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. select the correct choice below and, if necessary, fill in the answer box to complete the choice.
a. the equation of the slant asymptote is $y = x + 11$.
(type an equation.)
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.
origin symmetry
neither y - axis symmetry nor origin symmetry
y - axis symmetry

Explanation:

Step1: Determine the slant asymptote

For a rational function \(f(x)=\frac{N(x)}{D(x)}\) where \(N(x)\) is the numerator and \(D(x)\) is the denominator. If the degree of \(N(x)\) is one more than the degree of \(D(x)\), we use polynomial long - division.
Divide \(x^{2}+5x - 6\) by \(x - 6\).

$$
LATEXBLOCK0
$$

As \(x\to\pm\infty\), \(\frac{60}{x - 6}\to0\). So the slant asymptote is \(y=x + 11\).

Step2: Check for symmetry

• Check for \(y\) - axis symmetry: Replace \(x\) with \(-x\) in \(f(x)=\frac{x^{2}+5x - 6}{x - 6}\).

\(f(-x)=\frac{(-x)^{2}+5(-x)-6}{-x - 6}=\frac{x^{2}-5x - 6}{-x - 6}\)
And \(f(x)=\frac{x^{2}+5x - 6}{x - 6}\), \(f(-x)
eq f(x)\)

• Check for origin symmetry: Replace \(x\) with \(-x\)

\(f(-x)=\frac{x^{2}-5x - 6}{-x - 6}\), \(-f(x)=-\frac{x^{2}+5x - 6}{x - 6}=\frac{-x^{2}-5x + 6}{x - 6}\)
\(f(-x)
eq - f(x)\)

Answer:

a. A. The equation of the slant asymptote is \(y=x + 11\)
b. neither \(y\) - axis symmetry nor origin symmetry