Answer
(i) $f\left( {x,y} \right) = - x + {y^2}$ matches Figure (B).
(ii) $g\left( {x,y} \right) = x + {y^2}$ matches Figure (A).
Work Step by Step
(i) We have $f\left( {x,y} \right) = - x + {y^2}$. By setting $x=a$ we fix the $x$-coordinate and obtain the vertical trace curve $f\left( {a,y} \right) = z = - a + {y^2}$ that lies in the plane parallel to the $yz$-plane. As we increase the value of $a$, the curve moves down. So, it matches Figure (B).
(ii) We have $g\left( {x,y} \right) = x + {y^2}$. By setting $x=a$ we fix the $x$-coordinate and obtain the vertical trace curve $g\left( {a,y} \right) = z = a + {y^2}$ that lies in the plane parallel to the $yz$-plane. As we increase the value of $a$, the curve moves up. So, it matches Figure (A).
