Polynomial and Rational Functions

A horizontal line that the graph approaches but never touches.

A vertical line that the graph intersects infinitely many times.

A vertical line that the graph approaches but never touches.

A point where the function's value is zero.

$p(x) = 0$ and $q(x) \neq 0$

$q(x) = 0$ and $p(x) \neq 0$

$p(x) = 0$ and $q(x) = 0$

$p(x) = q(x)$

x = -1

x = 2

x = -1 and x = 2

No vertical asymptote

x = 2

x = -2

x = 2 and x = -2

x = 0

x = -2

x = -3

x = -2 and x = -3

x = 2

x = 1 and x = -1

x = 1

x = -1

No vertical asymptote

x = 1 and x = -2

x = -1 and x = 2

x = 1, x = -2, and x = 0

x = 1 (multiplicity 2) and x = -2

x = -1 and x = 2

x = 2

x = 3 and x = 2

x = 3, x = -1, and x = 2

A vertical asymptote at x = 2

A hole at x = -3

A hole at x = 2 and a vertical asymptote at x = -3

A vertical asymptote at x = -3

$f(x) = \frac{x-1}{x+1}$

$f(x) = \frac{x+1}{x-1}$

$f(x) = \frac{(x-1)^2}{x-1}$

$f(x) = \frac{x-1}{x^2+1}$