When the graph of the function $f ( x ) = 2 ^ { x }$ is translated by $m$ in the $x$-direction and by $n$ in the $y$-direction, the graph of the function $y = g ( x )$ is obtained. By this translation, point $\mathrm { A } ( 1 , f ( 1 ) )$ moves to point $\mathrm { A } ^ { \prime } ( 3 , g ( 3 ) )$. When the graph of the function $y = g ( x )$ passes through the point $( 0,1 )$, what is the value of $m + n$? [3 points]
(1) $\frac { 11 } { 4 }$
(2) 3
(3) $\frac { 13 } { 4 }$
(4) $\frac { 7 } { 2 }$
(5) $\frac { 15 } { 4 }$

On the coordinate plane, the graph of the exponential function $y = a ^ { x }$ is reflected about the $y$-axis, then translated 3 units in the $x$-direction and 2 units in the $y$-direction. The resulting graph passes through the point $( 1,4 )$. What is the value of the positive number $a$? [3 points]
(1) $\sqrt { 2 }$
(2) 2
(3) $2 \sqrt { 2 }$
(4) 4
(5) $4 \sqrt { 2 }$