Let's solve this problem step-by-step by applying the transformations to the given ordered pair [tex]\((1, -3)\)[/tex].
1. Reflection over the y-axis [tex]\( R_{y \text{-axis}} \)[/tex]:
- Reflection over the y-axis changes the sign of the x-coordinate while keeping the y-coordinate the same.
- So, reflecting the point [tex]\((1, -3)\)[/tex] over the y-axis gives us:
[tex]\[
(-1, -3)
\][/tex]
2. Dilation [tex]\( D_2 \)[/tex]:
- A dilation by a factor of 2, denoted as [tex]\( D_2 \)[/tex], scales both the x and y coordinates by 2.
- Applying this dilation to the reflected point [tex]\((-1, -3)\)[/tex] results in:
[tex]\[
(2 \cdot -1, 2 \cdot -3) = (-2, -6)
\][/tex]
Thus, after performing the reflection over the y-axis followed by the dilation by a factor of 2, the transformed image of the point [tex]\((1, -3)\)[/tex] will be [tex]\(\boxed{(-2, -6)}\)[/tex].
