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Sagot :
To determine which statements about the translation from point [tex]\( V(-2, 3) \)[/tex] to point [tex]\( V'(-2, 7) \)[/tex] are true, we need to analyze the changes in the coordinates step-by-step.
1. Initial and Final Coordinates:
- The original coordinates of point [tex]\( V \)[/tex] are [tex]\( (-2, 3) \)[/tex].
- The coordinates of the translated point [tex]\( V' \)[/tex] are [tex]\( (-2, 7) \)[/tex].
2. Translation Components:
- To find the translation components, we calculate the differences in the x and y coordinates between the original and translated points.
- Change in x-coordinate: [tex]\(\Delta x = -2 - (-2) = 0 \)[/tex]
- Change in y-coordinate: [tex]\(\Delta y = 7 - 3 = 4 \)[/tex]
3. Evaluating the Statements:
- Statement 1: "The point moves two units left and four units up."
- This is false because the translation in the x-direction ([tex]\(\Delta x\)[/tex]) is 0, not -2 (it does not move left or right at all). The movement is only in the y-direction ([tex]\(\Delta y\)[/tex]) by 4 units upwards.
- Statement 2: "The transformation rule is [tex]\( (x, y) \rightarrow (x+0, y+4) \)[/tex]."
- This is true. The translation rule correctly describes that [tex]\( x \)[/tex] remains unchanged ( [tex]\( x + 0 \)[/tex] ), and [tex]\( y \)[/tex] increases by 4 units ( [tex]\( y + 4 \)[/tex] ).
- Statement 3: "The transformation is a vertical translation."
- This is true. Since the x-coordinate remains unchanged ([tex]\(\Delta x = 0\)[/tex]) and the y-coordinate changes ([tex]\(\Delta y = 4\)[/tex]), this is a vertical translation.
- Statement 4: "The image is four units to the left of the pre-image."
- This is false because the x-coordinates of the pre-image and the image are both -2, which means there is no horizontal movement.
- Statement 5: "The translation can be described as [tex]\( (x, y) \rightarrow (x-2, y+7) \)[/tex]."
- This is false. The correct translation rule as identified in Statement 2 is [tex]\( (x, y) \rightarrow (x+0, y+4) \)[/tex]. The described rule here [tex]\( (x-2, y+7) \)[/tex] does not match our observed translation components ([tex]\(\Delta x = 0\)[/tex] and [tex]\(\Delta y = 4\)[/tex]).
In summary, the true statements about the translation are:
- The transformation rule is [tex]\( (x, y) \rightarrow (x+0, y+4) \)[/tex].
- The transformation is a vertical translation.
1. Initial and Final Coordinates:
- The original coordinates of point [tex]\( V \)[/tex] are [tex]\( (-2, 3) \)[/tex].
- The coordinates of the translated point [tex]\( V' \)[/tex] are [tex]\( (-2, 7) \)[/tex].
2. Translation Components:
- To find the translation components, we calculate the differences in the x and y coordinates between the original and translated points.
- Change in x-coordinate: [tex]\(\Delta x = -2 - (-2) = 0 \)[/tex]
- Change in y-coordinate: [tex]\(\Delta y = 7 - 3 = 4 \)[/tex]
3. Evaluating the Statements:
- Statement 1: "The point moves two units left and four units up."
- This is false because the translation in the x-direction ([tex]\(\Delta x\)[/tex]) is 0, not -2 (it does not move left or right at all). The movement is only in the y-direction ([tex]\(\Delta y\)[/tex]) by 4 units upwards.
- Statement 2: "The transformation rule is [tex]\( (x, y) \rightarrow (x+0, y+4) \)[/tex]."
- This is true. The translation rule correctly describes that [tex]\( x \)[/tex] remains unchanged ( [tex]\( x + 0 \)[/tex] ), and [tex]\( y \)[/tex] increases by 4 units ( [tex]\( y + 4 \)[/tex] ).
- Statement 3: "The transformation is a vertical translation."
- This is true. Since the x-coordinate remains unchanged ([tex]\(\Delta x = 0\)[/tex]) and the y-coordinate changes ([tex]\(\Delta y = 4\)[/tex]), this is a vertical translation.
- Statement 4: "The image is four units to the left of the pre-image."
- This is false because the x-coordinates of the pre-image and the image are both -2, which means there is no horizontal movement.
- Statement 5: "The translation can be described as [tex]\( (x, y) \rightarrow (x-2, y+7) \)[/tex]."
- This is false. The correct translation rule as identified in Statement 2 is [tex]\( (x, y) \rightarrow (x+0, y+4) \)[/tex]. The described rule here [tex]\( (x-2, y+7) \)[/tex] does not match our observed translation components ([tex]\(\Delta x = 0\)[/tex] and [tex]\(\Delta y = 4\)[/tex]).
In summary, the true statements about the translation are:
- The transformation rule is [tex]\( (x, y) \rightarrow (x+0, y+4) \)[/tex].
- The transformation is a vertical translation.
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