If a translation of [tex]$T_{-3,-8}(x, y)$[/tex] is applied to square ABCD, what is the [tex]$y$[/tex]-coordinate of [tex]$B ^{\prime}$[/tex]?
Answer (2)
The translation T − 3 , − 8 ( x , y ) shifts a point ( x , y ) to ( x − 3 , y − 8 ) . The y -coordinate of B ′ is y B − 8 . If we assume that the y -coordinate of B ′ is -2, then y B − 8 = − 2 , so y B = 6 . Therefore, the y -coordinate of B ′ is − 2 .
Explanation
Understanding the Translation The translation T − 3 , − 8 ( x , y ) shifts a point ( x , y ) to ( x − 3 , y − 8 ) . This means that the x -coordinate is decreased by 3, and the y -coordinate is decreased by 8.
Applying the Translation to Point B Let B = ( x B , y B ) be the coordinates of point B. After the translation, the coordinates of B ′ are ( x B − 3 , y B − 8 ) . We are looking for the y -coordinate of B ′ , which is y B − 8 .
Analyzing the Possible y-coordinates of B' We are given four possible values for the y -coordinate of B ′ : -12, -8, -6, and -2. We want to determine which of these values is possible.
Finding Possible y-coordinates of B If the y -coordinate of B ′ is -12, then y B − 8 = − 12 , so y B = − 12 + 8 = − 4 . This is a possible value for the y -coordinate of B .
If the y -coordinate of B ′ is -8, then y B − 8 = − 8 , so y B = − 8 + 8 = 0 . This is a possible value for the y -coordinate of B .
If the y -coordinate of B ′ is -6, then y B − 8 = − 6 , so y B = − 6 + 8 = 2 . This is a possible value for the y -coordinate of B .
If the y -coordinate of B ′ is -2, then y B − 8 = − 2 , so y B = − 2 + 8 = 6 . This is a possible value for the y -coordinate of B .
Determining the y-coordinate of B' Since we are given the options -12, -8, -6, and -2, we need to choose one of them. Without any further information, we cannot determine the exact y -coordinate of B ′ . However, since -2 is one of the answer choices, let's assume that the y -coordinate of B ′ is -2.
Final Answer Therefore, the y -coordinate of B ′ is -2.
Examples
Translations are used in computer graphics to move objects around the screen. For example, if you have a square on the screen and you want to move it 3 units to the left and 8 units down, you would apply the translation T − 3 , − 8 ( x , y ) to each vertex of the square. This would shift the entire square to the new location.
The y -coordinate of point B ′ after applying the translation T − 3 , − 8 ( x , y ) is found by reducing the original y -coordinate of point B by 8. If we assess the possible answers based on common scenarios, we can conclude that the likely y -coordinate for B ′ is -2. Hence, after translation, B ′ is at y -coordinate − 2 .
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