Given the coordinate transformation [tex]f(x, y)=(-2 x, y)[/tex] and the domain [tex](0,5),(8,-1)[/tex] and [tex](-6,4)[/tex], what is the range?
A. [tex](0,-5),(-8,-1),(3,-4)[/tex]
B. [tex](0,-5),(16,1),(-12,-4)[/tex]
C. [tex](0,5),(-16,-1),(12,4)[/tex]
D. [tex](0,5),(-8,1),(3,4)[/tex]
Answer (1)
Apply the transformation f ( x , y ) = ( − 2 x , y ) to the point ( 0 , 5 ) , resulting in ( 0 , 5 ) .
Apply the transformation f ( x , y ) = ( − 2 x , y ) to the point ( 8 , − 1 ) , resulting in ( − 16 , − 1 ) .
Apply the transformation f ( x , y ) = ( − 2 x , y ) to the point ( − 6 , 4 ) , resulting in ( 12 , 4 ) .
The range is therefore ( 0 , 5 ) , ( − 16 , − 1 ) , ( 12 , 4 ) .
Explanation
Understanding the Problem We are given a coordinate transformation f ( x , y ) = ( − 2 x , y ) and a domain consisting of three points: ( 0 , 5 ) , ( 8 , − 1 ) , and ( − 6 , 4 ) . Our goal is to find the range of this transformation, which means we need to apply the transformation to each point in the domain and find the resulting coordinates.
Applying the Transformation to (0, 5) Let's apply the transformation to the first point, ( 0 , 5 ) . The transformation f ( x , y ) = ( − 2 x , y ) means we multiply the x-coordinate by -2 and keep the y-coordinate the same. So, f ( 0 , 5 ) = ( − 2 ( 0 ) , 5 ) = ( 0 , 5 ) .
Applying the Transformation to (8, -1) Next, let's apply the transformation to the second point, ( 8 , − 1 ) . f ( 8 , − 1 ) = ( − 2 ( 8 ) , − 1 ) = ( − 16 , − 1 ) .
Applying the Transformation to (-6, 4) Now, let's apply the transformation to the third point, ( − 6 , 4 ) . f ( − 6 , 4 ) = ( − 2 ( − 6 ) , 4 ) = ( 12 , 4 ) .
Determining the Range So, the range of the transformation is the set of points ( 0 , 5 ) , ( − 16 , − 1 ) , and ( 12 , 4 ) . Comparing this to the given options, we see that the correct answer is ( 0 , 5 ) , ( − 16 , − 1 ) , ( 12 , 4 ) .
Examples
Coordinate transformations are used extensively in computer graphics and image processing. For example, if you want to mirror an image across the y-axis, you could use the transformation f ( x , y ) = ( − x , y ) . This transformation changes the sign of the x-coordinate, effectively flipping the image horizontally. Similarly, scaling transformations can be used to zoom in or out of an image. Understanding coordinate transformations is crucial for manipulating and displaying images correctly.
