Final answer:
The correct transformations to obtain the purple triangle from the blue triangle could be a reflection over the x-axis followed by a rotation clockwise by 180 degrees about the origin, which is option C.
Step-by-step explanation:
To determine which transformations could have occurred to obtain the purple triangle from the blue triangle, we need to understand the descriptions provided for each of the possible transformations in the given options. When a shape is translated horizontally to the right side of the coordinate system, it is moved to the right by a certain number of units without changing its orientation. If a shape is reflected over the x-axis, it is flipped vertically over the x-axis, which means that a point (x, y) would become (x, -y) after reflection. A rotation clockwise about the origin is a turn around the origin in the direction that moves from the positive y-axis towards the positive x-axis. A rotation of 90° clockwise would position what was at the positive y-axis to align with the positive x-axis and vice versa.
Based on the descriptions above, if we start with a reflection over the x-axis and then apply a rotation clockwise by 180 degrees, we are effectively turning the shape upside down from its reflected position. Therefore, one possible transformation sequence to obtain the purple triangle from the blue triangle could be reflected over the x-axis then rotated clockwise 180º about the origin, which corresponds to option C. This is because the reflection inverts the shape vertically, and then the rotation ensures the positioning relative to the original is flipped on both axes.