Answer
(a) center $C(0, 3)$, vertices $V(0, 3\pm\sqrt 3)$, foci $F(0, 3\pm\sqrt {19})$, asymptotes $y=\pm\frac{\sqrt 3}{4} x+3$
(b) See graph.
Work Step by Step
(a) Step 1. Write the given equation as $\frac{(y-3)^2}{4}-\frac{(x)^2}{16}=1$
Step 2. Identify the center as $C(0, 3)$
Step 3. Identify $a=\sqrt 3, b=4, c=\sqrt {3+16}=\sqrt {19}$
Step 4. The original vertices are $(0, \pm\sqrt 3)$, vertices after the shift $V(0, 3\pm\sqrt 3)$
Step 5. Original foci $(0, \pm\sqrt {19})$, foci after shift $F(0, 3\pm\sqrt {19})$
Step 6. Original asymptotes $y=\pm\frac{\sqrt 3}{4} x$, new asymptotes $y=\pm\frac{\sqrt 3}{4} x+3$
(b) See graph.
