Question

hw 9 rational functions
due wednesday by 11:59pm points 100 submitting an external tool
consider the graph of the function $f(x)=\frac{x - 1}{x^{2}+2x - 4}$. which is true of this graph and its asymptotes? select all that apply
select all that apply:
it has a horizontal asymptote at $y = 0$.
it has a horizontal asymptote at $y = 1$.
it has no horizontal asymptote.
it has a slant asymptote at $y = x$.
it has a slant asymptote at $y=x + 4$

Explanation:

Step1: Recall horizontal - asymptote rule

For a rational function $f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$, if $n\lt m$, the horizontal asymptote is $y = 0$. In the function $f(x)=\frac{x - 1}{x^2+2x - 4}$, the degree of the numerator $n = 1$ and the degree of the denominator $m=2$. Since $1\lt2$, the horizontal asymptote is $y = 0$.

Step2: Recall slant - asymptote rule

A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. Here, the degree of the numerator is 1 and the degree of the denominator is 2, so there is no slant asymptote.

Answer:

It has a horizontal asymptote at $y = 0$.