Question

name: _______________
representing and combining transformations
evidence of student success
assessment activity
part 1
which of the following describes the sequence of transformations shown?
i. reflect across the x-axis and then rotate 90° clockwise around the origin.
ii. rotate 90° degrees around the origin and then translate right.
iii. reflect across the x-axis and then reflect across the y-axis.
iv. translate up and then rotate 90° counterclockwise about the origin.

Explanation:

Step1: Analyze Transformation from A to A'

To get from triangle \( A \) to \( A' \), we observe the vertical movement. Reflecting across the \( x \)-axis would invert the \( y \)-coordinates. For a point \((x, y)\) in \( A \), reflecting over \( x \)-axis gives \((x, -y)\). Then, translating up (or considering the vertical shift) aligns with moving from the lower part to the upper part (since \( A \) is at \( y \approx -5 \) and \( A' \) at \( y \approx 3 \), but more accurately, reflecting over \( x \)-axis (flips \( y \)) and then rotating? Wait, no, let's check each option:

- Option I: Reflect across \( x \)-axis (changes \( y \) to \( -y \)) then rotate \( 90^\circ \) clockwise. A \( 90^\circ \) clockwise rotation matrix is \( \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \), but let's take a vertex of \( A \), say \( (4, -5) \) (approximate). Reflect over \( x \)-axis: \( (4, 5) \). Rotate \( 90^\circ \) clockwise: \( (5, -4) \), which doesn't match \( A'' \) or \( A' \). Wait, maybe I mixed up. Wait, \( A \) to \( A' \): \( A \) is at the bottom right, \( A' \) is at the top right. So reflecting over \( x \)-axis (flips \( y \) from negative to positive) would take \( A \) (with \( y \)-coordinates negative) to \( A' \) (with \( y \)-coordinates positive). Then, from \( A' \) to \( A'' \): rotating \( 90^\circ \) clockwise? Wait, no, the options are about the sequence to get from \( A \) to \( A'' \)? Wait, the diagram has \( A \), \( A' \), \( A'' \). Wait, the question is "the sequence of transformations shown"—probably from \( A \) to \( A'' \), or \( A \) to \( A' \) to \( A'' \)? Wait, let's re-express:

Wait, let's take a vertex of \( A \): let's say the rightmost vertex of \( A \) is \( (4, -5) \), bottom vertex \( (2, -5) \), top vertex \( (4, -3) \) (approx).

- Option IV: Translate up (so add to \( y \)-coordinate) then rotate \( 90^\circ \) counterclockwise. Translating up: suppose we translate \( A \) up by 8 units (from \( y=-5 \) to \( y=3 \), which is \( A' \)). Then rotating \( 90^\circ \) counterclockwise. A \( 90^\circ \) counterclockwise rotation matrix is \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \). Take a vertex of \( A' \), say \( (4, 3) \) (rightmost), \( (2, 3) \) (leftmost), \( (4, 5) \) (top). Rotating \( (4, 3) \) \( 90^\circ \) counterclockwise: \( (-3, 4) \), which is close to \( A'' \)'s vertex (which is at \( x=-3, y=2 \)? Wait, maybe my coordinates are off. Let's look at the grid:

\( A \) is in the fourth quadrant (positive \( x \), negative \( y \)), \( A' \) is in the first quadrant (positive \( x \), positive \( y \)), \( A'' \) is in the second quadrant (negative \( x \), positive \( y \)).

- Option I: Reflect across \( x \)-axis (takes \( A \) (4th quad) to \( A' \) (1st quad, since \( y \) becomes positive)), then rotate \( 90^\circ \) clockwise (which would take 1st quad to 2nd quad? Wait, a \( 90^\circ \) clockwise rotation about origin: \((x, y) \to (y, -x)\). So a point in 1st quad \((x, y)\) (x>0, y>0) becomes \((y, -x)\), which is 4th quad? No, that's not right. Wait, \( 90^\circ \) clockwise: (x,y) → (y, -x). So (2,3) → (3, -2) (4th quad), (4,3) → (3, -4) (4th quad), (4,5) → (5, -4) (4th quad). But \( A'' \) is in 2nd quad (x negative, y positive). So that's not.

- Option II: Rotate \( 90^\circ \) around origin then translate right. Rotating \( A \) (4th quad) \( 90^\circ \) clockwise: (x,y)→(y, -x). So (4, -5)→(-5, -4) (3rd quad), (2, -5)→(-5, -2) (3rd quad), (4, -3)→(-3, -4) (3rd quad). Then translating right would move to 4th or 1st, not 2nd. Rotating \( 90^\circ \) counterclockwise: (x,y)→(-y, x). So (4, -5)→(5, 4) (1st quad), (2, -5)→(5, 2) (1st quad), (4, -3)→(3, 4) (1st quad). Then translating right: more right, but \( A'' \) is left. So no.

