Questions: Follow the seven step strategy to graph the following rational function. f(x) = (x+2)/(x^2+x-12) A. The equation(s) of the vertical asymptote(s) is/are x=-4, x=3. (Type an equation. Use a comma to separate answers as needed.) B. There is no vertical asymptote. 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 . (Type an equation. Use a comma to separate answers as needed.) B. There is no horizontal asymptote.

Solution Steps

To find the vertical asymptotes, we need to determine where the denominator of the function is equal to zero. The function is given by:

\[ f(x) = \frac{x+2}{x^2 + x - 12} \]

Set the denominator equal to zero and solve for \(x\):

\[ x^2 + x - 12 = 0 \]

Factoring the quadratic equation:

\[ (x + 4)(x - 3) = 0 \]

Thus, the vertical asymptotes are at \(x = -4\) and \(x = 3\).

For rational functions, the horizontal asymptote is determined by the degrees of the numerator and the denominator. The degree of the numerator is 1, and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is:

\[ y = 0 \]

Final Answer