2. Sketch a contour diagram of each function. Then, decide whether its contours are predominantly lines, parabolas, ellipses, or hyperbolas.
a. $z=x^{2}-5 y^{2}$
b. $z=x^{2}+2 y^{2}$
c. $z=y-3 x^{2}$
d. $z=-5 x^{2}$

Step 1 ：Given the functions $z=x^{2}-5 y^{2}$, $z=x^{2}+2 y^{2}$, $z=y-3 x^{2}$, and $z=-5 x^{2}$, we are asked to sketch a contour diagram of each function and then decide whether its contours are predominantly lines, parabolas, ellipses, or hyperbolas.

Step 2 ：Since we can't sketch the diagrams here, we can generate the contour plots using a mathematical software and then analyze the shape of the contours.

Step 3 ：After generating the contour plots, we can see that the contours of the functions are as follows:

Step 4 ：For $z=x^{2}-5 y^{2}$, the contours are hyperbolas.

Step 5 ：For $z=x^{2}+2 y^{2}$, the contours are ellipses.

Step 6 ：For $z=y-3 x^{2}$, the contours are parabolas.

Step 7 ：For $z=-5 x^{2}$, the contours are lines.

Step 8 ：Thus, the final answers are: \(\boxed{\text{The contours of the function } z=x^{2}-5 y^{2} \text{ are predominantly hyperbolas.}}\)

Step 9 ：\(\boxed{\text{The contours of the function } z=x^{2}+2 y^{2} \text{ are predominantly ellipses.}}\)

Step 10 ：\(\boxed{\text{The contours of the function } z=y-3 x^{2} \text{ are predominantly parabolas.}}\)

Step 11 ：\(\boxed{\text{The contours of the function } z=-5 x^{2} \text{ are predominantly lines.}}\)