Questions: Graph the rational function. f(x) = 12 / (x^2 + x - 6) Start by drawing the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote. Finally, click on the graph-a-function button.

Graph the rational function.
f(x) = 12 / (x^2 + x - 6)

Start by drawing the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote. Finally, click on the graph-a-function button.

Transcript text: Graph the rational function. \[ f(x)=\frac{12}{x^{2}+x-6} \] Start by drawing the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote. Finally, click on the graph-a-function button.

Solution

Solution Steps

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. For the function \( f(x) = \frac{12}{x^2 + x - 6} \), set the denominator equal to zero and solve for \( x \): \[ x^2 + x - 6 = 0 \] Factor the quadratic equation: \[ (x + 3)(x - 2) = 0 \] So, the vertical asymptotes are at: \[ x = -3 \] \[ x = 2 \]

Step 2: Identify Horizontal Asymptote

Horizontal asymptotes are determined by the degrees of the numerator and the denominator. Here, the degree of the numerator (0) is less than the degree of the denominator (2). Therefore, the horizontal asymptote is: \[ y = 0 \]

Step 3: Find Intercepts

Y-Intercept:

Set \( x = 0 \) in the function: \[ f(0) = \frac{12}{0^2 + 0 - 6} = \frac{12}{-6} = -2 \] So, the y-intercept is at: \[ (0, -2) \]

X-Intercept:

Set the numerator equal to zero and solve for \( x \): \[ 12 = 0 \] Since this is never true, there are no x-intercepts.