To determine the image of any point [tex]\((x,y)\)[/tex] under the translation that maps [tex]\((3, -4)\)[/tex] to its image [tex]\((1, 0)\)[/tex], we need to figure out the translation vector.
A translation vector [tex]\((\delta_x, \delta_y)\)[/tex] will map an original point [tex]\((x_1, y_1)\)[/tex] to a new point [tex]\((x_2, y_2)\)[/tex] by adding [tex]\(\delta_x\)[/tex] to [tex]\(x_1\)[/tex] and [tex]\(\delta_y\)[/tex] to [tex]\(y_1\)[/tex]. Mathematically, this is expressed as:
[tex]\[
(x_2, y_2) = (x_1 + \delta_x, y_1 + \delta_y)
\][/tex]
Given the coordinates of the original point [tex]\((3, -4)\)[/tex] and its image [tex]\((1, 0)\)[/tex], we can set up the following equations to find [tex]\(\delta_x\)[/tex] and [tex]\(\delta_y\)[/tex]:
[tex]\[
1 = 3 + \delta_x
\][/tex]
[tex]\[
0 = -4 + \delta_y
\][/tex]
Solving for [tex]\(\delta_x\)[/tex]:
[tex]\[
\delta_x = 1 - 3 = -2
\][/tex]
Solving for [tex]\(\delta_y\)[/tex]:
[tex]\[
\delta_y = 0 + 4 = 4
\][/tex]
Thus, the translation vector is [tex]\((-2, 4)\)[/tex].
To find the image of any point [tex]\((x, y)\)[/tex] under this translation, we apply the translation vector to [tex]\((x, y)\)[/tex] as follows:
[tex]\[
(x', y') = (x + \delta_x, y + \delta_y) = (x - 2, y + 4)
\][/tex]
Therefore, the correct image of the point [tex]\((x, y)\)[/tex] under this translation is:
D. [tex]\((x-2, y+4)\)[/tex]
