The rule $r_{y=,} \circ r_{4, b}(x, y)$ is applied to trapezoid $A B C D$ to produce the final image $A^{"} B^{"} C^{"} D^{"}$. Which ordered pairs name the coordinates of vertices of the pre-image, trapezoid $A B C D$ ? Select two options.
$(-1,0)$
$(-1,-5)$
$(1,1)$
$(7,0)$
$(7,-5)$

Analyze the given transformation r y = 0 ​ ∘ r 4 , b ​ ( x , y ) as a reflection over x = 4 , followed by a translation ( 0 , b ) , and then a reflection over the x-axis.
Determine the combined transformation rule: ( x ′′ , y ′′ ) = ( 8 − x , − y − b ) .
Find the inverse transformation to express the pre-image coordinates in terms of the final image coordinates: ( x , y ) = ( 8 − x ′′ , − y ′′ − b ) .
Test the given options to find two ordered pairs that could be vertices of the pre-image. The two ordered pairs are ( 1 , 1 ) and ( 7 , 0 ) .
The coordinates of vertices of the pre-image trapezoid A BC D are ( 1 , 1 ) , ( 7 , 0 ) ​ .

Explanation

Analyze the transformations Let's analyze the given transformations and the options for the coordinates of the pre-image trapezoid vertices. The transformation applied is r y = 0 ​ ∘ r 4 , b ​ ( x , y ) , which means a reflection over the line x = 4 followed by a translation by the vector ( 0 , b ) , and then a reflection over the x-axis. We need to find two ordered pairs from the given options that could be vertices of the pre-image trapezoid.

Determine the combined transformation Let ( x ′ , y ′ ) be the coordinates after the transformation r 4 , b ​ ( x , y ) . Then x ′ = 8 − x and y ′ = y + b . So, r 4 , b ​ ( x , y ) = ( 8 − x , y + b ) .


Let ( x ′′ , y ′′ ) be the coordinates after the transformation r y = 0 ​ ∘ r 4 , b ​ ( x , y ) . Then x ′′ = 8 − x and y ′′ = − ( y + b ) = − y − b . So, r y = 0 ​ ∘ r 4 , b ​ ( x , y ) = ( 8 − x , − y − b ) .

Find the inverse transformation We are given the final image A ′ B ′ C ′ D ′ and we want to find the pre-image A BC D . Let ( x ′′ , y ′′ ) be the coordinates of a vertex in the final image and ( x , y ) be the coordinates of the corresponding vertex in the pre-image. Then x ′′ = 8 − x and y ′′ = − y − b .

Solve for x and y in terms of x ′′ and y ′′ : x = 8 − x ′′ and y = − y ′′ − b . So, ( x , y ) = ( 8 − x ′′ , − y ′′ − b ) .

Test each option Now, let's test each of the given options to see if they can be vertices of the pre-image. We need to find two options that satisfy the equation ( x , y ) = ( 8 − x ′′ , − y ′′ − b ) for some value of b .

Option 1: ( − 1 , 0 ) . If ( x ′′ , y ′′ ) = ( − 1 , 0 ) , then ( x , y ) = ( 8 − ( − 1 ) , − 0 − b ) = ( 9 , − b ) .
Option 2: ( − 1 , − 5 ) . If ( x ′′ , y ′′ ) = ( − 1 , − 5 ) , then ( x , y ) = ( 8 − ( − 1 ) , − ( − 5 ) − b ) = ( 9 , 5 − b ) .
Option 3: ( 1 , 1 ) . If ( x ′′ , y ′′ ) = ( 1 , 1 ) , then ( x , y ) = ( 8 − 1 , − 1 − b ) = ( 7 , − 1 − b ) .
Option 4: ( 7 , 0 ) . If ( x ′′ , y ′′ ) = ( 7 , 0 ) , then ( x , y ) = ( 8 − 7 , − 0 − b ) = ( 1 , − b ) .
Option 5: ( 7 , − 5 ) . If ( x ′′ , y ′′ ) = ( 7 , − 5 ) , then ( x , y ) = ( 8 − 7 , − ( − 5 ) − b ) = ( 1 , 5 − b ) .

Analyze possible pairs Notice that if we choose options ( 1 , 1 ) and ( 7 , 0 ) , we have ( 7 , − 1 − b ) and ( 1 , − b ) as vertices of the final image.

If we choose options ( 7 , − 5 ) and ( 7 , 0 ) , we have ( 1 , 5 − b ) and ( 1 , − b ) as vertices of the pre-image.
If we choose options ( 1 , 1 ) and ( 7 , − 5 ) , we have ( 7 , − 1 − b ) and ( 1 , 5 − b ) as vertices of the pre-image.

Determine the correct options Let's analyze options ( 1 , 1 ) and ( 7 , 0 ) further. If ( 1 , 1 ) and ( 7 , 0 ) are vertices of the final image, then the corresponding vertices of the original trapezoid are ( 8 − 1 , − 1 − b ) = ( 7 , − 1 − b ) and ( 8 − 7 , − 0 − b ) = ( 1 , − b ) . So, the original trapezoid has vertices ( 7 , − 1 − b ) and ( 1 , − b ) . Since we are looking for the coordinates of the vertices of the pre-image , the correct options are ( 1 , 1 ) and ( 7 , 0 ) .

State the final answer Therefore, the two ordered pairs that name the coordinates of vertices of the pre-image trapezoid A BC D are ( 1 , 1 ) and ( 7 , 0 ) .


Examples
Understanding transformations like reflections and translations is crucial in various fields. For instance, in computer graphics, these transformations are used to manipulate objects on the screen, such as rotating, scaling, or moving them. In architecture, reflections and translations can be used to create symmetrical designs or to repeat patterns in a building's facade. These concepts also form the basis for more advanced topics like linear algebra and group theory, which have applications in physics and engineering.

Answered by GinnyAnswer | 2025-07-08