- Option III: Reflect across \( x \)-axis (takes \( A \) to \( A' \) (1st quad)), then reflect across \( y \)-axis (which takes (x,y)→(-x,y), so 1st quad to 2nd quad). Let's check: \( A \) vertex (4, -5) → reflect x-axis: (4, 5) → reflect y-axis: (-4, 5). \( A \) vertex (2, -5) → (2, 5) → (-2, 5). \( A \) vertex (4, -3) → (4, 3) → (-4, 3). Now, \( A'' \) is in 2nd quad, with vertices around (-4,5), (-2,5), (-4,2)? Wait, the diagram shows \( A'' \) with a vertex at (-3,2) maybe? Wait, maybe my coordinate estimation is off, but the key is: reflecting over x-axis (flips y) then over y-axis (flips x) would take 4th quad (A) to 2nd quad (A''), but \( A' \) is in 1st quad. Wait, the sequence is from A to A' to A''? Wait, the question is "the sequence of transformations shown"—probably A to A' (reflect x-axis) then A' to A'' (reflect y-axis), which is Option III? No, Option III is reflect x-axis then reflect y-axis, which would go from A (4th) to A' (1st) to A'' (2nd), which matches the diagram (A in 4th, A' in 1st, A'' in 2nd). Wait, but let's check Option IV: Translate up (A to A' is translating up 8 units, since y goes from -5 to 3, difference 8) then rotate 90° counterclockwise. Translating A up 8: (4, -5)→(4, 3), (2, -5)→(2, 3), (4, -3)→(4, 5) (which is A'). Then rotating 90° counterclockwise: (x,y)→(-y, x). So (4,3)→(-3,4), (2,3)→(-3,2), (4,5)→(-5,4). Which matches \( A'' \)'s position (in 2nd quad, x negative, y positive). Let's check the coordinates:

- \( A \) vertices: Let's take exact coordinates. Let's assume each grid square is 1 unit. \( A \): bottom left (2, -5), bottom right (4, -5), top right (4, -3). So three vertices: (2, -5), (4, -5), (4, -3).

- \( A' \): bottom left (2, 3), bottom right (4, 3), top right (4, 5). So translating up 8 units (from y=-5 to y=3: 3 - (-5) = 8). So translation up 8.

- Then rotating \( 90^\circ \) counterclockwise about origin: The rotation formula for \( 90^\circ \) counterclockwise is \( (x, y) \to (-y, x) \).

- For (2, 3): \( (-3, 2) \)
- For (4, 3): \( (-3, 4) \)
- For (4, 5): \( (-5, 4) \)

Now, looking at \( A'' \) in the diagram: it's a triangle in the second quadrant, with vertices around (-3,2), (-3,4), (-5,4) (matching the rotated points). So the sequence is: Translate up (from A to A') then rotate \( 90^\circ \) counterclockwise (from A' to A''), which is Option IV.

Wait, let's recheck other options:

- Option I: Reflect x-axis (A to A' would be reflect x-axis: (2, -5)→(2,5), (4, -5)→(4,5), (4, -3)→(4,3). Then rotate 90° clockwise: (2,5)→(5, -2), (4,5)→(5, -4), (4,3)→(3, -4). These are in 4th quad, not matching A'' (2nd quad). So I is wrong.

- Option II: Rotate 90° around origin (clockwise or counterclockwise?) then translate right. Rotating A (2,-5) 90° counterclockwise: (-(-5), 2)=(5,2); (4,-5)→(5,4); (4,-3)→(3,4). Then translating right: (5,2)→(6,2), etc. Not matching A''. Rotating 90° clockwise: (-5, -2), (-5, -4), (-3, -4). Translating right: (-4, -2), etc. Not matching. So II is wrong.

- Option III: Reflect x-axis (A to A' as (2,5), (4,5), (4,3)) then reflect y-axis: (-2,5), (-4,5), (-4,3). These are in 2nd quad, but \( A'' \) in the diagram has vertices at (-3,2), (-3,4), (-5,4), which is different from (-2,5), (-4,5), (-4,3). So III is wrong.

- Option IV: Translate up (A to A' by 8 units: (2,-5)→(2,3), (4,-5)→(4,3), (4,-3)→(4,5)) then rotate 90° counterclockwise: (2,3)→(-3,2), (4,3)→(-3,4), (4,5)→(-5,4). Which matches the position of \( A'' \) (since \( A'' \) is in 2nd quad with those x and y coordinates). So IV is correct.

Step2: Confirm the Correct Option

After analyzing each transformation sequence, Option IV (Translate up and then rotate \( 90^\circ \) counterclockwise about the origin) matches the transformation from \( A \) to \( A' \) (translation up) and then \( A' \) to \( A'' \) (rotation \( 90^\circ \) counterclockwise).

Answer:

IV. Translate up and then rotate \( 90^\circ \) counterclockwise about the origin.