Question

1. find the equation(s) of all horizontal asymptotes, if any.

f(x) = \frac{5x^{2}-5}{x^{2}-5x + 1}
g) find the equation(s) of all oblique asymptotes, if any.
h.) carefully graph the function using a - h. label your axes and all information found in a - g.

Explanation:

Step1: Recall horizontal - asymptote rules

For a rational function $y = \frac{f(x)}{g(x)}=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$, if $n = m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$. Here, $f(x)=5x^2 - 5$, $a_2 = 5$, $g(x)=x^2-5x + 11$, $b_2=1$, and $n = m=2$.

Step2: Calculate the horizontal - asymptote

$y=\frac{5}{1}=5$. So the equation of the horizontal asymptote is $y = 5$.

Step3: Recall oblique - asymptote rules

For a rational function $y=\frac{f(x)}{g(x)}$, an oblique asymptote exists when $n=m + 1$. Since $n=m = 2$ in $y=\frac{5x^2-5}{x^2-5x + 11}$, there is no oblique asymptote.

Answer:

1f.) $y = 5$
1g.) None