Final answer:
The horizontal asymptote of the function y = (2x^2 + 7) / (3x^2 - x - 1) is y = 2/3. The vertical asymptotes are located at approximately x = 1.0 and x = -1/3.
Step-by-step explanation:
Finding Horizontal and Vertical Asymptotes
To find the horizontal asymptote of the function y = (2x^2 + 7) / (3x^2 - x - 1), we look at the degrees of the polynomials in the numerator and denominator. Since the degrees are equal (both are 2), the horizontal asymptote will be the ratio of the leading coefficients: y = 2/3.
To find the vertical asymptotes, we need to determine where the function is undefined, which is where the denominator equals zero. Factoring the denominator, we find the roots of the equation 3x^2 - x - 1 = 0. These roots are the x-values of the vertical asymptotes. After factoring, we find the vertical asymptotes to be at approximately x = 1.0 and x = -1/3.