To find the coordinates of [tex]\( H' \)[/tex] after translating point [tex]\( H \)[/tex] according to the given rule, we follow a step-by-step process.
Step 1: Understand the translation rule
The translation rule [tex]\( T_{-5, 9}(x, y) \)[/tex] means that we need to move each point by adding [tex]\(-5\)[/tex] to the [tex]\( x \)[/tex]-coordinate and adding [tex]\( 9 \)[/tex] to the [tex]\( y \)[/tex]-coordinate.
Step 2: Identify the coordinates of the pre-image
The coordinates of point [tex]\( H \)[/tex] are given as [tex]\((-2, -3)\)[/tex]. This means:
- [tex]\( x_{\text{pre}} = -2 \)[/tex]
- [tex]\( y_{\text{pre}} = -3 \)[/tex]
Step 3: Apply the translation to the [tex]\( x \)[/tex]-coordinate
To obtain the new [tex]\( x \)[/tex]-coordinate, we add [tex]\(-5\)[/tex] to the original [tex]\( x \)[/tex]-coordinate:
[tex]\[ x_{\text{post}} = x_{\text{pre}} + (-5) = -2 + (-5) = -7 \][/tex]
Step 4: Apply the translation to the [tex]\( y \)[/tex]-coordinate
To obtain the new [tex]\( y \)[/tex]-coordinate, we add [tex]\( 9 \)[/tex] to the original [tex]\( y \)[/tex]-coordinate:
[tex]\[ y_{\text{post}} = y_{\text{pre}} + 9 = -3 + 9 = 6 \][/tex]
Step 5: Write the coordinates of [tex]\( H' \)[/tex]
After the translation, the coordinates of [tex]\( H' \)[/tex] are:
[tex]\[ (-7, 6) \][/tex]
Thus, the correct answer is:
[tex]\[ (-7, 6) \][/tex]
