We have a translation T that transform (-2, 3) into (3,-1).
We can think of the translation as adding "a" units to the x-coordinate and "b" units to the y-coordinate.
So we can express it as:
[tex](x^{\prime},y^{\prime})=(x+a,y+b)[/tex]As the translated point is (3,-1) and the original point is (-2,3), we should have:
[tex]\begin{gathered} (3,-1)=(-2+a,3+b) \\ 3=-2+a\longrightarrow a=3+2=5 \\ -1=3+b\longrightarrow b=-1-3=-4 \end{gathered}[/tex]Knowing "a" and "b", we can describe the transformation as:
[tex](x^{\prime},y^{\prime})=(x+5,y-4)[/tex]Then, for the point (4,2) the translation will result in:
[tex]\begin{gathered} (x^{\prime},y^{\prime})=(4+5,2-4) \\ (x^{\prime},y^{\prime})=(9,-2) \end{gathered}[/tex]Answer: (9,-2) [Fourth option]