Final answer:
To determine the vertical and horizontal asymptotes of the given functions, analyze the behavior of the function as x approaches infinity and negative infinity. For each function, factor the denominator to find the vertical asymptotes and compare the degrees of the numerator and denominator to find the horizontal asymptote.
Step-by-step explanation:
To determine the vertical and horizontal asymptotes of a function, we need to analyze the behavior of the function as x approaches infinity and negative infinity. For y = x^2 / (x^2 - 5x + 6), we can factor the denominator to (x-2)(x-3). The vertical asymptotes occur when the denominator equals zero, so x = 2 and x = 3 are the vertical asymptotes. Since the degree of the numerator and denominator is the same, the horizontal asymptote is determined by the ratio of the leading coefficients, which in this case is 1/1, so the horizontal asymptote is y = 1.
For y = (x + 1)^2 / (x^2 - 1), we can factor the denominator to (x+1)(x-1). The vertical asymptotes occur when the denominator equals zero, so x = -1 and x = 1 are the vertical asymptotes. Again, the horizontal asymptote is determined by the ratio of the leading coefficients, which is 1/1, so the horizontal asymptote is y=1.
For y = 3 / (x + 6) - 9, there are no vertical asymptotes since the denominator is a constant. The horizontal asymptote is determined by the ratio of the leading coefficients, which is 3/1, so the horizontal asymptote is y=3.