To determine the image of any point [tex]\((x, y)\)[/tex] under the translation that maps the point [tex]\((3, -4)\)[/tex] to its image [tex]\((1, 0)\)[/tex], we need to find the translation vector [tex]\((t_x, t_y)\)[/tex]. This vector helps us understand how much the point is being shifted in the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] directions.
### Step-by-Step Solution:
1. Calculate Translation in the x-direction:
- The original point in the [tex]\(x\)[/tex]-direction is [tex]\(3\)[/tex], and it is mapped to [tex]\(1\)[/tex].
- To find [tex]\(t_x\)[/tex]:
[tex]\[
1 = 3 + t_x
\][/tex]
- Solving for [tex]\(t_x\)[/tex]:
[tex]\[
t_x = 1 - 3 = -2
\][/tex]
2. Calculate Translation in the y-direction:
- The original point in the [tex]\(y\)[/tex]-direction is [tex]\(-4\)[/tex], and it is mapped to [tex]\(0\)[/tex].
- To find [tex]\(t_y\)[/tex]:
[tex]\[
0 = -4 + t_y
\][/tex]
- Solving for [tex]\(t_y\)[/tex]:
[tex]\[
t_y = 0 + 4 = 4
\][/tex]
3. Formulate the Translation Rule:
- A point [tex]\((x, y)\)[/tex] will be translated by [tex]\((t_x, t_y) = (-2, 4)\)[/tex].
- Therefore, the new coordinates after translation will be:
[tex]\[
(x + t_x, y + t_y) = (x - 2, y + 4)
\][/tex]
### Conclusion:
The image of any point [tex]\((x, y)\)[/tex] under this translation is:
[tex]\[
(x - 2, y + 4)
\][/tex]
Thus, the correct answer is:
C. [tex]\((x - 2, y + 4)\)[/tex]
