Example 5. Given $z=y^{2}-2 x$, when $|x+y| \leqslant 1$, find $z_{max}$ and $z_{min}$.
Solve for the domain of $x$ and $y$ (Figure 5), which is a square with vertices at A, B, $A^{\prime}$, and $B^{\prime}$. By the same method, we can derive
$$
\begin{array}{l}
z_{\text {max }}=2, \\
z_{\text {min }}=-2 .
\end{array}
$$

From examples 3, 4, and 5, it can be seen that for a general bivariate function
$z=a x^{2}+b x y+c y^{2}+d x+$
$e y+f \quad\left(a^{2}+b^{2}+c^{2} \neq 0\right)$,
by coordinate transformation in analytic geometry, it can always be reduced to one of the functions in examples 3, 4, and 5 (degenerate cases excluded). If the closed domain D is simple, then this analytical method is effective. This method can also be used to study the extremum problems of irrational functions such as $z=\sqrt{x^{2}+y^{2}}$, $z=\sqrt{\mathrm{x}^{2}+\frac{\mathrm{y}^{2}}{\mathrm{a}^{2}} \pm}, \quad z=\sqrt{\mathrm{y}^{2}}-2 \mathrm{px}$. Readers are encouraged to construct their own functions and closed regions to appreciate the ideas and methods of solving extremum problems of bivariate functions using this approach.