To find the image of the point [tex]\( P(3, -5) \)[/tex] under the translation [tex]\( T(x, y) = (x + 4, y - 3) \)[/tex], follow these steps:
1. Identify the coordinates of the original point: The original point is [tex]\( P(3, -5) \)[/tex], so the coordinates are [tex]\( x = 3 \)[/tex] and [tex]\( y = -5 \)[/tex].
2. Identify the translation components: The translation rule given is [tex]\( T(x, y) = (x + 4, y - 3) \)[/tex]. This means we need to add 4 to the x-coordinate and subtract 3 from the y-coordinate.
3. Apply the translation to the x-coordinate:
- Original x-coordinate: [tex]\( x = 3 \)[/tex]
- Translation component for x: [tex]\( +4 \)[/tex]
- New x-coordinate after translation: [tex]\( 3 + 4 = 7 \)[/tex]
4. Apply the translation to the y-coordinate:
- Original y-coordinate: [tex]\( y = -5 \)[/tex]
- Translation component for y: [tex]\( -3 \)[/tex]
- New y-coordinate after translation: [tex]\( -5 - 3 = -8 \)[/tex]
5. Write the coordinates of the new point: After applying the translation, the new coordinates [tex]\( P' \)[/tex] are [tex]\( (7, -8) \)[/tex].
Therefore, the image of the given point after the translation is [tex]\( P'(7, -8) \)[/tex].
