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The following graph represents a rational function.  Identify the horizontal and vertical asymptotes which the vertical lines are: t(x = 1) and (x = 5),. Explain how you would find horizontal and vertical asymptotes of any rational function mathematically.

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.

Learn how to identify vertical and horizontal asymptotes in rational functions using a step-by-step approach. Understand the rules for determining asymptotes based on the behavior of the numerator and denominator degrees. Suitable for high school students studying advanced algebra and calculus concepts.

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