Answer
(a) center $C(0, 3)$, vertices $V_1(0,0)$ and $V_2(0,6)$, foci $F(0, 3\pm3\sqrt {2})$, asymptotes $y=\pm x+3$
(b) See graph.
Work Step by Step
(a) Step 1. Rewrite the given equation as $-(x^2)+(y^2-6y+9)=9$ or $-(x^2)+(y-3)^2=9$ which gives $\frac{(y-3)^2}{9}-\frac{(x)^2}{9}=1$
Step 2. Identify the center as $C(0, 3)$
Step 3. Identify $a=3, b=3, c=\sqrt {9+9}=3\sqrt {2}$
Step 4. The original vertices are $(0, \pm3)$, vertices after the shift $V(0, 3\pm3)$ or $V_1(0,0)$ and $V_2(0,6)$
Step 5. Original foci $(0, \pm3\sqrt {2})$, foci after shift $F(0, 3\pm3\sqrt {2})$
Step 6. Original asymptotes $y=\pm x$, new asymptotes$y=\pm x+3$
(b) See graph.
