Obtain the standard form

$2 x^2\+y^2-4 x\+2 y-z\+4\=0$$\stackrel{\text{manipulate equation}}{\to }$${\left(y\+1\right)}^2\+2{\left(x-1\right)}^2\=z-1$

Obtain the equivalent of the surfaces in Figures 3.3.9(a, b)

$2 x^2\+y^2-4 x\+2 y-z\+4\=0$$\to$

Define $f$ so that the graph of $f\=0$ is a quadric surface

$f≔2 x^2\+y^2-4 x\+2 y-z\+4\:$

Complete the square and put $f$ into standard form

$\mathrm{Student}:-\mathrm{Precalculus}:-\mathrm{CompleteSquare}\left(f\=0\,\left[x\,y\right]\right)$

$- \left(-z\+1\right)$

$\mathrm{rhs}\left(\right)\=\mathrm{lhs}\left(\right)$

Obtain the equivalent of the surfaces drawn in Figures 3.3.9(a, b)

$\mathrm{plots}:-\mathrm{implicitplot3d}\left(f\=0\,x\=-1..3\,y\=-3..1\,z\=0..5\, \mathrm{scaling}\=\mathrm{constrained}\,\mathrm{axes}\=\mathrm{frame}\,\mathrm{orientation}\=\left[-50\,60\,0\right]\,\mathrm{style}\=\mathrm{surfacecontour}\,\mathrm{tickmarks}\=\left[4\,4\,5\right]\,\mathrm{view}\=0..5\,\mathrm{labels}\=\left[x\,y\,z\right]\,\mathrm{grid}\=\left[20\,20\,20\right]\,\mathrm{lightmodel}\=\mathrm{none}\right)$