Answer
(a) $x^2=-2y$. $x^2=-4y, x^2=-16y$ and $x^2=-32y$
(b) See graphs and explanations below.
Work Step by Step
(a) Since the vertex is at the origin and the directrixes are parallel to the x-axis, we can write the general equation as $x^2=4py$ with directrix at $y=-p$. Given $y=\frac{1}{2}$ as the directrix, we have $p=-\frac{1}{2}$ and the equation becomes $x^2=-2y$. Similarly, the equations for others directrixes are $x^2=-4y, x^2=-16y$ and $x^2=-32y$ for drectrixes of $y=1, 4$ and $8$ respectively.
(b) See graphs, we can conclude that with the increasing of the absolution value of the directrix, the opening of the parabola will become wider.
