Question 1: Two-step transformations involve applying one transformation followed by another, such as a translation then a rotation. The order of these steps is crucial because changing it can lead to a different final position.

Answer: Order matters because transformations are not commutative; changing steps changes the result.

Question 1: Perform a translation of (3, 4) by (2, -3), then reflect across the x-axis. What are the coordinates of the final point?

Answer: (5, 1)

Question 2: Translate point (−1, 5) by (4, 2) and then rotate 90° clockwise about the origin. What are the new coordinates?

Answer: (7, -2)

Question 3: A figure is translated 5 units right and 3 units up, then scaled by a factor of 2 from the origin. What is the scale of the final transformation?

Answer: Scaling by 2 after translation

Question 1: A triangle is translated 3 units left, then rotated 180° about point (0,0). The original vertices are (2, 3), (4, 5), and (2, 5). What are the coordinates of the vertices after both transformations?

Answer: Vertices after transformations: (-2, -3), (-4, -5), (-2, -5).

Question 2: Explain why performing a reflection then a translation yields a different final position than translating then reflecting, using an example.

Answer: Because transformations are not commutative; the order affects the final position, e.g., reflecting across a line then translating results in a different point than translating then reflecting.

Question 1: A robot moves 4 meters forward, then turns right 90°, and moves another 3 meters. Describe the path and the final position relative to its starting point.

Answer: Path: forward then right turn; final position 4 meters forward, 3 meters to the right from start.

Question 2: A painting is shifted 2 meters up, then rotated 45° clockwise. How would this double transformation affect its position on the wall?

Answer: The painting moves upward and then rotates, changing its orientation and position relative to the original point.

Question 1: Construct a triangle with vertices at (1, 1), (4, 1), and (1, 4). Perform a dilation by factor 2 from the origin, then rotate 135° about the origin. What are the new vertices?

Answer: Vertices after dilation and rotation: (−1, 3), (−4, 4), (−3, 0).

Question 2: A shape is translated 6 units down, then reflected across the y-axis. If a point was originally at (5, 2), where will it be after both transformations?

Answer: Point after transformations: (−5, -2).

Question 1: A student claims that translating a figure then rotating it will give the same result as rotating first then translating. Is this correct? Explain why or why not, providing an example.

Answer: Incorrect; transformations are not commutative. For example, translating then rotating produces a different position than rotating then translating.

Question 2: A common mistake is performing two transformations in the wrong order. Given a point at (2,3), if a student reflects across the y-axis then translates 3 units right, what should be the correct final position? Identify the mistake in order.

Answer: Correct order: reflect first (−2, 3), then translate right 3 units to (1, 3). Mistake: doing translation before reflection changes the final position.