Sketch the level curves f(x, y) = c of the following functions for c = −3, −2, −1, 0, 1, 2, and 3:

$f(x,y)=-\frac{2y}{x}$

given,

$f(x,y)=-\frac{2y}{x}$

$\frac{1}{2}$Then for $c=-3,-\frac{2y}{x}=-3\text{that is}2y-3x=0$

Which is the equation of a straight line passing through the origin with a slope $\frac{3}{2}$

For $c=-2,-\frac{2y}{x}=-2\text{that is}y-x=0$

Which is the equation of a straight line passing through the origin with slope 1

For $c=-1,-\frac{2y}{x}=-1\text{that is}2y-x=0$

Which is the equation of a straight line passing through the origin with a slope$\frac{1}{2}$

For $c=0,-\frac{2y}{x}=0thatisy=0$

Which is the x-axis

For $c=1,-\frac{2y}{x}=1\text{that is}2y+x=0$

Which is the equation of a straight line passing through the origin with a slope $\frac{-1}{2}$

For $c=2,-\frac{2y}{x}=2\text{that is}y+x=0$

Which is the equation of a straight line passing through the origin with slope $-1$

For $c=3,-\frac{2y}{x}=3\text{that is}2y+3x=0$

Which is the equation of a straight line passing through the origin with a slope$-\frac{3}{2}$