preimage and the image:

1. Identify corresponding points in the preimage and image. For example, point $C$ in the preimage corresponds to point $C'$ in the image.

2. Measure the horizontal (x-direction) and vertical (y-direction) changes between these corresponding points.

   For $C$ to $C'$:

   • Horizontal change (x): From $(4, 6)$ in the preimage to $(0, 9)$ in the image. The x-coordinate shifts by $0 - 4 = -4$.
   • Vertical change (y): From $(4, 6)$ in the preimage to $(0, 9)$ in the image. The y-coordinate shifts by $9 - 6 = 3$.

Thus, the translation vector is $\langle -4, 3 \rangle$.

Would you like further clarification or assistance with another point on this problem?

Here are five related questions to deepen your understanding:

1. How would you verify a translation using a different pair of corresponding points, such as $D$ and $D'$?
2. What effect does a translation have on the orientation and size of a figure?
3. Can you describe how translations differ from other transformations, such as rotations or reflections?
4. What are real-world applications of translation vectors in mapping or graphic design?
5. How would you graphically demonstrate a translation vector on a grid?

Tip: Always double-check your calculations for the translation vector by using at least two corresponding points for accuracy.