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.
$f(x)=\frac{x^{2}+5x - 6}{x - 6}$

origin symmetry
neither y - axis symmetry nor origin symmetry
y - axis symmetry
what is the y - intercept? select the correct choice below and, if necessary, fill in the answer box to complete the choice.
a. the y - intercept is 1.
(type an integer or a simplified fraction.)
b. there is no y - intercept.
what is/are the x - intercept(s)? select the correct choice below and, if necessary, fill in the answer box within your choice.
a. the x - intercept(s) is/are - 6,1.
(type an integer or a simplified fraction. use a comma to separate answers if needed.)
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.
a. the equation of the vertical asymptote is
(type an equation.)
b. there is no vertical asymptote.

Explanation:

Step1: Perform polynomial long - division

We divide \(x^{2}+5x - 6\) by \(x - 6\).

$$
LATEXBLOCK0
$$

As \(x\to\pm\infty\), \(\frac{60}{x - 6}\to0\).

Step2: Determine the slant asymptote

The slant asymptote is \(y=x + 11\) since the non - remainder part of the division \(\frac{x^{2}+5x - 6}{x - 6}=x + 11+\frac{60}{x - 6}\) is \(y=x + 11\)

Answer:

The slant asymptote is \(y=x + 11\)