1. ## z^2 = x^3+2xy^2-x

For the surface defined by z^2 = x^3+2xy^2-x, describe as best you can the traces in planes parallel to the yz-plane.

Please teach me how to do this question. Thank you very much.

2. Hello, Jenny!

For the surface defined by $\displaystyle z^2 \:= \:x^3+2xy^2-x$,
describe as best you can the traces in planes parallel to the yz-plane.

A plane parallel to the yz-plane has the equation: .$\displaystyle x \,=\,k$ (constant).

In the equation let $\displaystyle x = k\!:\;\;z^2\:=\:k^3 + 2ky^2 - k$

. . So we have: .$\displaystyle z^2\:=\:2ky^2 + k^3 - k$

$\displaystyle 2k$ is a constant; call it $\displaystyle a$.
$\displaystyle k^3 - k$ is a constant; call it $\displaystyle b$.

The equation has the form: .$\displaystyle z^2\:=\:ay^2 + b\quad\Rightarrow\quad z^2 - ay^2 \:=\:b$

The traces are hyperbolas.

If $\displaystyle b$ is positive, the hyperbola is "vertical": $\displaystyle \begin{array}{cc}\cup \\ \cap\end{array}$

If $\displaystyle b$ is negative, the hyperbola is "horizontal": $\displaystyle \supset\;\subset$

If $\displaystyle b = 0$, we have a pair of intersecting lines: $\displaystyle z \:=\:\pm\sqrt{a}\,y$