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 equation of the slant asymptote is $y = x + 11$.
(type an equation.)
\\(\bigcirc\\) b. there is no slant asymptote.

b. to graph the function, first determine the symmetry of the graph of $f$. choose the correct answer below.
\\(\bigcirc\\) origin symmetry
\\(\bigcirc\\) neither y-axis symmetry nor origin symmetry
\\(\bigcirc\\) 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.)
\\(\bigcirc\\) 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.
\\(\bigcirc\\) a. the x-intercept(s) is/are $\square$.
(type an integer or a simplified fraction. use a comma to separate answers if needed.)
\\(\bigcirc\\) b. there are no x-intercepts.

Explanation:

Step1: Find x - intercepts

Set \(y = f(x)=0\), so \(\frac{x^{2}+5x - 6}{x - 6}=0\). A fraction is zero when the numerator is zero (and the denominator is not zero).
Solve \(x^{2}+5x - 6=0\).
Factor the quadratic: \(x^{2}+5x - 6=(x + 6)(x - 1)\).
Set \((x + 6)(x - 1)=0\).
Using the zero - product property \(x+6 = 0\) or \(x - 1=0\).
So \(x=-6\) or \(x = 1\).
Check that these values do not make the denominator zero. When \(x=-6\), \(x-6=-6 - 6=-12
eq0\); when \(x = 1\), \(x-6=1 - 6=-5
eq0\).

Answer:

A. The \(x\) - intercept(s) is/are \(-6,1\)