Answer:
Option D is correct.
The rule describe a composition is, [tex]r_{y-axis} o R_{0,270^{\circ}}(x, y)[/tex]
Step-by-step explanation:
From the given figure:
The coordinates of PQRS are;
P = (1,1)
Q= (1, 5)
R=(3, 5)
S = (3, 1)
First apply the rule of 270° counterclockwise rotation about the origin is given by;
[tex]R_{0,270^{\circ}}(x, y) \rightarrow (y , -x)[/tex]
Apply the rule of 270 degree counterclockwise rotation on PQRS:
[tex]P(1, 1) \rightarrow (1 , -1)[/tex]
[tex]Q(1, 5) \rightarrow (5, -1)[/tex]
[tex]R(3, 5) \rightarrow (5, -3)[/tex]
[tex]S(3, 1) \rightarrow (1 , -3)[/tex]
Next apply y-axis the rule of reflection i.e
[tex]r_{y-axis}(x, y) \rightarrow (-x, y)[/tex]
[tex](1, -1) \rightarrow (-1 , -1)[/tex]=P"
[tex](5 -1) \rightarrow (-5 , -1)[/tex] =Q"
[tex](5, -3) \rightarrow (-5 , -3)[/tex]=R"
[tex](1, -3) \rightarrow (-1 , -3)[/tex]=S"
Therefore, the rule describe a composition of transformation that maps pre-image PQRS to image P"Q"R"S" is; [tex]r_{y-axis} o R_{0,270^{\circ}}(x, y)[/tex]
The correct option is Option D [tex]\boxed{{r_{y-axis}}o{R_{0,270^\circ }}\left({x,y}\right)}[/tex] .
Further explanation:
A translation is a transformation that transforms the figure with a fixed distance in the same direction.
A rotation is the transformation that rotates the figure with given angles.
Given:
It is given that the two transformations that maps pre-image PQRS to image [tex]{\text{P''Q''R''S''}}[/tex] .
Step by step explanation:
Step 1:
It can be seen from the given figure that that the pre image is in the first quadrant and the image is the third quadrant.
The coordinates in the second quadrant represents as [tex]\left({-x,y}\right)[/tex] and in the third quadrant represents as [tex]\left({-x,-y}\right)[/tex] if [tex]x,y[/tex] are positive.
Therefore, the rotation is in the counter clockwise direction of [tex]270^\circ[/tex] .
Step 2:
The rotation of [tex]270^\circ[/tex] in the counter clockwise direction represents the coordinates as,
[tex]\left({x,y}\right)\to\left({y,-x}\right)[/tex]
It can be seen that the coordinate of [tex]{\text{PQRS}}[/tex] are as follows,
[tex]\begin{aligned}P=\left({1,1}\right)\hfill\\Q=\left(1,5}\right)\hfill\\R=\left({3,5}\right)\hfill\\S=\left({3,1}\right)\hfill\\\end{aligned}[/tex]
Then after rotation of [tex]270^\circ[/tex] counterclockwise on [tex]{\text{PQRS}}[/tex] [tex]\left( {x,y}\right)\to\left({y,-x}\right)[/tex] as,
[tex]\begin{gathered}P\left({1,1}\right)\to\left({1,-1}\right)\hfill\\Q\left({1,5}\right)\to\left({5,1}\right)\hfill\\R\left({3,5}\right)\to\left({5,-3}\right)\hfill\\S\left({3,1}\right)\to\left({1,-3}\right)\hfill\\\end{gathered}[/tex]
Step 3:
Now apply the rule of [tex]y[/tex] axis of reflection [tex]{R_{y-axis}}\left({x,y}\right)\to\left({-x,y}\right)[/tex] on the above transformation as,
[tex]\begin{gathered}\left({1,-1}\right)\to\left({-1,1}\right)=P''\hfill\\\left({5,1}\right)\to\left({-5,-1}\right)=Q''\hfill\\\left({5,-3}\right)\to\left({-5,-3}\right)=R''\hfill\\\left({1,-3}\right)\to\left({-1,-3}\right)=S''\hfill\\\end{gathered}[/tex]
Therefore, the given transformation is the rotation of [tex]270^\circ[/tex] counterclockwise followed by [tex]y[/tex] axis of reflection.
Therefore, this is the composition of transformation.
The composition of the given transformation can be written as,
[tex]{r_{y-axis}}o{R_{0,270^\circ}}\left({x,y}\right)[/tex]
Therefore, option D [tex]{r_{y-axis}}o{R_{0,270^\circ}}\left({x,y}\right)[/tex] is correct.
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Answer details:
Grade: High school
Subject: Mathematics
Chapter: Transformations
Keywords: transformations, dilation, translation, rotation, counterclockwise, angle, clockwise, coordinates, mapping, rigid transformation, right side, left side, quadrant, composition.
