We have a piecewise function consisting of a parabolic graph and a linear graph.
When the parabolic graph of
is elevated by 6 units, the new equation becomes
.
Given that the parabola aligns with this requirement, its equation can be expressed as

The parabola extends to x=3, where there is an open circle at this point. Thus, we need to use a less than sign.
Next, the line's equation is determined as 
Since there is a solid dot, we must employ the greater than or equal to sign.
Consequently, the piecewise function's equation is derived as

C is the right choice.