Let's analyze the given translation which moves point [tex]\( V(-2, 3) \)[/tex] to [tex]\( V'(-2, 7) \)[/tex]. Based on this translation, we'll evaluate each statement.
1. The point moves two units left and four units up.
To determine the movement, we calculate the difference in the coordinates:
- The x-coordinate change: [tex]\( \Delta x = -2 - (-2) = 0 \)[/tex]
- The y-coordinate change: [tex]\( \Delta y = 7 - 3 = 4 \)[/tex]
Therefore, the point does not move horizontally (left or right), but it does move vertically (up) by 4 units. Hence, this statement is false.
2. The transformation rule is [tex]\((x, y) \rightarrow (x + 0, y + 4)\)[/tex].
The transformation rule can be determined by the changes in coordinates:
- Since there is no change in the x-coordinate: [tex]\( x \rightarrow x \)[/tex]
- The y-coordinate increases by 4 units: [tex]\( y \rightarrow y + 4 \)[/tex]
Thus, the transformation rule is indeed [tex]\((x, y) \rightarrow (x, y + 4)\)[/tex], making this statement true.
3. The transformation is a vertical translation.
Since we determined that the point only moved vertically (by 4 units up) without any horizontal movement, this statement is true.
4. The image is four units to the left of the pre-image.
Since there is no movement in the x-direction (horizontal movement is [tex]\(\Delta x = 0\)[/tex]), the image is directly above the pre-image, not to the left. Thus, this statement is false.
5. The translation can be described as [tex]\((x, y) \rightarrow (x - 2, y + 7)\)[/tex].
This description does not match the actual translation. We have:
- No change in x: so, [tex]\( x \rightarrow x \)[/tex], not [tex]\( x - 2 \)[/tex]
- An increase of 4 in y, not 7
Therefore, this translation rule is incorrect, making this statement false.
Summarizing the true statements:
- The transformation rule is [tex]\((x, y) \rightarrow (x + 0, y + 4)\)[/tex]
- The transformation is a vertical translation.
