We can look at examples of quadric surfaces with centers or vertices at points other than the origin by employing a change of coordinates of the form $\bar{x}=x-x_{0}, \bar{y}=y-y_{0}$, and $\bar{z}=z-z_{0} .$ This coordinate change simply puts the point $\left(x_{0}, y_{0}, z_{0}\right)$ of the $x y z$ -coordinate system at the origin of the $\bar{x} \bar{y} \bar{z}$ -coordinate system by a translation of axes. Then, for example, the surface having equation
$$
\frac{(x-1)^{2}}{4}+\frac{(y+2)^{2}}{9}+(z-5)^{2}=1
$$
can be identified by setting $\bar{x}=x-1, \bar{y}=y+2,$ and $\bar{z}=$ $z-5,$ so that we obtain
$$
\frac{\bar{x}^{2}}{4}+\frac{\bar{y}^{2}}{9}+\bar{z}^{2}=1
$$
which is readily seen to be an ellipsoid centered at (1,-2,5) of the $x$ yz-coordinate system. By completing the square in $x$, y, or z as necessary, identify and sketch the quadric surfaces.
$$
(x-1)^{2}+(y+1)^{2}=(z+3)^{2}
$$