To find the horizontal asymptote of a rational function, we need to compare the degrees of the numerator and the denominator.

The function is: $f(x) = \frac{1 - 5x - 2x^2}{3x^2 + 7}$

Step 1: Identify the degrees

• The degree of the numerator $1 - 5x - 2x^2$ is 2 (since the highest power of $x$ is $x^2$).
• The degree of the denominator $3x^2 + 7$ is also 2 (since the highest power of $x$ is $x^2$).

Step 2: Compare the degrees

• When the degree of the numerator equals the degree of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients.

The leading term in the numerator is $-2x^2$, and in the denominator, it's $3x^2$. Thus, the horizontal asymptote is:

$y = \frac{-2}{3}$

Therefore, the horizontal asymptote to the graph of $f(x)$ is $y = \frac{-2}{3}$.

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Would you like further details on horizontal asymptotes or any related concepts? Here are five related questions to consider:

1. What happens when the degree of the numerator is greater than the degree of the denominator?
2. Can there be more than one horizontal asymptote?
3. How are oblique asymptotes different from horizontal asymptotes?
4. How can we determine vertical asymptotes for a rational function?
5. What does the behavior near an asymptote tell us about the function?

Tip: Always check the leading terms when determining horizontal asymptotes for rational functions.