Question: Follow the steps for graphing a rational function to graph the function \( R(x) = \frac{x^2 + 5x…

Follow the steps for graphing a rational function to graph the function \( R(x) = \frac{x^2 + 5x - 36}{x^2 - x - 12} \).

Determine the behavior of the graph of \( R \) at any x-intercepts. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

A. The graph will cross the x-axis at \( x = \) and touch but not cross the x-axis at \( x = \). (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.)

B. The graph will touch but not cross the x-axis at \( x = \). (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.)

C. The graph will cross the x-axis at \( x = \). (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.)

D. There is no x-intercept.

Solution

To determine the behavior of the graph of \( R(x) = \frac{x^2 + 5x - 36}{x^2 - x - 12} \) at any x-intercepts, we need to find the x-values where the numerator equals zero because the function is zero when the numerator is zero and the denominator is not zero. First, factor the numerator and denominator: Factor the numerator: \(x^2 + 5x - 36\). \[ x^2 + 5x - 36 = (x + 9)(x - 4) \] Factor the denominator: \(x^2 - x - 12\). \[ x^2 - x - 12 = (x - 4)(x + 3) \] Set the numerator equal to zero to find x-intercepts: \[ \begin{cases} x + 9 = 0 & \\ x - 4 = 0 & \end{cases} \] Solve for \(x\): \[ \begin{cases} x = -9 & \\ x = 4 & \end{cases} \] Check these values against the denominator to find if they are x-intercepts: At \(x = 4\), the denominator is zero, so there is no x-intercept at \(x = 4\). At \(x = -9\), the denominator is not zero. Thus, \(x = -9\) is an x-intercept. Therefore, the graph will cross the x-axis at \(x = -9\) but not at \(x = 4\). The correct choice is \( C \). The graph will cross the x-axis at \( x = -9 \). So, the graph’s behavior is that it crosses the x-axis at \( x = -9 \).