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