To find the hyperbola of the function \( f(x) = \frac{x^2 - 4}{x^2 - x - 6} \), let's analyze and rewrite the function. We will first factor the numerator and the denominator.

1. **Factor the numerator**:
   \[
   x^2 - 4 = (x - 2)(x + 2)
   \]

2. **Factor the denominator**:
   \[
   x^2 - x - 6 = (x - 3)(x + 2)
   \]

3. **Rewrite the function**:
   \[
   f(x) = \frac{(x - 2)(x + 2)}{(x - 3)(x + 2)}
   \]

4. **Simplify the function** by canceling out the common factor \((x + 2)\) in the numerator and the denominator, noting that \(x \neq -2\) (since division by zero is undefined):
   \[
   f(x) = \frac{x - 2}{x - 3}, \quad x \neq -2
   \]

To understand the hyperbola, consider the simplified form:

### Asymptotes
Asymptotes are lines that the graph of the function approaches but never touches.

- **Vertical Asymptote**: Occurs where the denominator is zero and the numerator is non-zero. For \( f(x) = \frac{x - 2}{x - 3} \), the vertical asymptote is at:
  \[
  x - 3 = 0 \implies x = 3
  \]

- **Horizontal Asymptote**: For rational functions where the degree of the numerator and the denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. Both the numerator and denominator are first-degree polynomials with leading coefficient 1, so:
  \[
  y = \frac{1}{1} = 1
  \]

### Critical Points and Analysis
- **Zeros of the function**: The function is zero where the numerator is zero:
  \[
  x - 2 = 0 \implies x = 2
  \]
  Thus, \( f(2) = 0 \).

- **Behavior near the asymptotes**: Analyze the limits as \( x \) approaches the vertical asymptote \( x = 3 \) and as \( x \) approaches \( \pm \infty \):

  \[
  \lim_{{x \to 3^-}} f(x) = -\infty \quad \text{and} \quad \lim_{{x \to 3^+}} f(x) = +\infty
  \]
  
  \[
  \lim_{{x \to \pm \infty}} f(x) = 1
  \]

### Graph
The graph of \( f(x) = \frac{x - 2}{x - 3} \) is a hyperbola with the following characteristics:
- Vertical asymptote at \( x = 3 \)
- Horizontal asymptote at \( y = 1 \)
- A zero at \( x = 2 \)

Would you like a detailed step-by-step graphing or have any other questions?

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**Questions for further exploration:**

1. How do you find the intercepts of a rational function?
2. What is the general process to identify vertical and horizontal asymptotes?
3. Can you explain the significance of the behavior near the vertical asymptote?
4. How do you determine the end behavior of a rational function?
5. What happens to the graph of \( f(x) \) as \( x \to \pm \infty \)?
6. How do you analyze the domain of a rational function?
7. What other types of asymptotes can a function have, apart from vertical and horizontal?
8. How would you find and interpret slant (oblique) asymptotes?

**Tip:** When graphing rational functions, always start by finding and plotting the asymptotes, as they provide a framework for understanding the function's behavior.

Explore the hyperbola of the rational function f(x) = (x^2 - 4) / (x^2 - x - 6). Learn how to analyze asymptotes, critical points, and graphing techniques. Suitable for advanced high school students.

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