Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of $$ y=f(x) $$. $$ f(x)=\frac{x^{2}+9 x+18}{x^{2}+6 x+9} $$ A. The $$ y $$-intercept(s) is/are at $$ y=\square $$ (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no y-intercepts. Find any horizontal asymptotes of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, $$ \square . ~ $$ (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is $$ \square $$ and the bottom asymptote is $$ \square $$. (Type equations.) C. There are no horizontal asymptotes.

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### Step 1. Factor and Simplify

The function is  
$$
f(x)=\frac{x^2+9x+18}{x^2+6x+9}.
$$
Factor the numerator and the denominator.

- Numerator:  
  $$
  x^2+9x+18 = (x+3)(x+6).
  $$

- Denominator:  
  $$
  x^2+6x+9 = (x+3)^2.
  $$

Thus, for $$ x\neq -3 $$ (to avoid division by zero),
$$
f(x)=\frac{(x+3)(x+6)}{(x+3)^2}=\frac{x+6}{x+3}.
$$

Note that $$ x=-3 $$ is a point of removable discontinuity (a hole).

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### Step 2. Identify the $$ y $$-intercept

To find the $$ y $$-intercept, set $$ x=0 $$:
$$
f(0)=\frac{0+6}{0+3}=\frac{6}{3}=2.
$$

Thus, the $$ y $$-intercept is at  
$$
y=2.
$$

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### Step 3. Determine the Horizontal Asymptote

The degrees of the numerator and denominator (in the simplified form) are both 1. For rational functions where the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.

The leading coefficient of the numerator (in $$ x+6 $$) is 1, and the leading coefficient of the denominator (in $$ x+3 $$) is also 1. Therefore,
$$
y = \frac{1}{1} = 1.
$$

The function has one horizontal asymptote at  
$$
y=1.
$$

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### Final Answers

- A. The $$ y $$-intercept is at $$ y=2 $$.

- For the horizontal asymptote:  
  A. The function has one horizontal asymptote, $$ y=1 $$.
- A. The $$ y $$-intercept is at $$ y=2 $$. - A. The function has one horizontal asymptote, $$ y=1 $$.

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