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To analyze the given rational function \( f(x) = \frac{-3}{x + 2} \), let's identify the horizontal and vertical asymptotes.

### 1. **Horizontal Asymptote:**
The horizontal asymptote for a rational function \( f(x) = \frac{a}{x + b} \) (where \( a \) and \( b \) are constants and \( x \) appears only in the denominator) is \( y = 0 \). This is because as \( x \) approaches infinity or negative infinity, \( f(x) \) approaches zero.

For this function:
\[
y = 0
\]
So, you should draw a horizontal line along the \( x \)-axis in the left graph and write \( y = 0 \) as the horizontal asymptote.

### 2. **Vertical Asymptote:**
The vertical asymptote occurs where the denominator equals zero, as this makes the function undefined. 

Set the denominator equal to zero:
\[
x + 2 = 0 \Rightarrow x = -2
\]
Thus, \( x = -2 \) is the vertical asymptote. Draw a vertical line at \( x = -2 \) in the right graph and write \( x = -2 \) as the vertical asymptote.

Would you like more details on asymptotes or have any questions?

### Follow-up Questions:
1. What are asymptotes, and how do they affect the shape of a graph?
2. How can we determine if a function has an oblique asymptote?
3. How do horizontal and vertical asymptotes influence the limits of a function?
4. How do we find asymptotes for more complex rational functions?
5. What are some real-world examples where asymptotes appear in graphs?

### Tip:
For rational functions, remember that horizontal asymptotes depend on the degrees of the numerator and denominator, while vertical asymptotes are found by setting the denominator equal to zero.

Graph the asymptotes for the rational function f(x) = -3 / (x + 2). Identify and draw the horizontal and vertical asymptotes.

Learn how to identify and graph the asymptotes for the rational function f(x) = -3 / (x + 2). This problem covers horizontal and vertical asymptote concepts and provides step-by-step guidance suitable for high school math students (Grades 9-12).

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