The problem asks you to sketch the level curves of the function $h(x, y) = xy - x$ for three different values of $c$: $c = -2$, $c = 1$, and $c = 4$.

Step 1: Level Curve Equation

For level curves, we set the function $h(x, y) = xy - x$ equal to the constant $c$, meaning: $xy - x = c$ We can factor out $x$: $x(y - 1) = c$ So, the level curves are described by the equation: $x = \frac{c}{y - 1} \quad \text{(for} \ y \neq 1 \text{)}.$ This is a family of hyperbolas for different values of $c$.

Step 2: Plot for Each Value of $c$

For each given value of $c$, the equation of the level curve becomes:

• For $c = -2$: $x = \frac{-2}{y - 1}$
• For $c = 1$: $x = \frac{1}{y - 1}$
• For $c = 4$: $x = \frac{4}{y - 1}$

These are all hyperbolas that shift depending on the value of $c$.

Key Characteristics

• For $c = -2$, the curve will reflect across the origin relative to $c = 1$.
• For $c = 4$, the curve will be steeper and farther from the origin than the curve for $c = 1$.

Instructions for Sketching

1. Select the curve drawing tool.
2. Use the fact that $x = \frac{c}{y - 1}$ to plot points for each value of $c$.
3. Draw the hyperbolas for $c = -2$, $c = 1$, and $c = 4$.

Would you like more detailed guidance on how to sketch the curves, or do you have any specific questions?

Related Questions:

1. What is the significance of level curves in multivariable functions?
2. How do level curves of hyperbolas differ from those of ellipses?
3. Can we use polar coordinates to simplify the graphing of hyperbolic functions?
4. How does changing $c$ affect the geometry of the level curves in general?
5. What happens to the level curves if we add a constant to the function $h(x, y)$?

Tip: When sketching, always plot a few points for clarity, especially near the key value $y = 1$, where the curve has asymptotic behavior.