Question

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

to graph the function, first determine the symmetry of the graph of $f$. choose the correct answer below.
\bigcirc y-axis symmetry
\bigcirc origin symmetry
\bigcirc neither y-axis symmetry nor origin symmetry

what is the y-intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\bigcirc a. the y-intercept is \square.
\quad \quad (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 to complete your choice.
\bigcirc a. the x-intercept(s) is/are \square.
\quad \quad (type an integer or a simplified fraction. use a comma to separate answers as needed.)
\bigcirc 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 your choice.
\bigcirc a. the equation(s) of the vertical asymptote(s) is/are \square.
\quad \quad (type an equation. use a comma to separate answers as needed.)
\bigcirc b. there are no vertical asymptotes.

Explanation:

Symmetry

Step1: Check for y - axis symmetry

Replace \(x\) with \(-x\) in \(f(x)=\frac{x^{2}+x - 12}{x^{2}-4}\).
\(f(-x)=\frac{(-x)^{2}+(-x)-12}{(-x)^{2}-4}=\frac{x^{2}-x - 12}{x^{2}-4}
eq f(x)\)

Step2: Check for origin symmetry

Check if \(f(-x)=-f(x)\). \(-f(x)=-\frac{x^{2}+x - 12}{x^{2}-4}=\frac{-x^{2}-x + 12}{x^{2}-4}
eq f(-x)\)

y - intercept

Step1: Set \(x = 0\)

\(f(0)=\frac{0^{2}+0-12}{0^{2}-4}=\frac{- 12}{-4}=3\)

x - intercept

Step1: Set \(y = 0\) (i.e., numerator \(=0\))

\(x^{2}+x - 12=0\)
Factor: \((x + 4)(x - 3)=0\)
\(x=-4\) or \(x = 3\)

Vertical asymptote

Step1: Set denominator \(=0\)

\(x^{2}-4=0\)
Factor: \((x + 2)(x - 2)=0\)
\(x=-2\) or \(x = 2\)

Answer:

• Symmetry: neither y - axis symmetry nor origin symmetry
• y - intercept: A. The y - intercept is \(3\)
• x - intercept: A. The x - intercept(s) is/are \(-4,3\)
• Vertical asymptote: A. The equation(s) of the vertical asymptote(s) is/are \(x=-2,x = 2\)