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The given function is:  

\[
y = \frac{4x^2 + 12x + 1}{x - 3}
\]

This is a **rational function** where the numerator \( 4x^2 + 12x + 1 \) is a quadratic polynomial and the denominator \( x - 3 \) is a linear polynomial. Let's analyze it:

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### Key Steps to Analyze the Function:

1. **Domain**:  
   A rational function is undefined when the denominator equals zero.  
   - Solve \( x - 3 = 0 \), so \( x = 3 \) is not in the domain.  
   - The domain of \( y \) is all real numbers except \( x = 3 \).

2. **Vertical Asymptote**:  
   A vertical asymptote occurs when the denominator is zero and the numerator is non-zero.  
   - As \( x \to 3 \), the denominator \( x - 3 \to 0 \), so there is a **vertical asymptote** at \( x = 3 \).

3. **Simplification** (if possible):  
   Check if the numerator and denominator have any common factors:  
   The numerator \( 4x^2 + 12x + 1 \) cannot be factored to include \( x - 3 \), so it does not simplify further.

4. **End Behavior and Horizontal Asymptote**:  
   For large values of \( x \), the behavior of \( y \) depends on the degrees of the numerator and denominator:  
   - Degree of the numerator \( = 2 \) (from \( 4x^2 \))  
   - Degree of the denominator \( = 1 \) (from \( x \))  

   Since the degree of the numerator is **greater** than the denominator, there is no horizontal asymptote. Instead, the function has a **slant (oblique) asymptote**.

5. **Finding the Slant Asymptote**:  
   To find the slant asymptote, perform polynomial division of \( 4x^2 + 12x + 1 \) by \( x - 3 \):

   - Divide \( 4x^2 \div x = 4x \)  
   - Multiply back: \( 4x \cdot (x - 3) = 4x^2 - 12x \)  
   - Subtract: \( (4x^2 + 12x + 1) - (4x^2 - 12x) = 24x + 1 \)  

   - Divide \( 24x \div x = 24 \)  
   - Multiply back: \( 24 \cdot (x - 3) = 24x - 72 \)  
   - Subtract: \( (24x + 1) - (24x - 72) = 73 \)  

   The result of the division is:

   \[
   y = 4x + 24 + \frac{73}{x - 3}
   \]

   As \( x \to \infty \), the term \( \frac{73}{x - 3} \to 0 \), so the slant asymptote is:

   \[
   y = 4x + 24
   \]

---

### Summary:
1. **Domain**: \( x \in \mathbb{R}, x \neq 3 \)  
2. **Vertical Asymptote**: \( x = 3 \)  
3. **Slant Asymptote**: \( y = 4x + 24 \)  
4. **Simplification**: The function does not simplify further.  

Would you like a graph or further clarification on any step?  

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### Related Questions:
1. What are vertical and horizontal asymptotes for rational functions?  
2. How do you perform polynomial long division to find slant asymptotes?  
3. How does the degree of the numerator and denominator affect asymptotes?  
4. What happens to the graph near a vertical asymptote?  
5. How do you determine where a rational function is increasing or decreasing?  

### Tip:
When analyzing rational functions, always check for simplification before determining asymptotes.

y = (4x^2 + 12x + 1) / (x - 3); Analyze the rational function.

Learn how to analyze the rational function y = (4x^2 + 12x + 1) / (x - 3). Find its domain, vertical asymptotes, and slant asymptote through polynomial division. Suitable for high school math students in Grades 10-12.

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