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

• y - axis symmetry
• origin symmetry
• 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.

• a. the y - intercept is 3

(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 to complete your choice.

• a. the x - intercept(s) is/are - 4,3

(type an integer or a simplified fraction. use a comma to separate answers as 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 your choice.

• a. the equation(s) of the vertical asymptote(s) is/are

(type an equation. use a comma to separate answers as needed)

• b. there are no vertical asymptotes

Explanation:

Step1: Factor the numerator and denominator

$$\begin{align*} x^{2}+x - 12&=(x + 4)(x-3)\\ x^{2}-4&=(x + 2)(x - 2) \end{align*}$$

So \(f(x)=\frac{(x + 4)(x-3)}{(x + 2)(x - 2)}\)

Step2: Check for symmetry

• Y - axis symmetry: Replace \(x\) with \(-x\)

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

• Origin symmetry: Replace \(x\) with \(-x\) and check if \(f(-x)=-f(x)\)

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

Step3: Find the \(y\) - intercept

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

Step4: Find the \(x\) - intercepts

Set \(y = 0\), then \(x^{2}+x-12=0\) (since the numerator of a rational function \(y = \frac{N(x)}{D(x)}\) is zero when \(y = 0\) and \(D(x)
eq0\))
\((x + 4)(x - 3)=0\) gives \(x=-4\) or \(x = 3\)

Step5: Find the vertical asymptotes

Set the denominator \(x^{2}-4=0\) (since \(x^{2}-4=(x + 2)(x - 2)\) and the numerator is non - zero at \(x=-2\) and \(x = 2\))
\(x=-2\) and \(x = 2\)

Answer:

• Symmetry: neither \(y\) - axis symmetry nor origin symmetry
• \(y\) - intercept: \(3\)
• \(x\) - intercepts: \(-4,3\)
• Vertical asymptotes: \(x=-2,x = 2\)