Question 1: Calculate the resulting coordinates if a point at (3, 2) is translated by the vector \,\begin{bmatrix} 4 \\ -1 \\end{bmatrix}.

Answer: 7, 1

Question 2: Construct the image of the point (1, 5) after applying the translation vector \,\begin{bmatrix} -2 \\ 3 \\end{bmatrix}.

Answer: (-1, 8)

Question 3: Plot the point (2, 3) on the grid. Now, translate it using the vector \,\begin{bmatrix} 0 \\ -4 \\end{bmatrix}. What are the new coordinates?

Answer: 2, -1

Question 4: Identify the translation vector that moved the point (6, 2) to (3, 5).

Answer: \begin{bmatrix} -3 \\ 3 \end{bmatrix}

Question 5: A point at (4, -2) is translated by \,\begin{bmatrix} 3 \\ 4 \\). What are the new coordinates?

Answer: 7, 2

Question 6: A triangle has vertices at (1, 2), (3, 4), and (5, 2). If it is translated by \,\begin{bmatrix} 2 \\ -3 \\), what are the coordinates of the translated vertices?

Answer: 3, -1; 5, 1; 7, -1

Question 7: A real-world scenario: A boat at position (10, 15) moves south by 5 units and east by 3 units. Represent this movement as a translation vector and find the new position.

Answer: \begin{bmatrix} 3 \\ -5 \end{bmatrix}, new position (13, 10)

Question 8: Construct a translation that shifts the point (2, 7) to (5, 3). What is the vector?

Answer: \begin{bmatrix} 3 \\ -4 \end{bmatrix}

Question 9: Identify the error in this translation: Moving point (4, 5) using vector \,\begin{bmatrix} 2 \\ 2 \end{bmatrix} results in (4, 5).

Answer: The translation should move the point to (6, 7). The error is that the vector was not added correctly.

Question 10: Challenge: Given the translation vector \,\begin{bmatrix} -3 \\ 4 \end{bmatrix}, if a point at (x, y) is translated to (x + a, y + b), find the values of a and b.

Answer: a = -3, b = 4