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

$f(x,y)=y^2-x^2$

given,

$f(x,y)=y^2-x^2$

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

Which is an equation of two hyperbolas with a vertex on the -axis at localid="1649872245591" $x=\sqrt{3},-\sqrt{3}$

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

Which is an equation of two hyperbolas with a vertex on the -axis at localid="1649872254756" $x=\sqrt{2},-\sqrt{2}$

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

Which is an equation of two hyperbolas with a vertex on the -axis atlocalid="1649872264402" $x=1,-1$

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

Which is the equation of two straight lines passing through the origin.

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

Which is an equation of two hyperbolas with a vertex on the -axis at $y=1,-1$

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

Which is an equation of two hyperbolas with a vertex on the -axis at $y=\sqrt{2},-\sqrt{2}$

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

Which is an equation of two hyperbolas with a vertex on the -axis at $\sqrt{3},-\sqrt{3}$