To determine the horizontal and vertical asymptotes of a rational function, let's follow these steps:

Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function is zero and the numerator is not zero at those points. For a rational function $f(x) = \frac{P(x)}{Q(x)}$, the vertical asymptotes can be found by solving the equation $Q(x) = 0$.

From the given graph:

• The function appears to have vertical asymptotes at $x = 2$ and $x = 4$.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as $x$ approaches positive or negative infinity. For the rational function $f(x) = \frac{P(x)}{Q(x)}$, we compare 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 $a$ and $b$ are the leading coefficients of $P(x)$ and $Q(x)$, respectively.
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.

From the given graph:

• The function approaches $y = 0$ as $x$ approaches positive and negative infinity, indicating a horizontal asymptote at $y = 0$.

Revisiting the Graph and Analysis

Given your correction, the vertical asymptotes are at $x = 1$ and $x = 5$.

Steps to Find Asymptotes Mathematically

1. Horizontal Asymptotes:

   • Compare the degrees of the numerator and the denominator.
   • Use the rules mentioned to determine the horizontal asymptote.

2. Vertical Asymptotes:

   • Set the denominator $Q(x)$ equal to zero and solve for $x$.
   • Verify these $x$ values do not also make the numerator $P(x)$ zero.

Example Calculation

Let's consider a general rational function: $f(x) = \frac{x^2 - 1}{x^2 - 6x + 8}$.

1. Vertical Asymptotes:

   • Set the denominator equal to zero: $x^2 - 6x + 8 = 0$.
   • Solve for $x$: $(x - 2)(x - 4) = 0$, so $x = 2$ and $x = 4$.

2. Horizontal Asymptotes:

   • The degrees of the numerator and denominator are equal (both are 2).
   • The leading coefficients are both 1.
   • Therefore, the horizontal asymptote is $y = \frac{1}{1} = 1$.

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.