Question

1. which sequence of transformations takes δa to its image, δb? image of a coordinate grid with triangles and axes options: a reflection over the x - axis and translation 2 units down; b reflection over the y - axis and translation 3 units down; c translation 2 units down and 90° rotation about the origin; d translation 13 units right and 90° rotation about the origin.

Explanation:

Step1: Analyze reflection over y - axis

A reflection over the \( y \) - axis changes the \( x \) - coordinate of a point \((x,y)\) to \((-x,y)\). We can check the key points of \(\triangle A\) and see how they transform.

Step2: Analyze translation 2 units down

A translation 2 units down changes the \( y \) - coordinate of a point \((x,y)\) to \((x,y - 2)\). After reflecting over the \( y \) - axis, we then move the points 2 units down. By comparing the coordinates of the vertices of \(\triangle A\) and \(\triangle B\), we find that first reflecting over the \( y \) - axis (changing the sign of the \( x \) - coordinate) and then translating 2 units down (subtracting 2 from the \( y \) - coordinate) maps \(\triangle A\) to \(\triangle B\).
Let's check the other options:

• Option A: Reflection over \( x \) - axis changes \((x,y)\) to \((x, - y)\), which does not match the transformation from \(\triangle A\) to \(\triangle B\).
• Option C: A \(90^{\circ}\) rotation about the origin would change the coordinates in a different way (e.g., \((x,y)\) to \((-y,x)\) for \(90^{\circ}\) counter - clockwise), and the translation and rotation combination does not match.
• Option D: A \(90^{\circ}\) rotation about the origin and translation 13 units right does not match the transformation from \(\triangle A\) to \(\triangle B\).

Answer:

B. reflection over the \( y \) - axis and translation 2 units down