Question

graph all vertical and horizontal asymptotes of the rational function.
$f(x) = \frac{3x^2 - 12}{2x^2 + 9}$

Explanation:

Step1: Find vertical asymptotes

Set denominator equal to 0:
$$2x^2 + 9 = 0$$
$$2x^2 = -9$$
$$x^2 = -\frac{9}{2}$$
This has no real solutions, so there are no vertical asymptotes.

Step2: Find horizontal asymptote

Compare degrees of numerator/denominator (both degree 2). Take ratio of leading coefficients:
$$y = \frac{3}{2}$$

Answer:

Horizontal asymptote: $y=\frac{3}{2}$; No vertical asymptotes.
To graph this: draw the horizontal dashed line $y=\frac{3}{2}$ on the coordinate plane, and note there are no vertical dashed lines to draw.