A segment with endpoints at $A(2, -2)$ and $B(14, 4)$ is extended through $B$ to point $C$. If $BC = \frac{1}{3} \cdot AB$,

To get the point that divides AB so that AP:PB = r, you multiply the coordiantes of A by r and the coordinates of B by 1 - r, and add them.  For example, for r = 1/2, you just take the average of the coordinates of A and the coordinates of B.

So for point C, we take r = 1 + 1/3 = 4/3.  So

C = 4/3 (2,-2) + (-1/3) (14,4) = (-2,-4).

Thanks for trying but the answer is (18,6)