# How do you find the vertical, horizontal or slant asymptotes for f(x) = (x^2) / (x-2)?

Dec 17, 2016

Slant: y = x + 2.
Vertical: x = 2.

#### Explanation:

By actual division,

$y = f \left(x\right) = x + 2 + \frac{4}{x - 2}$

This form reveals ths the asymptotes as follows.

y = quotient = x + 2 gives the slant asymptote.

The denominator in the remainder,

$x - 2 = 0$ gives the vertical asymptote.

There is no horizontal asymptote.

A reorganization of the equation gives the form

$\left(y - x - 2\right) \left(x - 2\right) = c o n s \tan t = 4$ that represents a hyperbola,

with the pair of asymptotes

( y-x-2)(x-2)= 0

graph{y(x-2)-x^2=0 [-80, 80, -40, 40]}