Using the translation that maps [tex]\((3, -4)\)[/tex] to its image [tex]\((1, 0)\)[/tex], what is the image of any point [tex]\((x, y)\)[/tex]?

A. [tex]\((x - 2, y + 4)\)[/tex]

B. [tex]\((x + 2, y + 4)\)[/tex]

C. [tex]\((x + 2, y - 4)\)[/tex]

D. [tex]\((x - 2, y + 4)\)[/tex]



Answer :

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]