To determine the vertical and horizontal asymptotes from the graph of $y = f(x)$:

Vertical Asymptotes:

A vertical asymptote occurs where the function tends to infinity (positive or negative) as $x$ approaches a certain value. In the graph:

• As $x$ approaches $-2$ from both sides, the function tends to negative infinity. This indicates a vertical asymptote at $x = -2$.
• As $x$ approaches $3$, the function grows without bound as well. This shows a vertical asymptote at $x = 3$.

Horizontal Asymptotes:

A horizontal asymptote occurs when the function approaches a specific $y$-value as $x$ tends to infinity or negative infinity. In the graph:

• As $x$ approaches negative infinity, the function appears to approach $y = 1$. This indicates a horizontal asymptote at $y = 1$.
• As $x$ approaches positive infinity, the function appears to approach $y = -1$. This indicates a horizontal asymptote at $y = -1$.

Conclusion:

• Vertical asymptotes: $x = -2$ and $x = 3$
• Horizontal asymptotes: $y = 1$ and $y = -1$

Would you like any more details on this explanation?

Related Questions:

1. What is the difference between a vertical and horizontal asymptote?
2. How do you determine asymptotes algebraically from a rational function?
3. Why can't a function cross a vertical asymptote but can cross a horizontal asymptote?
4. How do you find oblique asymptotes, and are there any in this graph?
5. How would a logarithmic function's asymptotes appear differently compared to this graph?

Tip: For rational functions, vertical asymptotes occur at values of $x$ that make the denominator zero (without canceling with the numerator). Horizontal asymptotes depend on the degrees of the numerator and denominator.