Final answer:
The question seeks to identify the reflection and translation transformations that would map triangle ABC onto itself in the Cartesian coordinate system.
Step-by-step explanation:
The question is asking about a sequence of transformations that maps triangle ABC onto itself. Specifically, it involves a reflection followed by a translation. Considering the options given, we need to choose the correct reflection and translation that would result in triangle ABC being mapped onto itself.
Reflection across the x-axis followed by a translation 3 units to the right would not map ABC onto itself unless ABC was symmetric with respect to the x-axis and then moved horizontally.
Reflection across the y-axis followed by a translation 3 units to the left is essentially the same as the first option but mirrored on the y-axis.
Reflection across the y-axis followed by a translation 3 units upward maps ABC onto itself if ABC is symmetric with respect to the y-axis and is located below the y-axis.
Reflection across the x-axis followed by a translation 3 units downwards is similar to the third option but along the x-axis. This would map ABC onto itself if ABC is symmetric with respect to the x-axis and located above it.
The response to the question depends on the initial position of triangle ABC in the coordinate system. In a Cartesian coordinate system, a reflection is a transformation that flips a figure over a line. After reflection, a translation moves every point of a figure or space by the same distance in a given direction.