To find the coordinates of [tex]\( H' \)[/tex] after a translation, we need to apply the translation rule [tex]\( T_{-5,9}(x, y) \)[/tex] to the original coordinates of point [tex]\( H \)[/tex]. Here, the pre-image coordinates of point [tex]\( H \)[/tex] are given as [tex]\((2, -3)\)[/tex].
The translation rule [tex]\( T_{-5,9}(x, y) \)[/tex] means that we will translate the point by moving it [tex]\(-5\)[/tex] units in the [tex]\( x \)[/tex]-direction and [tex]\(9\)[/tex] units in the [tex]\( y \)[/tex]-direction.
Let's break down the translation step-by-step:
1. Starting coordinates: The pre-image point [tex]\( H \)[/tex] has coordinates [tex]\((2, -3)\)[/tex].
2. Translation in the [tex]\( x \)[/tex]-direction:
- The original [tex]\( x \)[/tex]-coordinate is [tex]\(2\)[/tex].
- According to the rule, we need to subtract [tex]\(5\)[/tex] from the [tex]\( x \)[/tex]-coordinate.
- So, [tex]\( 2 - 5 = -3 \)[/tex].
3. Translation in the [tex]\( y \)[/tex]-direction:
- The original [tex]\( y \)[/tex]-coordinate is [tex]\(-3\)[/tex].
- According to the rule, we need to add [tex]\(9\)[/tex] to the [tex]\( y \)[/tex]-coordinate.
- So, [tex]\(-3 + 9 = 6\)[/tex].
Hence, the new coordinates of [tex]\( H' \)[/tex] are [tex]\( (-3, 6) \)[/tex].
Therefore, the correct answer is [tex]\( (-3, 6) \)[/tex].
