To graph the function

\[
f(x, y) = \frac{y - 2x}{3y + x + 10},
\]

we'll first examine its structure and behavior.

### Steps to Graph the Function

1. **Rewrite in terms of \( y \):** This function is implicitly defined in terms of both \( y \) and \( x \), making it more complex to isolate \( y \) alone.
  
2. **Analyze for Horizontal and Vertical Asymptotes:**
   - The function has a potential vertical asymptote where the denominator \( 3y + x + 10 = 0 \). Solving for \( y \) in terms of \( x \) here will give:
     \[
     y = -\frac{x + 10}{3}.
     \]
   - There is no clear horizontal asymptote since the function's form doesn't allow for easy limit-taking as \( x \) or \( y \) goes to infinity directly.

3. **Plot Sample Points:**
   To get an initial idea of the function's behavior, try a range of values for \( x \) and \( y \). These points can reveal regions where the function increases, decreases, or approaches infinity.

4. **Contour Plotting:**
   For functions of two variables, a contour plot (showing curves of constant \( f(x, y) \)) is a common way to visualize the relationship between \( x \), \( y \), and \( f(x, y) \).

Would you like me to generate a contour plot or provide a step-by-step approach to graph this function more manually?

### Follow-up Questions

1. How do you determine asymptotes for more complex rational functions?
2. Would you like assistance plotting this with specific values?
3. Do you need help with contour plotting for multivariable functions?
4. Would you like a breakdown of limits in this function?
5. Are you interested in the function's domain and range analysis?

**Tip:** For multivariable functions, contour plots and examining asymptotes can reveal critical behavior even without isolating one variable.

Graph the function \( \frac{y - 2x}{3y + x + 10} \)

Explore how to graph the function \( \frac{y - 2x}{3y + x + 10} \), analyzing vertical asymptotes, behavior, and contour plots. This guide breaks down rational function analysis for advanced high school or college students, covering essential multivariable calculus techniques.

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