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The images you uploaded are of the same type, showing the rational function \( f(x) = \frac{2x^2 + 7x + 3}{x^2 - x - 12} \) and part (b) asking about vertical asymptotes.

To summarize part (b):

### Vertical Asymptotes

A rational function has vertical asymptotes where the denominator equals zero and the numerator does not. 

1. **Factor the denominator**:
   \[
   x^2 - x - 12 = (x - 4)(x + 3)
   \]
   This shows the function has potential vertical asymptotes at \( x = 4 \) and \( x = -3 \).

2. **Check the numerator at these values**:
   \[
   2x^2 + 7x + 3
   \]
   At \( x = 4 \), the numerator is \( 2(4)^2 + 7(4) + 3 = 63 \), which is non-zero. Hence, there is a **vertical asymptote** at \( x = 4 \).

   At \( x = -3 \), the numerator is \( 2(-3)^2 + 7(-3) + 3 = 0 \), so there's no vertical asymptote here, but a **hole** at \( x = -3 \).

Thus, **the vertical asymptote** is at \( x = 4 \), and the function has a **hole** at \( x = -3 \).

Would you like a deeper explanation of the limits or the behavior of the function around these points? Here are five follow-up questions:

1. What are the horizontal asymptotes for this function?
2. Can you explain how limits help determine the type of asymptote?
3. What does it mean graphically when there's a hole in the function?
4. How do you find the range of this function?
5. How can the function's end behavior be analyzed using limits?

**Tip**: Always check for both vertical asymptotes and holes in rational functions by testing where both the denominator and numerator are zero.

Consider the rational function f(x) = (2x^2 + 7x + 3) / (x^2 - x - 12). Does f(x) have any vertical asymptotes? If so, what are they? Show your analysis, justifying your answer using limits.

Explore the vertical asymptotes and holes of the rational function f(x) = (2x^2 + 7x + 3) / (x^2 - x - 12). Learn step-by-step how to factor the denominator, use limits to identify vertical asymptotes, and determine where the function has a hole. This detailed breakdown is perfect for high school students studying algebra and calculus.

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