Here is the sketch of the level curves for the function $f(x, y) = xy - x$ corresponding to $c = -5$, $c = -2$, and $c = 1$. The different colors represent the curves for each value of $c$:

• The blue curve corresponds to $c = -5$,
• The green curve corresponds to $c = -2$,
• The red curve corresponds to $c = 1$.

I have also highlighted specific points on the curves to illustrate them clearly. Let me know if you'd like more details or if you have any questions!

Here are some follow-up questions to deepen your understanding:

1. How does changing the value of $c$ affect the shape of the hyperbola?
2. What happens to the curve as $x$ approaches zero?
3. Can you find the asymptotes of these hyperbolas?
4. What does the graph look like if $c = 0$?
5. How does the graph change if the function is modified to $f(x, y) = xy + x$?

Tip: Level curves can provide valuable insight into the behavior of a multivariable function without needing to visualize the full 3D surface.