Question

graph all vertical and horizontal asymptotes of the rational function.
$f(x) = \frac{-2x - 7}{-4x - 2}$

Explanation:

Step1: Find vertical asymptote

Set denominator equal to 0:
$-4x - 2 = 0$
Solve for $x$:
$-4x = 2$
$x = \frac{2}{-4} = -\frac{1}{2}$

Step2: Find horizontal asymptote

Compare leading coefficients:
For rational function $\frac{ax+b}{cx+d}$, horizontal asymptote is $y=\frac{a}{c}$.
Here $a=-2$, $c=-4$, so:
$y = \frac{-2}{-4} = \frac{1}{2}$

Answer:

Vertical asymptote: $x=-\frac{1}{2}$ (a vertical dashed line at $x=-0.5$ on the graph)
Horizontal asymptote: $y=\frac{1}{2}$ (a horizontal dashed line at $y=0.5$ on the graph)