Question

a. find the slant asymptote of the graph of the rational function.
b. follow the seven-step strategy and use the slant asymptote to graph the rational function.
$f(x) = \frac{x^2 + 5x - 6}{x - 6}$

a. the x-intercept(s) is/are $-6,1$.
(type an integer or a simplified fraction. use a comma to separate answers if needed.)
\\(\bigcirc\\) b. there are no x-intercepts.
find the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete the choice.
\\(\bigcirc\\) a. the equation of the vertical asymptote is $x = 6$.
(type an equation.)
\\(\bigcirc\\) b. there is no vertical asymptote.
find the horizontal asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete the choice.
\\(\bigcirc\\) a. the equation of the horizontal asymptote is \\(\square\\).
(type an equation.)
\\(\bigcirc\\) b. there is no horizontal asymptote.
plot points between and beyond the x-intercepts and the vertical asymptote. evaluate the function at $-8$, $-7$, $-3$, $4$, and $9$.
\

$$\begin{tabular}{|c|c|c|c|c|c|} \\hline $x$ & $-8$ & $-7$ & $-3$ & $4$ & $9$ \\\\ \\hline $f(x) = \\frac{x^2 + 5x - 6}{x - 6}$ & $\\square$ & $\\square$ & $\\square$ & $\\square$ & $\\square$ \\\\ \\hline \\end{tabular}$$

(simplify your answers.)

Explanation:

Step1: Substitute \(x = - 8\) into \(f(x)=\frac{x^{2}+5x - 6}{x - 6}\)

$$
LATEXBLOCK0
$$

Step2: Substitute \(x=-7\) into \(f(x)=\frac{x^{2}+5x - 6}{x - 6}\)

$$
LATEXBLOCK1
$$

Step3: Substitute \(x = - 3\) into \(f(x)=\frac{x^{2}+5x - 6}{x - 6}\)

$$
LATEXBLOCK2
$$

Step4: Substitute \(x = 4\) into \(f(x)=\frac{x^{2}+5x - 6}{x - 6}\)

$$
LATEXBLOCK3
$$

Step5: Substitute \(x = 9\) into \(f(x)=\frac{x^{2}+5x - 6}{x - 6}\)

$$
LATEXBLOCK4
$$

Answer:

When \(x=-8\), \(f(x)=-\frac{9}{7}\); when \(x = - 7\), \(f(x)=-\frac{8}{13}\); when \(x=-3\), \(f(x)=\frac{4}{3}\); when \(x = 4\), \(f(x)=-15\); when \(x = 9\), \(f(x)=40\)