Find all asymptotes (including vertical, horizontal, and slant) of the function $y = \frac { 3 x + 2 } { x ^ { 2 } - 2 x + 1 }$ .
A) No asymptote
B) Horizontal:
$y = 0$ ; vertical:
$x = \frac { 5 } { 2 }$
C) Horizontal:
$y = 2$ ; vertical:
$x = \frac { 5 } { 2 }$
D) Horizontal:
$y = 0$ ; vertical:
$x = 1$
E) Horizontal:
$y = \pm 1$ ; vertical:
$x = - 1$

Question

Find all asymptotes (including vertical, horizontal, and slant) of the function $y = \frac { 3 x + 2 } { x ^ { 2 } - 2 x + 1 }$ . 
A) No asymptote
B) Horizontal:
 $y = 0$ ; vertical:
 $x = \frac { 5 } { 2 }$ 
C) Horizontal:
 $y = 2$ ; vertical:
 $x = \frac { 5 } { 2 }$ 
D) Horizontal:
 $y = 0$ ; vertical:
 $x = 1$ 
E) Horizontal:
 $y = \pm 1$ ; vertical:
 $x = - 1$

Quizplus · Accepted Answer

The denominator $x^2 - 2x + 1$ can be factored as $(x-1)^2$, indicating a vertical asymptote at $x = 1$. For horizontal asymptotes, we compare the degrees of the polynomial in the numerator and the denominator. Since the degree of the denominator (2) is greater than the degree of the numerator (1), the horizontal asymptote is $y = 0$. There are no slant asymptotes because the degree of the numerator is not exactly one more than the degree of the denominator.

Find All Asymptotes (Including Vertical, Horizontal, and Slant) of the Function