Answer
(a)
Vertex: $V(0,3)$
Focus: $F(-\frac{1}{2},3)$
Directrix: $x=\frac{1}{2}$
(b)
Work Step by Step
Equation of a parabola with horizontal axis and vertex at $(h,k)$:
$(y-k)^2=4p(x-h)$
$\frac{1}{2}(y-3)^2+x=0$
$\frac{1}{2}(y-3)^2=-x$
$(y-3)^2=-2(x-0)$
$h=0$
$k=3$
Vertex: $V(h,k)=V(0,3)$
$4p=-2$
$p=-\frac{1}{2}$
The given equation can be obtained by shifting
$y^2=-2x$
upward 3 units. In this equation:
Focus: $F(p,0)=F(-\frac{1}{2},0)$
Directrix: $x=-p=\frac{1}{2}$
Now, shift the focus 3 units upward:
Focus: $F(-\frac{1}{2},3)$
Directrix: $x=\frac{1}{2}$
