To find the horizontal asymptote of the function:

$f(x) = \frac{-3x + 2}{7x + 6}$

we need to examine the degrees of the polynomial in the numerator and the denominator:

1. Degree of the numerator is 1 (the highest power of $x$ in $-3x + 2$).
2. Degree of the denominator is also 1 (the highest power of $x$ in $7x + 6$).

When the degrees of the numerator and denominator are the same, the horizontal asymptote is determined by dividing the leading coefficients of the numerator and the denominator.

Here, the leading coefficient of the numerator is $-3$, and the leading coefficient of the denominator is $7$. Thus, the horizontal asymptote is:

$y = \frac{-3}{7}$

Answer:

The horizontal asymptote is: $y = -\frac{3}{7}$

Would you like further details, or do you have any questions?

Here are five related questions that expand on this concept:

1. How do you determine if a rational function has no horizontal asymptote?
2. What happens if the degree of the numerator is greater than the degree of the denominator?
3. How would the horizontal asymptote change if the numerator and denominator degrees were different?
4. Can a function have both a horizontal and an oblique asymptote?
5. How do vertical asymptotes differ from horizontal asymptotes?

Tip: Always check the degrees of the numerator and denominator when determining horizontal asymptotes.