what is the horizontal asymptote of the functionf(x)=(1)/(-2x^(2)-2x+12)

To determine the horizontal asymptote of the function \( f(x) = \frac{1}{-2x^2 - 2x + 12} \), you need to analyze the behavior of the function as \( x \) approaches infinity (both positive and negative). Horizontal asymptotes are concerned with the end behavior of a function as \( x \) becomes very large or very small.

### Step 1: Analyze the degrees of the numerator and denominator
The function is given by:
\[
f(x) = \frac{1}{-2x^2 - 2x + 12}
\]
Here, the numerator is a constant (1), which is a polynomial of degree 0. The denominator is a quadratic polynomial of degree 2.

### Step 2: Determine the horizontal asymptote
For rational functions, the horizontal asymptote is determined by the degrees of the numerator and the denominator:

- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (but there might be an oblique asymptote).

In this case, the degree of the numerator (0) is less than the degree of the denominator (2). Therefore, the horizontal asymptote is:
\[
y = 0
\]

### Conclusion
The horizontal asymptote of the function \( f(x) = \frac{1}{-2x^2 - 2x + 12} \) is \( y = 0 \).

Would you like to explore this concept further or have any additional questions?

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Here are 5 related questions to consider:

1. How do you determine the vertical asymptotes of a rational function?
2. What is the significance of the leading coefficient in finding horizontal asymptotes?
3. How do you find the horizontal asymptote for a function where the numerator and denominator have the same degree?
4. What are oblique asymptotes, and when do they occur?
5. How does the presence of horizontal asymptotes affect the graph of a function?

**Tip:** When determining horizontal asymptotes, focus on the leading terms of the numerator and denominator, as they dominate the behavior of the function for large values of \( x \).

Learn how to find the horizontal asymptote of the function f(x) = 1 / (-2x^2 - 2x + 12) with a detailed step-by-step solution. Understand the behavior of rational functions, asymptotes, and their significance in mathematical analysis. Suitable for high school students.

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