Question 1: Given the column vector v = [3; -2], translate the point P(1, 4) using v. What are the coordinates of the translated point P'?

Answer: [4; 2]

Question 2: If a point Q(−1, 5) is translated by the vector v = [−4; 3], what are the coordinates of the image Q'?

Answer: [-5; 8]

Question 3: Construct a translation of point R(2, −3) using the vector v = [5; 4]. Write the new coordinates.

Answer: [7; 1]

Question 4: Plot the graph of y = 2x and then translate the point (2, 4) by the vector [−3; 1]. What are the new coordinates?

Answer: (-1, 5)

Question 1: A triangle with vertices A(1, 2), B(4, 3), and C(2, 5) is translated by the vector v = [−2; 3]. Write the coordinates of the new vertices and explain each step of your translation.

Answer: A'(-1, 5), B'(2, 6), C'(0, 8)

Question 2: Explain why adding the vector [3; -1] to a point (x, y) results in the translated point, and identify a common mistake students make when applying this translation.

Answer: Adding [3; -1] shifts the point by 3 units right and 1 unit down. A common mistake is to multiply the vector rather than add it, or to add the wrong components.

Question 1: A delivery robot moves from its starting position at (5, 7) to deliver a package. It moves by the vector [−2; 4]. What is its new position? Why might understanding translation be important in robotics?

Answer: (3, 11)

Question 1: Construct a parallelogram on the grid with vertices at (0, 0), (4, 0), (1, 3), and (−3, 3). Use a translation vector to produce a congruent parallelogram shifted 5 units right and 2 units up. Write the coordinates of the new vertices.

Answer: (5, 2), (9, 2), (6, 5), (2, 5)

Question 2: A square has vertices at (1, 1), (3, 1), (3, 3), and (1, 3). If the square is translated by the vector [−4; −2], what are the new vertices? Is the shape still a square? Explain.

Answer: (−3, −1), (−1, −1), (−1, 1), (−3, 1); Yes, translation preserves shape and size.

Question 1: A student translates point P(2, 5) using the vector [−2; 3], but writes the new point as (0, 8). Identify and explain the mistake.

Answer: They forgot to add the components properly; the correct translation is (0, 8), so their answer is correct. (No mistake).

Question 2: A student incorrectly applies the translation vector [4; −3] by adding the components in reverse order to point (−1, 2). What is the error, and what should be the correct translation?

Answer: They swapped the components; correct translation: (−1 + 4, 2 − 3) = (3, -1).