To find the image of the point [tex]\( P(8, -3) \)[/tex] under the translation defined by the transformation [tex]\( (x-4, y+7) \)[/tex], we follow these steps:
1. Identify the coordinates of the given point:
The point [tex]\( P \)[/tex] has coordinates [tex]\( x = 8 \)[/tex] and [tex]\( y = -3 \)[/tex].
2. Apply the translation to the x-coordinate:
We need to translate the x-coordinate by subtracting 4.
[tex]\[
x' = x - 4 = 8 - 4 = 4
\][/tex]
3. Apply the translation to the y-coordinate:
We need to translate the y-coordinate by adding 7.
[tex]\[
y' = y + 7 = -3 + 7 = 4
\][/tex]
4. Combine the translated coordinates:
The new coordinates after applying the translation are [tex]\( (4, 4) \)[/tex].
Therefore, the image of the point [tex]\( P(8, -3) \)[/tex] under the translation [tex]\( (x-4, y+7) \)[/tex] is [tex]\( P'(4, 4) \)[/tex].
