A polygon is graphed on a coordinate plane. The polygon is transformed according to the rule [tex]$T_{(2,3)}$[/tex].
Which transformations produce the same result?
[tex]$T_{(2,2)} \circ T_{(3,3)}$[/tex]
[tex]$T_{(2,3)} \circ T_{(2,3)}$[/tex]
[tex]$T_{(1,-4)} \circ T_{(-3,1)}$[/tex]
[tex]$T_{(-1,4)} \circ T_{(3,-1)}$[/tex]
Answer (2)
Calculate the resulting transformation for each option by adding the components of the individual transformations.
Option 1: T ( 2 , 2 ) ∘ T ( 3 , 3 ) = T ( 5 , 5 ) .
Option 2: T ( 2 , 3 ) ∘ T ( 2 , 3 ) = T ( 4 , 6 ) .
Option 3: T ( 1 , − 4 ) ∘ T ( − 3 , 1 ) = T ( − 2 , − 3 ) .
Option 4: T ( − 1 , 4 ) ∘ T ( 3 , − 1 ) = T ( 2 , 3 ) .
The transformation that matches T ( 2 , 3 ) is the equivalent transformation: T ( − 1 , 4 ) ∘ T ( 3 , − 1 ) .
Explanation
Analyze the problem and available data. The problem asks us to find which of the given transformations is equivalent to the transformation T ( 2 , 3 ) . The transformation T ( a , b ) represents a translation by a units in the x-direction and b units in the y-direction. The composition of transformations T ( a , b ) ∘ T ( c , d ) is equivalent to T ( a + c , b + d ) . We need to calculate the resulting transformation for each option by adding the components of the individual transformations and compare the result with the original transformation T ( 2 , 3 ) .
Calculate the resulting transformation for each option. Let's analyze each option:
Option 1: T ( 2 , 2 ) ∘ T ( 3 , 3 ) To find the equivalent transformation, we add the components: T ( 2 + 3 , 2 + 3 ) = T ( 5 , 5 ) This is not equal to T ( 2 , 3 ) .
Option 2: T ( 2 , 3 ) ∘ T ( 2 , 3 ) To find the equivalent transformation, we add the components: T ( 2 + 2 , 3 + 3 ) = T ( 4 , 6 ) This is not equal to T ( 2 , 3 ) .
Option 3: T ( 1 , − 4 ) ∘ T ( − 3 , 1 ) To find the equivalent transformation, we add the components: T ( 1 + ( − 3 ) , − 4 + 1 ) = T ( − 2 , − 3 ) This is not equal to T ( 2 , 3 ) .
Option 4: T ( − 1 , 4 ) ∘ T ( 3 , − 1 ) To find the equivalent transformation, we add the components: T ( − 1 + 3 , 4 + ( − 1 )) = T ( 2 , 3 ) This is equal to T ( 2 , 3 ) .
Determine the equivalent transformation. By comparing the resulting transformations with the original transformation T ( 2 , 3 ) , we find that only Option 4, T ( − 1 , 4 ) ∘ T ( 3 , − 1 ) , produces the same result.
Therefore, the transformation that produces the same result as T ( 2 , 3 ) is T ( − 1 , 4 ) ∘ T ( 3 , − 1 ) .
Examples
Imagine you're programming a robot to move across a grid. Each transformation T ( a , b ) represents a command to move the robot a steps horizontally and b steps vertically. If you want the robot to end up at a specific location (e.g., 2 steps right and 3 steps up), you can combine multiple movement commands. This problem shows how different combinations of commands can achieve the same final position, helping you optimize the robot's path.
After analyzing all options, only Option 4, T ( − 1 , 4 ) ∘ T ( 3 , − 1 ) , results in the same transformation as T ( 2 , 3 ) . The other options do not yield the same result. Therefore, the correct answer is Option 4.
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