Question

follow the seven step strategy to graph the following rational function.
$f(x) = \frac{x^2 + x - 12}{x^2 - 4}$

find the horizontal asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the equation(s) of the horizontal asymptote(s) is/are $y = 1$.
(type an equation. use a comma to separate answers as needed.)
b. there is no horizontal asymptote.

plot points between and beyond each x - intercept and vertical asymptote. find the value of the function at the given value of x.
$x$: -10, -8, -1, 1, 7, 8
$f(x) = \frac{x^2 + x - 12}{x^2 - 4}$: $\frac{13}{18}$, $\frac{11}{15}$, 4, $\frac{10}{3}$, $\frac{44}{45}$, 1
(simplify your answers.)

use the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. choose the correct graph below.
a. graph a
b. graph b
c. graph c
d. graph d

Explanation:

Step1: Determine the degrees of numerator and denominator

The degree of the numerator \(n\) for \(f(x)=\frac{x^{2}+x - 12}{x^{2}-4}\) is \(n = 2\) (since the highest - power of \(x\) in \(x^{2}+x - 12\) is \(x^{2}\)), and the degree of the denominator \(m\) is \(m=2\) (since the highest - power of \(x\) in \(x^{2}-4\) is \(x^{2}\)).

Step2: Use the horizontal asymptote rule

When \(n = m\), the horizontal asymptote \(y\) is given by \(y=\frac{a_{n}}{b_{m}}\), where \(a_{n}\) is the leading coefficient of the numerator and \(b_{m}\) is the leading coefficient of the denominator. For the numerator \(x^{2}+x - 12\), \(a_{n}=1\); for the denominator \(x^{2}-4\), \(b_{m}=1\). So \(y=\frac{1}{1}=1\).

Step3: Calculate \(f(8)\)

Substitute \(x = 8\) into \(f(x)=\frac{x^{2}+x - 12}{x^{2}-4}\).

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Answer:

A. The equation(s) of the horizontal asymptote(s) is/are \(y = 1\). For \(x = 8\), \(f(8)=1\).