Final answer:
The equation for the horizontal asymptote of the given rational function is y = -1/7.
Step-by-step explanation:
The equation for the horizontal asymptote of a rational function is determined by the degrees of the numerator and denominator polynomials. 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 y = a/b, where a and b are the leading coefficients of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
In this case, the degree of the numerator is 2 and the degree of the denominator is 2 as well. So the horizontal asymptote of the function f(x) = (2x² − 6x − 4) / (1 − 5x − 14x²) is y = a/b, where a and b are the leading coefficients of the numerator and denominator.
Therefore, the equation for the horizontal asymptote is y = 2/(-14) which simplifies to y = -1/7.