Question

a sequence of transformations is performed on δabc, resulting in δabc. both triangles are shown on the coordinate plane below. some transformations are listed below: - ( r_x ): a reflection across the x - axis - ( r_y ): a reflection across the y - axis - ( t_{-1, -7} ): a translation so that ( (x, y)\to(x - 1, y - 7) ) which of these describes the sequence of transformations performed on δabc that results in δabc? a. ( t_{-1, -7} ) followed by ( r_x ) b. ( t_{-1, -7} ) followed by ( r_y ) c. ( r_x ) followed by ( t_{-1, -7} ) d. ( r_y ) followed by ( t_{-1, -7} )

Explanation:

Step1: Identify coordinates of \( \triangle ABC \)

Let's find coordinates of \( A, B, C \) in \( \triangle ABC \). From the graph, \( A(1,6) \), \( B(5,2) \), \( C(2,2) \).

Step2: Analyze transformation options

Option C: \( R_x \) (reflection over x - axis) followed by \( T_{-1,-7} \)

• Step 2.1: Apply \( R_x \) to \( \triangle ABC \)

Reflection over x - axis: \( (x,y)\to(x, - y) \)
For \( A(1,6) \): \( A_1(1,-6) \)
For \( B(5,2) \): \( B_1(5,-2) \)
For \( C(2,2) \): \( C_1(2,-2) \)

• Step 2.2: Apply \( T_{-1,-7} \) (translation \( (x,y)\to(x - 1,y - 7) \)) to \( A_1,B_1,C_1 \)

For \( A_1(1,-6) \): \( A'=1 - 1,-6 - 7=(0,-13) \)? Wait, no, let's check the graph of \( \triangle A'B'C' \). Wait, maybe I made a mistake. Let's check option D.

Option D: \( R_y \) (reflection over y - axis) followed by \( T_{-1,-7} \)

• Step 2.3: Apply \( R_y \) to \( \triangle ABC \)

Reflection over y - axis: \( (x,y)\to(-x,y) \)
For \( A(1,6) \): \( A_1(-1,6) \)
For \( B(5,2) \): \( B_1(-5,2) \)
For \( C(2,2) \): \( C_1(-2,2) \)

• Step 2.4: Apply \( T_{-1,-7} \) (translation \( (x,y)\to(x - 1,y - 7) \)) to \( A_1,B_1,C_1 \)

For \( A_1(-1,6) \): \( A'=-1-1,6 - 7=(-2,-1) \)
For \( B_1(-5,2) \): \( B'=-5 - 1,2 - 7=(-6,-5) \)
For \( C_1(-2,2) \): \( C'=-2 - 1,2 - 7=(-3,-5) \)
Now, let's check the graph of \( \triangle A'B'C' \). From the graph, \( A'(-2,-1) \), \( B'(-6,-5) \), \( C'(-3,-5) \) which matches the coordinates after \( R_y \) followed by \( T_{-1,-7} \).

Let's re - check option C:
After \( R_x \) (reflection over x - axis) on \( A(1,6) \) we get \( (1,-6) \), then translation \( T_{-1,-7} \): \( (1 - 1,-6 - 7)=(0,-13) \), which does not match the graph.

Option A: \( T_{-1,-7} \) followed by \( R_x \)

• Translate \( A(1,6) \): \( (1 - 1,6 - 7)=(0,-1) \), then reflect over x - axis: \( (0,1) \), not matching.

Option B: \( T_{-1,-7} \) followed by \( R_y \)

• Translate \( A(1,6) \): \( (0,-1) \), then reflect over y - axis: \( (0,-1) \), not matching.

Answer:

D. \( R_y \) followed by \( T_{-1,-7} \)