A triangle has vertices at $B(-3,0), C(2,-1), D(-1,2)$. Which transformation would produce an image with vertices $B^{\prime}(-2,1), C^{\prime}(3,2), D^{\prime}(0,-1)$?

A. $(x, y) \rightarrow(x,-y) \rightarrow(x+1, y+1)$
B. $(x, y) \rightarrow(-x, y) \rightarrow(x+1, y+1)$
C. $(x, y) \rightarrow(x,-y) \rightarrow(x+2, y+2)$
D. $(x, y) \rightarrow(-x, y) \rightarrow(x+2, y+2)$

Reflect the triangle across the x-axis: ( x , y ) → ( x , − y ) .
Translate the reflected triangle by adding 1 to both the x and y coordinates: ( x , − y ) → ( x + 1 , y + 1 ) .
Applying the transformation to the vertices B ( − 3 , 0 ) , C ( 2 , − 1 ) , D ( − 1 , 2 ) results in B ′ ( − 2 , 1 ) , C ′ ( 3 , 2 ) , D ′ ( 0 , − 1 ) .
Therefore, the correct transformation is: ( x , y ) → ( x , − y ) → ( x + 1 , y + 1 ) ​ .

Explanation

Problem Analysis We are given a triangle with vertices B ( − 3 , 0 ) , C ( 2 , − 1 ) , D ( − 1 , 2 ) , and we want to find the transformation that produces an image with vertices B ′ ( − 2 , 1 ) , C ′ ( 3 , 2 ) , D ′ ( 0 , − 1 ) . We are given four possible transformations, and we will test each one to see which one maps the original vertices to the transformed vertices.

Testing Transformation 1 Transformation 1: ( x , y ) → ( x , − y ) → ( x + 1 , y + 1 ) .
Applying this transformation to B ( − 3 , 0 ) : First, ( x , − y ) gives ( − 3 , 0 ) . Then, ( x + 1 , y + 1 ) gives ( − 3 + 1 , 0 + 1 ) = ( − 2 , 1 ) , which is B ′ ( − 2 , 1 ) .
Applying this transformation to C ( 2 , − 1 ) : First, ( x , − y ) gives ( 2 , 1 ) . Then, ( x + 1 , y + 1 ) gives ( 2 + 1 , 1 + 1 ) = ( 3 , 2 ) , which is C ′ ( 3 , 2 ) .
Applying this transformation to D ( − 1 , 2 ) : First, ( x , − y ) gives ( − 1 , − 2 ) . Then, ( x + 1 , y + 1 ) gives ( − 1 + 1 , − 2 + 1 ) = ( 0 , − 1 ) , which is D ′ ( 0 , − 1 ) .
Since this transformation maps B to B ′ , C to C ′ , and D to D ′ , this is the correct transformation.

Testing Other Transformations Transformation 2: ( x , y ) → ( − x , y ) → ( x + 1 , y + 1 ) .
Applying this transformation to B ( − 3 , 0 ) : First, ( − x , y ) gives ( 3 , 0 ) . Then, ( x + 1 , y + 1 ) gives ( 3 + 1 , 0 + 1 ) = ( 4 , 1 ) . This is not B ′ ( − 2 , 1 ) , so this transformation is incorrect. Transformation 3: ( x , y ) → ( x , − y ) → ( x + 2 , y + 2 ) .
Applying this transformation to B ( − 3 , 0 ) : First, ( x , − y ) gives ( − 3 , 0 ) . Then, ( x + 2 , y + 2 ) gives ( − 3 + 2 , 0 + 2 ) = ( − 1 , 2 ) . This is not B ′ ( − 2 , 1 ) , so this transformation is incorrect. Transformation 4: ( x , y ) → ( − x , y ) → ( x + 2 , y + 2 ) .
Applying this transformation to B ( − 3 , 0 ) : First, ( − x , y ) gives ( 3 , 0 ) . Then, ( x + 2 , y + 2 ) gives ( 3 + 2 , 0 + 2 ) = ( 5 , 2 ) . This is not B ′ ( − 2 , 1 ) , so this transformation is incorrect.

Final Answer The transformation that produces the image with vertices B ′ ( − 2 , 1 ) , C ′ ( 3 , 2 ) , D ′ ( 0 , − 1 ) is ( x , y ) → ( x , − y ) → ( x + 1 , y + 1 ) .


Examples
Transformations are used extensively in computer graphics and animation. For example, when creating a 3D model of a car, you might start with a basic shape and then apply a series of transformations (rotations, translations, scaling) to position the wheels correctly, adjust the height of the roof, and so on. Each transformation changes the coordinates of the vertices of the model, ultimately creating the final, detailed design. Understanding transformations allows designers to manipulate objects in a virtual space with precision.

Answered by GinnyAnswer | 2025-07-08