Question 1: Write the vector notation for the translation moving a point from (3, 4) to (7, 9).

Answer: →v = (4, 5)

Question 2: If a point P has coordinates (x, y) and is translated by vector →v = (−2, 3), write the new coordinates in terms of x and y.

Answer: New point: (x - 2, y + 3)

Question 3: Express the translation of point (5, -1) by vector →v = (−3, 4) using vector notation.

Answer: →(5, -1) + (−3, 4)

Question 4: Construct the vector notation to translate the point (2, 3) to (−1, 7).

Answer: →v = (−3, 4)

Question 5: Explain what the notation →v = (a, b) represents in the context of translation.

Answer: It represents the vector that describes the horizontal and vertical shifts for translation, moving points by a units horizontally and b units vertically.

Question 6: Perform the translation of point (−4, 2) by vector →v = (6, −3). Write the new coordinates.

Answer: (2, -1)

Question 7: Given the vector notation →v = (x, y), describe how to find the coordinates of a translated point starting from (a, b).

Answer: Add x to the original x-coordinate and y to the original y-coordinate to find the new point.

Question 8: If a translation moves the point (1, 1) to (4, 5), what is the vector notation for this translation?

Answer: →v = (3, 4)

Question 9: Construct a translation vector that moves the point (0, 0) to (−2, 3).

Answer: →v = (−2, 3)

Question 10: Identify the mistake in interpreting the vector notation →v = (a, b) as a translation, and explain how to correct it.

Answer: A common mistake is confusing the vector with a position coordinate. To correct this, understand that →v = (a, b) shows the shift from the original point, not a new position.

Question 11: Given the translation vector →v = (−5, 2), what are the new coordinates of a point originally at (8, 4)?

Answer: (3, 6)

Question 12: Challenge: Construct a triangle with vertices at (1, 2), (4, 3), and (2, 5). Then, translate the entire triangle by vector →v = (−2, 4). Write the new coordinates of each vertex.

Answer: Vertices after translation: (−1, 6), (2, 7), (0, 9)