The rule $r_{y=,}{ }^{\circ} T_{4,0}(x, y$ is applied to trapezoid ABCD to produce the final image $A^{\prime \prime} B^{\prime \prime} C^{\prime \prime} D^{\prime \prime}$.
Which ordered pairs name the coordinates of vertices of the pre-image, trapezoid ABCD? Select two options.
(-1,0)
(-1,-5)
(1,1)
(7,0)
(7, -5)
Answer (1)
Apply the inverse transformation T − 4 , 0 ∘ r y = x to the given options.
The inverse transformation maps ( x , y ) to ( y − 4 , x ) .
Apply the inverse transformation to each option: (-1,0) -> (-4, -1), (-1,-5) -> (-9, -1), (1,1) -> (-3, 1), (7,0) -> (-4, 7), (7,-5) -> (-9, 7).
Since none of the transformed points match the original options, it's not possible to determine the pre-image vertices with the given information.
Explanation
Analyze the problem Let's analyze the given problem. We have a trapezoid ABCD that undergoes a transformation r y = x ∘ T 4 , 0 ( x , y ) to produce the final image A ′′ B ′′ C ′′ D ′′ . Our goal is to find which of the given ordered pairs could be vertices of the original trapezoid ABCD.
Understand the transformation The transformation r y = x ∘ T 4 , 0 ( x , y ) means we first apply the translation T 4 , 0 ( x , y ) which maps ( x , y ) to ( x + 4 , y ) , and then we apply the reflection r y = x which maps ( x + 4 , y ) to ( y , x + 4 ) . So, the composite transformation maps ( x , y ) to ( y , x + 4 ) .
Find the inverse transformation To find the pre-image coordinates, we need to apply the inverse transformation. The inverse of the reflection r y = x is the reflection itself, r y = x . The inverse of the translation T 4 , 0 ( x , y ) is T − 4 , 0 ( x , y ) , which maps ( x , y ) to ( x − 4 , y ) . Therefore, the inverse transformation is T − 4 , 0 ∘ r y = x , which means we first reflect across y = x and then translate by ( − 4 , 0 ) . So, ( x , y ) becomes ( y , x ) after reflection, and then ( y − 4 , x ) after translation.
Apply the inverse transformation to the options Now, let's apply the inverse transformation T − 4 , 0 ∘ r y = x to each of the given options:
( − 1 , 0 ) becomes ( 0 − 4 , − 1 ) = ( − 4 , − 1 )
( − 1 , − 5 ) becomes ( − 5 − 4 , − 1 ) = ( − 9 , − 1 )
( 1 , 1 ) becomes ( 1 − 4 , 1 ) = ( − 3 , 1 )
( 7 , 0 ) becomes ( 0 − 4 , 7 ) = ( − 4 , 7 )
( 7 , − 5 ) becomes ( − 5 − 4 , 7 ) = ( − 9 , 7 )
Analyze the results Since none of the resulting points match the original options, it seems there might be an issue with the problem statement or the provided options. However, based on the given information and the inverse transformation we derived, we can't directly determine which two options are vertices of the pre-image. Let's consider the possibility that two of the transformed points could be the pre-image coordinates. However, this approach doesn't lead to a definitive answer with the given options.
Trying another approach Let's try a different approach. Assume the given points are the final image points. Then ( x ′′ , y ′′ ) = ( y − 4 , x ) . Thus x ′′ = y − 4 and y ′′ = x . So y = x ′′ + 4 and x = y ′′ . The original point is ( y ′′ , x ′′ + 4 ) .
Applying this to the options:
(-1, 0) becomes (0, -1+4) = (0, 3)
(-1, -5) becomes (-5, -1+4) = (-5, 3)
(1, 1) becomes (1, 1+4) = (1, 5)
(7, 0) becomes (0, 7+4) = (0, 11)
(7, -5) becomes (-5, 7+4) = (-5, 11)
Again, none of the transformed points match the original options.
Conclusion Given the available options and the transformations, it's not possible to definitively determine which two ordered pairs are the coordinates of the vertices of the pre-image trapezoid ABCD. There might be an error in the problem statement or the options provided.
Examples
In computer graphics, transformations like translation and reflection are fundamental for manipulating objects in a virtual space. Understanding how to apply and reverse these transformations is crucial for tasks such as animation, object placement, and creating realistic visual effects. For example, when designing a game, you might need to reflect a character's movement across a symmetry axis or translate it to a new location on the screen. The principles used here are directly applicable in those scenarios.
