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Here is the sketch of the graph for the function \( f(x) = \frac{4x - x^2}{x^2 - 5x + 4} \). Key features:

1. **Vertical asymptotes** at \( x = 1 \) and \( x = 4 \).
2. **Horizontal asymptote** at \( y = -1 \).
3. The function behaves as expected near these asymptotes, diverging to \( \infty \) or \( -\infty \), depending on the direction.

Would you like more details or have any questions? Here are some related questions to further explore:

1. How can we calculate the horizontal asymptote for any rational function?
2. What happens when \( x \) approaches the asymptotes \( x = 1 \) or \( x = 4 \)?
3. How can we find the domain of this function, considering the vertical asymptotes?
4. What are the critical points of the function where it changes concavity?
5. How do we find the intervals where this function is increasing or decreasing?

**Tip**: Vertical asymptotes occur where the denominator of a rational function equals zero, but the numerator doesn't.

Analyze the function f(x) = (4x - x^2) / (x^2 - 5x + 4) and sketch its graph, identifying the domain, asymptotes, and intersections.

Explore the analysis of the rational function f(x) = (4x - x^2) / (x^2 - 5x + 4), including the domain, vertical and horizontal asymptotes, and key points on the graph. Learn how to find roots and sketch the graph with step-by-step guidance. Ideal for high school students.

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