Question

what composition of transformation is being demonstrated in the graph below?
reflection across the x - axis, then a reflection across the y - axis.
translation left 5 down, then a rotation 90 degrees clockwise around the origin.
rotation 180° around the origin, then reflection across the y - axis.
translation right 5 & up 1, then rotation 180 degrees around the origin.

Explanation:

1. Analyze the transformation of vertices: Observe the change in position of the vertices of the original triangle $\triangle ABC$ to the final triangle $\triangle A''B''C''$. A $180^{\circ}$ rotation around the origin changes the sign of both $x$ - and $y$ - coordinates of each point. Then, a reflection across the $y$ - axis changes the sign of the $x$ - coordinate of each point.
2. Check each option:

• Option 1: A reflection across the $x$ - axis changes the sign of the $y$ - coordinate and then a reflection across the $y$ - axis changes the sign of the $x$ - coordinate, which is not the case here.
• Option 2: A translation left 5 and down changes the position by subtracting 5 from the $x$ - coordinate and subtracting from the $y$ - coordinate, and a $90^{\circ}$ clock - wise rotation has a different transformation rule, not matching the graph.
• Option 3: First, a $180^{\circ}$ rotation around the origin flips the triangle to the opposite side of the origin. Then, a reflection across the $y$ - axis flips it to the final position as shown in the graph.
• Option 4: A translation right 5 and up 1 and then a $180^{\circ}$ rotation does not match the given transformation.

Answer:

Rotation $180^{\circ}$ around the origin, then reflection across the $y$-axis.