Answer
$ a.\qquad$
New center: $\quad (4,3)$
New foci: $\quad (4\pm\sqrt{7}, 3)$
New vertices: $\quad (0, 3)$ and $(8, 3)$
$ b.\qquad$
See graph.
Work Step by Step
$ a.\qquad$
$\displaystyle \frac{x^{2}}{16}+\frac{y^{2}}{9}=1$
Which is of the form :
Foci on the x-axis: $\quad \displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad (a\gt b)$
Center: $(0,0),\quad a=4,\quad b=2$
Center-to-focus distance: $\quad c=\sqrt{a^{2}-b^{2}}=\sqrt{16-9}=\sqrt{7}$
Foci: $\quad(\pm c, 0)= \quad(\pm\sqrt{7}, 0)$
Vertices: $\quad (\pm a, 0)= \quad (\pm 4, 0)$
The translations are such that $(x,y)\rightarrow(x',y')$, where
$\left\{\begin{array}{ll}
x'=x+4 & \text{... shift right}\\
y'=y+3 & \text{... shift up}
\end{array}\right.$
New center: $(4+0,3+0)=(4,3)$
New foci: $\quad(4\pm c, 0+3)= \quad(4\pm\sqrt{7}, 3)$
New vertices: $\quad (4\pm a, 0+3)= \quad (4\pm 4, 3)$
$ b.\qquad$
See graph.
