To determine the coordinates of the translated point [tex]\( H' \)[/tex] given the translation rule [tex]\( T_{-5,9}(x, y) \)[/tex] and the pre-image coordinates of point [tex]\( H \)[/tex] as [tex]\((-2, -3)\)[/tex], follow these steps:
1. Identify the pre-image coordinates: The initial coordinates of point [tex]\( H \)[/tex] are given as [tex]\((-2, -3)\)[/tex].
2. Understand the translation rule: The translation rule [tex]\( T_{-5,9} \)[/tex] means that every point [tex]\((x, y)\)[/tex] is translated by [tex]\(-5\)[/tex] units along the x-axis and [tex]\(9\)[/tex] units along the y-axis.
3. Apply the translation rule to the x-coordinate:
[tex]\[
x_{H'} = x_H + (-5)
\][/tex]
Substituting the x-coordinate of point [tex]\( H \)[/tex]:
[tex]\[
x_{H'} = -2 + (-5) = -2 - 5 = -7
\][/tex]
4. Apply the translation rule to the y-coordinate:
[tex]\[
y_{H'} = y_H + 9
\][/tex]
Substituting the y-coordinate of point [tex]\( H \)[/tex]:
[tex]\[
y_{H'} = -3 + 9 = 6
\][/tex]
5. Write the new coordinates of [tex]\( H' \)[/tex]: After applying the translation rule to both coordinates, we obtain:
[tex]\[
H' = (-7, 6)
\][/tex]
Therefore, the coordinates of [tex]\( H' \)[/tex] after the translation are [tex]\((-7, 6)\)[/tex].
Among the given options, the correct choice is:
[tex]\[
\boxed{(-7, 6)}
\][/tex]
