To determine the coordinates of the image [tex]\( H' \)[/tex] after the translation, we need to apply the given translation rule to the coordinates of the pre-image of point [tex]\( H \)[/tex]. Here are the detailed steps to follow:
1. Identify the coordinates of pre-image [tex]\( H \)[/tex]:
The coordinates of the pre-image of point [tex]\( H \)[/tex] are [tex]\( (-2, -3) \)[/tex].
2. Understand the translation rule:
The translation rule provided is [tex]\( T_{-5,9}(x, y) \)[/tex]. This means:
- [tex]\( x \)[/tex]-coordinate is translated by [tex]\(-5\)[/tex] units.
- [tex]\( y \)[/tex]-coordinate is translated by [tex]\(+9\)[/tex] units.
3. Apply the translation to the [tex]\( x \)[/tex]-coordinate:
- The original [tex]\( x \)[/tex]-coordinate of [tex]\( H \)[/tex] is [tex]\(-2\)[/tex].
- Applying the translation: [tex]\[ x' = -2 + (-5) \][/tex]
- Simplifying, we get: [tex]\[ x' = -2 - 5 = -7 \][/tex]
4. Apply the translation to the [tex]\( y \)[/tex]-coordinate:
- The original [tex]\( y \)[/tex]-coordinate of [tex]\( H \)[/tex] is [tex]\(-3\)[/tex].
- Applying the translation: [tex]\[ y' = -3 + 9 \][/tex]
- Simplifying, we get: [tex]\[ y' = 6 \][/tex]
5. Determine the coordinates of point [tex]\( H' \)[/tex]:
- After applying the translation, the new coordinates of [tex]\( H' \)[/tex] are: [tex]\((-7, 6)\)[/tex].
Thus, the correct answer is:
[tex]\[ (-7, 6) \][/tex]
