Graphing a function with two variables

Question

The function $f(x,y)=y^2-x^2$ has the geometric figure of a hyperbolic paraboloid.

How do I graph its curves if a) $x=-50$, b) $x=5$, c) $y=-5$, d) $y=5$?

Answer 1

You will want to draw an $xyz$-axis and then plot the level curves for each of the values of $x$ and $y$ on the $yz$- and $xz$-planes, respectively.

So, for $x=-50$ and $x=5$ you will substitute these values into the equation and then sketch the resulting 2D curve. For example, substituting $x=5$ into the $f(x,y)$ gives $$z=y^2-25$$ which is a parabola in the $yz$-plane with y-intercepts at $y=5$ and $y=-5$. Similarly, substituting $y=5$ into $f(x,y)$ gives $$z=25-x^2$$ which is a parabola in the $xz$-plane. You would then annotate each level curve with the corresponding value of $x$ or $y$. Does this help?