To determine the correct answer, we will revisit the steps to find the translation vector and how it is applied to any point $(x, y)$.

Given that the point $(3, -4)$ maps to the point $(1, 0)$, we first find the translation vector $(a, b)$ as follows: $(a, b) = (1 - 3, 0 - (-4))$ $(a, b) = (-2, 4)$

This translation vector indicates that any point $(x, y)$ is translated by $-2$ units in the $x$-direction and $4$ units in the $y$-direction. The translated point $(x', y')$ is given by: $(x', y') = (x + a, y + b)$ $(x', y') = (x - 2, y + 4)$

Therefore, the image of any point $(x, y)$ under this translation is: $(x - 2, y + 4)$

Comparing this result with the given options:

A. $(x + 2, y - 4)$

B. $(x - 2, y - 4)$

C. $(x - 2, y + 4)$

D. $(x + 2, y + 4)$

The correct answer is: C. $(x - 2, y + 4)$

If you have any questions or would like more details, feel free to ask.

Here are 8 related questions for further exploration:

1. What would be the new coordinates of the point $(6, -2)$ under the same translation?
2. How do you reverse the translation to find the original point from its image?
3. How would you graphically represent the translation of a point in the coordinate plane?
4. What is the effect of this translation on the equation of a line?
5. If another translation maps $(3, -4)$ to $(5, -6)$, what would be the translation vector?
6. How would you apply this translation to a triangle with vertices $(0, 0)$, $(1, 0)$, and $(0, 1)$?
7. How do you combine multiple translations into a single translation vector?
8. How would the translation affect the area and perimeter of a geometric shape?

Tip: When solving translation problems, carefully calculate the translation vector and apply it systematically to each coordinate to ensure accuracy.