What are the asymptotes of the rational function y = (x^2 - 2x + 1) / (x - 4)?

To determine the asymptotes of the rational function ( y = \frac{x^2 - 2x + 1}{x - 4} ), we analyze both vertical and horizontal asymptotes.

Firstly, vertical asymptotes occur where the denominator of the fraction is equal to zero, provided that the numerator is not also zero at that point. Here, the denominator ( x - 4 = 0 ) when ( x = 4 ). Since the numerator does not equal zero when ( x = 4 ) (as it simplifies to ( (4^2 - 2(4) + 1) = 1 )), there is a vertical asymptote at ( x = 4 ).

Next, we examine horizontal asymptotes, which are determined by analyzing the degrees of the polynomial in the numerator and denominator. In this case, the numerator ( x^2 - 2x + 1 ) is a quadratic polynomial (degree 2), and the denominator ( x - 4 ) is a linear polynomial (degree 1). When the degree of the numerator is greater than that of the denominator, the horizontal asymptote is considered to be at infinity, and thus