Let's break down the sequence of transformations and apply them step-by-step to find the coordinates of point [tex]\( Y \)[/tex].
### Given:
1. The final coordinates of [tex]\( Y'' \)[/tex] are [tex]\( (3, 0) \)[/tex].
2. The transformation sequence is [tex]\(R_{0,90^\circ} r_{X \text{ -axis }}\)[/tex].
### Steps:
1. First Transformation: Reflection over the X-axis
The reflection over the X-axis changes the [tex]\(y\)[/tex]-coordinate of a point while keeping the [tex]\(x\)[/tex]-coordinate the same. Let's apply this to [tex]\( Y'' \)[/tex]:
[tex]\[
Y'' = (3, 0)
\][/tex]
The reflection over the X-axis is:
[tex]\[
(x, y) \rightarrow (x, -y)
\][/tex]
Applying this transformation:
[tex]\[
Y' = (3, -0) = (3, 0)
\][/tex]
2. Second Transformation: Rotation by 90 degrees about the origin
The 90-degree rotation about the origin swaps the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates and changes the sign of the [tex]\( y \)[/tex]-coordinate (the new [tex]\( x \)[/tex]-coordinate). Let's apply this to [tex]\( Y' \)[/tex]:
[tex]\[
Y' = (3, 0)
\][/tex]
The 90-degree rotation about the origin is:
[tex]\[
(x, y) \rightarrow (-y, x)
\][/tex]
Applying this transformation:
[tex]\[
Y = (-0, 3) = (0, 3)
\][/tex]
### Conclusion:
The coordinates of [tex]\( Y \)[/tex] after applying the transformations are [tex]\((0, 3)\)[/tex].
So, the coordinates of [tex]\( Y \)[/tex] are:
[tex]\[
Y \ ( \ 0, \ 3 \ )
\][/tex]
