To determine the translation rule that correctly describes how point [tex]\( A(7,3) \)[/tex] is translated to [tex]\( A^{\prime}(16,-9) \)[/tex], follow these step-by-step instructions:
1. Initial Coordinates and Translated Coordinates:
- The initial coordinates of point [tex]\( A \)[/tex] are [tex]\( (7, 3) \)[/tex].
- The coordinates of the translated point [tex]\( A^{\prime} \)[/tex] are [tex]\( (16, -9) \)[/tex].
2. Calculate the Translation in the x-direction:
- Subtract the initial x-coordinate from the translated x-coordinate:
[tex]\[
x_{\text{translation}} = x_{\text{translated}} - x_{\text{initial}} = 16 - 7 = 9
\][/tex]
Hence, the translation in the x-direction is 9 units to the right.
3. Calculate the Translation in the y-direction:
- Subtract the initial y-coordinate from the translated y-coordinate:
[tex]\[
y_{\text{translation}} = y_{\text{translated}} - y_{\text{initial}} = -9 - 3 = -12
\][/tex]
Hence, the translation in the y-direction is 12 units downward.
4. Form the Translation Rule:
- The translation described by [tex]\( x_{\text{translation}} \)[/tex] and [tex]\( y_{\text{translation}} \)[/tex]:
[tex]\[
(x, y) \rightarrow (x + x_{\text{translation}}, y + y_{\text{translation}})
\][/tex]
Substituting the values we calculated:
[tex]\[
(x, y) \rightarrow (x + 9, y - 12)
\][/tex]
Therefore, the correct rule that describes the translation of point [tex]\( A(7, 3) \)[/tex] to point [tex]\( A^{\prime}(16, -9) \)[/tex] is:
[tex]\[
(x, y) \rightarrow (x + 9, y - 12)
\][/tex]
