Question 1: Express the translation of point P(3, 5) by vector v = (2, -3) using vector notation. What are the coordinates of the translated point?

Answer: (5, 2)

Question 2: If a point Q has coordinates (x, y) and is translated by vector  = (-1, 4), write an expression for the coordinates of the translated point in vector notation.

Answer: Q +  = (x - 1, y + 4)

Question 3: Calculate the new point when the point (7, -2) is translated by the vector  = (3, 5). Show your working.

Answer: (10, 3)

Question 4: Construct a translation on the grid that moves point A(1, 2) to point B(4, 6). Write the translation vector in vector notation.

Answer:  = (3, 4)

Question 5: Explain why the vector notation  = (x_2 - x_1, y_2 - y_1) is useful for representing translations between two points.

Answer: It provides a clear, formulaic way to find the translation needed to move from one point to another, directly giving the vector components.

Question 6: Given a point R(2, -3) and a translation vector  = (-4, 2), find the coordinates of the translated point using vector notation.

Answer: (-2, -1)

Question 7: A translation moves the point (5, 7) to (9, 2). Write the translation vector, and verify the translation by calculation.

Answer: Translation vector: (4, -5); Verified: (5+4, 7-5) = (9, 2)

Question 8: Construct a scenario where a translation vector is incorrectly written as (x, y) = (3, 5) instead of using vector notation. Identify and correct the mistake.

Answer: Incorrect: (x, y) = (3, 5) implies a point, not a vector. Correct notation:  = (3, 5).

Question 9: For the points A(1, 2) and B(4, 5), write the translation vector that moves A to B. Then, use it to find the image of point C(6, 1).

Answer: Vector: (3, 3); C translated: (6+3, 1+3) = (9, 4)

Question 10: Explain how vector notation simplifies the process of performing multiple translations on a point, with an example.

Answer: Using vector notation, multiple translations can be combined by adding their vectors, simplifying calculations. For example, translating by (2, 1) then by (3, -2) is equivalent to translating by (5, -1).

Question 11: Identify the common mistakes students make when writing translation vectors and suggest how to avoid them.

Answer: Common mistakes include confusing points with vectors, mixing coordinate order, or writing the vector as a point. To avoid this, students should remember vectors are direction and magnitude, not points.

Question 12: Challenge: Given points D(2, 3), E(7, 8), and F(4, 1), find a translation vector that moves D to E and then F to a new point. Write the combined translation vector.

Answer: D to E: (5, 5); F to new point: (4+5, 1+5) = (9, 6); Combined vector: (5, 5)