Let's analyze the presence and equations of horizontal asymptotes for the functions provided in the image you've uploaded:

Analysis

For rational functions, a horizontal asymptote is determined based on the degrees of the numerator and denominator:

1. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
2. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is $y = \frac{a}{b}$ where $a$ and $b$ are the leading coefficients of the numerator and the denominator respectively.
3. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Now, let's apply this to each function provided:

1. $k(x) = \frac{6x^3 + 2x + 3}{2x^2 - 11x + 4}$

• Degree of numerator = 3
• Degree of denominator = 2
• Result: No horizontal asymptote (N)

2. $r(x) = \frac{(2x-1)(5x+6)}{(x+3)(x-6)}$

• Degree of numerator = 2
• Degree of denominator = 2
• Leading coefficients: $10$ (numerator), $1$ (denominator)
• Result: Horizontal asymptote at $y = 10$ (Y)

3. $q(x) = \frac{(x^2-3)^2}{3x^3 + 4x^2 + 7}$

• Degree of numerator = 4
• Degree of denominator = 3
• Result: No horizontal asymptote (N)

4. $p(x) = \frac{(3x-1)^2}{2x^2 + 3x + 5}$

• Degree of numerator = 2
• Degree of denominator = 2
• Leading coefficients: $9$ (numerator), $2$ (denominator)
• Result: Horizontal asymptote at $y = \frac{9}{2}$ (Y)

5. $y = \frac{(x-2)(4-x)}{(x+3)^2}$

• Degree of numerator = 2
• Degree of denominator = 2
• Leading coefficients: $-1$ (numerator), $1$ (denominator)
• Result: Horizontal asymptote at $y = -1$ (Y)

6. $s(x) = \frac{(2x^2+3)^2}{(x^2+5)(x-2)}$

• Degree of numerator = 4
• Degree of denominator = 3
• Result: No horizontal asymptote (N)

Summary of Results

• $k(x)$: N
• $r(x)$: Y, $y = 10$
• $q(x)$: N
• $p(x)$: Y, $y = \frac{9}{2}$
• $y$ (function 8): Y, $y = -1$
• $s(x)$: N

Let me know if you want details about any of these functions or have other questions!

Here are some related questions:

1. What determines the presence of oblique asymptotes in rational functions?
2. How can the behavior at infinity influence the graph of a function with no horizontal asymptote?
3. How do you find vertical asymptotes and holes for these functions?
4. What impact does the location of zeros of the numerator have on the graph of a rational function?
5. How can the leading coefficients influence the end behavior of a rational function?

Tip: For more complex rational functions, sometimes it's useful to perform polynomial long division if the degrees of the numerator and denominator are close to better understand the function's behavior at infinity.