Identifying asymptotes through graphs

A vertical asymptote is a vertical line that a graph approaches ever more closely without ever crossing it.
It occurs at an $x$-value where the function blows up because it is undefined there.

For a rational function, "undefined" means the denominator equals $0$.
In $f(x) = \frac{x + 3}{x - 2}$, the denominator $x - 2$ is $0$ when $x = 2$, so the graph has a vertical asymptote at $x = 2$.
Near $x = 2$ the curve shoots up or down toward the line but never meets it.

To read an asymptote off a graph, find the vertical line the curve hugs.
If two branches of the graph rush toward the line $x = 2$ from either side, that line is the vertical asymptote.
Reading it directly lets you match a graph to the function that produces it.

In general form $f(x) = \frac{x + a}{x + b}$, set the denominator to zero to locate the asymptote.
Since $x + b = 0$ gives $x = -b$, the vertical asymptote is $x = -b$, regardless of $a$.
So if $b < 0$, the asymptote is at a positive $x$-value, and if $b > 0$, it is at a negative one.

The graph below has a vertical asymptote.

What is the equation of the vertical asymptote?
(A) $x = 3$
(B) $x = -3$
The two branches of the graph rush toward the vertical line at $x = 3$ without ever crossing it.
Choice (A), $x = 3$, is the vertical asymptote, so (A) is correct.

The graph below has a vertical asymptote.

What is its equation?
(A) $x = 2$
(B) $x = -2$
The curve blows up around the vertical line $x = -2$, approaching it from both sides but never touching it.
Choice (B), $x = -2$, is the vertical asymptote, so (B) is correct.

The rational function graphed below has a vertical asymptote.

Where is it?
(A) $x = 1$
(B) $x = -1$
The graph approaches the vertical line $x = 1$ ever more closely but never crosses it.
Choice (A), $x = 1$, is the vertical asymptote, so (A) is correct.