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 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 x = 6.
(type an equation.)
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.

a. the equation of the horizontal asymptote is \square.
(type an equation.)
b. there is no horizontal 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 of a rational function \(y=\frac{f(x)}{g(x)}\) (where \(\text{deg}(f(x))=\text{deg}(g(x)) + 1\)) is given by the quotient of the polynomial long - division.
Since \(\frac{x^{2}+5x - 6}{x - 6}=x + 11+\frac{60}{x - 6}\), the slant asymptote is \(y=x + 11\).

Step3: Check for horizontal asymptote

The degree of the numerator \(n = 2\) and the degree of the denominator \(m=1\). When \(n>m\) (in this case \(n=m + 1\)), there is no horizontal asymptote.

Answer:

The slant asymptote is \(y=x + 11\) and there is no horizontal asymptote (Option B for the horizontal asymptote part).