Final answer:
The horizontal asymptote is y = 1 and the vertical asymptotes are x = -4 and x = 1.
Step-by-step explanation:
The horizontal asymptote of a rational function is the value that the function approaches as x approaches positive or negative infinity. The vertical asymptote is the vertical line that the function gets arbitrarily close to as x approaches a certain value.
To find the horizontal asymptote, we compare the degrees of the numerator and denominator. In this case, both have a degree of 2. To determine the horizontal asymptote, we divide the leading coefficients of the numerator and denominator. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is also 1. Therefore, the horizontal asymptote is y = 1.
To find the vertical asymptotes, we set the denominator equal to zero and solve for x. In this case, the denominator is given by x² + 3x - 4. Factoring it, we have (x + 4)(x - 1). Setting each factor equal to zero, we find that x = -4 and x = 1. Therefore, the vertical asymptotes are x = -4 and x = 1.