To determine the horizontal and vertical asymptotes of a rational function, we can follow these steps:

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as $x$ approaches positive or negative infinity. For a rational function $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials, the horizontal asymptote can be determined by comparing the degrees of $P(x)$ and $Q(x)$:

1. If the degree of $P(x)$ is less than the degree of $Q(x)$, the horizontal asymptote is $y = 0$.
2. If the degree of $P(x)$ is equal to the degree of $Q(x)$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ is the leading coefficient of $P(x)$ and $b$ is the leading coefficient of $Q(x)$.
3. If the degree of $P(x)$ is greater than the degree of $Q(x)$, there is no horizontal asymptote. Instead, there might be an oblique asymptote.

Vertical Asymptotes

Vertical asymptotes occur at values of $x$ that make the denominator zero, provided these values do not also make the numerator zero at the same points (i.e., provided the function does not have a hole at these points).

To find vertical asymptotes:

1. Set the denominator $Q(x)$ equal to zero and solve for $x$.
2. Ensure that these $x$ values do not also make the numerator $P(x)$ zero.

Analyzing the Given Graph

Based on the given graph:

• Vertical Asymptotes:

  • The graph approaches infinity near $x = 2$, indicating a vertical asymptote at $x = 2$.
  • The graph also has a vertical asymptote at $x = 4$, as the function approaches infinity there too.

• Horizontal Asymptotes:

  • As $x$ approaches positive or negative infinity, the function appears to approach $y = 0$, indicating a horizontal asymptote at $y = 0$.

So, for this function:

• Vertical asymptotes are at $x = 2$ and $x = 4$.
• Horizontal asymptote is at $y = 0$.

General Method to Find Asymptotes

1. Horizontal Asymptotes:
   • Compare the degrees of the numerator and denominator.
   • Use the rules mentioned to determine the horizontal asymptote.
2. Vertical Asymptotes:
   • Find the roots of the denominator.
   • Check these roots against the numerator to confirm they are not holes but true vertical asymptotes.

Would you like further details on this process or any specific questions? Here are some questions you might want to explore:

1. How do you determine if a rational function has an oblique asymptote?
2. Can you explain how to handle holes in the graph of a rational function?
3. What happens to the graph of a rational function near a hole?
4. How can we confirm the behavior of a function near its asymptotes using limits?
5. What is the significance of the leading coefficients in determining asymptotes?

Tip: When analyzing the asymptotes of a rational function, always start by factoring both the numerator and the denominator to identify potential holes and vertical asymptotes.