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The graph provided corresponds to the function \( f(x) = \frac{x^2 - 5x}{x^2 - 9} \).

We can analyze this function step by step:

### 1. **Vertical Asymptotes:**
   Vertical asymptotes occur where the denominator is zero but the numerator is not zero. The denominator \( x^2 - 9 \) equals zero when:
   \[
   x^2 - 9 = (x - 3)(x + 3) = 0
   \]
   So, \( x = 3 \) and \( x = -3 \) are the vertical asymptotes.

### 2. **Horizontal Asymptote:**
   To determine the horizontal asymptote, we analyze the behavior of the function as \( x \to \infty \) or \( x \to -\infty \). Since the degrees of the numerator and the denominator are both 2, the horizontal asymptote is determined by the ratio of the leading coefficients:
   \[
   \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{x^2 - 5x}{x^2 - 9} = \frac{1}{1} = 1
   \]
   Thus, the horizontal asymptote is \( y = 1 \).

### 3. **Intercepts:**
   - **X-intercepts:** These occur where the numerator is zero. Set \( x^2 - 5x = 0 \):
     \[
     x(x - 5) = 0
     \]
     So, the x-intercepts are at \( x = 0 \) and \( x = 5 \).
   - **Y-intercept:** The y-intercept occurs where \( x = 0 \). Substituting \( x = 0 \) into the function:
     \[
     f(0) = \frac{0^2 - 5(0)}{0^2 - 9} = \frac{0}{-9} = 0
     \]
     Thus, the y-intercept is at \( (0, 0) \), which is consistent with the x-intercept at \( x = 0 \).

### 4. **Behavior near the asymptotes:**
   - Near \( x = 3 \) and \( x = -3 \), the graph tends towards \( \pm \infty \), indicating the presence of vertical asymptotes at these points.
   - The graph approaches \( y = 1 \) as \( x \to \infty \) or \( x \to -\infty \), which is the horizontal asymptote.

From the graph and the analysis, we conclude that the key points of the function \( f(x) \) are as follows:

- **Vertical Asymptotes:** \( x = 3 \) and \( x = -3 \)
- **Horizontal Asymptote:** \( y = 1 \)
- **X-intercepts:** \( x = 0 \) and \( x = 5 \)
- **Y-intercept:** \( (0, 0) \)

Would you like further details or have any other questions?

### Additional Questions:
1. How do you determine the horizontal asymptote of a rational function in general?
2. Why do vertical asymptotes occur at \( x = 3 \) and \( x = -3 \)?
3. How can we identify holes (removable discontinuities) in a rational function?
4. What is the behavior of the function as \( x \to \pm \infty \)?
5. How do x-intercepts affect the graph of a rational function?

### Tip:
Always factor the numerator and denominator of a rational function to easily identify vertical asymptotes, holes, and x-intercepts!

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Explore the detailed analysis of the rational function f(x) = (x^2 - 5x) / (x^2 - 9), including the identification of vertical and horizontal asymptotes, and intercepts. This topic is suitable for high school students in Grades 10-12.

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