Which rule describes a composition of transformations that maps pre-image PQRS to image P"Q"R"S"?
A. [tex]R_{0,2700} \circ T_{-2,0}(x, y)[/tex]
B. [tex]T_{-2,0^{\circ}} R_{0,270^{\circ}}(x, y)[/tex]
C. [tex]R_{0,2700}{ }^{\circ} r_{y-2 x i s}(x, y)[/tex]
D. [tex]r_{y \text {-axis }}{ }^{\circ} R_{0,2700}(x, y)[/tex]
Answer (2)
Analyzes each transformation option by applying it to a general point (x, y).
Determines the resulting coordinates after each transformation.
Simplifies the rotations by recognizing that a 2700-degree rotation is equivalent to a 180-degree rotation.
Concludes that options 3 and 4, R 0 , 2700 ∘ r y − a x i s ( x , y ) and r y − a x i s ∘ R 0 , 2700 ( x , y ) , both result in the transformation (x, -y).
The final answer is either R 0 , 2700 ∘ r y − a x i s ( x , y ) or r y − a x i s ∘ R 0 , 2700 ( x , y ) .
Since both options 3 and 4 result in the same transformation, we can say the answer is: r y -axis ∘ R 0 , 2700 ( x , y )
Explanation
Analyze the problem and available data Let's analyze each option to determine the correct composition of transformations that maps the pre-image PQRS to the image P"Q"R"S". We will apply each transformation to a general point (x, y). Recall that a rotation of 2700 degrees is equivalent to a rotation of 180 degrees, since 2700 = 7 * 360 + 180.
Analyze Option 1 Option 1: R 0 , 2700 ∘ T − 2 , 0 ( x , y ) .
First, we translate (x, y) by (-2, 0) to get (x - 2, y). Then, we rotate (x - 2, y) by 180 degrees about the origin. A rotation of 180 degrees maps (x, y) to (-x, -y), so (x - 2, y) is mapped to (-(x - 2), -y) = (-x + 2, -y).
Analyze Option 2 Option 2: T − 2 , 0 ∘ R 0 , 270 ( x , y ) .
First, we rotate (x, y) by 270 degrees about the origin. A rotation of 270 degrees maps (x, y) to (y, -x). Then, we translate (y, -x) by (-2, 0) to get (y - 2, -x).
Analyze Option 3 Option 3: R 0 , 2700 ∘ r y − a x i s ( x , y ) .
First, we reflect (x, y) across the y-axis to get (-x, y). Then, we rotate (-x, y) by 180 degrees about the origin. A rotation of 180 degrees maps (x, y) to (-x, -y), so (-x, y) is mapped to (x, -y).
Analyze Option 4 Option 4: r y − a x i s ∘ R 0 , 2700 ( x , y ) .
First, we rotate (x, y) by 180 degrees about the origin. A rotation of 180 degrees maps (x, y) to (-x, -y). Then, we reflect (-x, -y) across the y-axis to get (x, -y).
Compare the results and conclude Comparing the results, we see that options 3 and 4 both result in the transformation (x, y) -> (x, -y). Without additional information about the specific coordinates of the pre-image PQRS and the image P"Q"R"S", we cannot definitively determine which option is correct. However, if we assume that the transformation is (x, y) -> (x, -y), then both options 3 and 4 are valid.
Examples
Understanding transformations is crucial in computer graphics for tasks like object animation and creating special effects. For instance, rotating an image or translating it across the screen involves applying these mathematical principles to each pixel's coordinates, ensuring smooth and accurate visual results.
After analyzing the transformations provided in the options, I conclude that Option D, which is the composition of the reflection across the y-axis followed by a 270-degree rotation, accurately maps the pre-image PQRS to the image P"Q"R"S". The transformations result in the correct changes in coordinates, supporting this conclusion. Therefore, the answer is D.
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