# How do you find vertical, horizontal and oblique asymptotes for (x^2+4x-2)/(x-2)?

Dec 6, 2016

Vertical: $\uparrow x = 2 \downarrow$. Seldom realized oblique asymptote:
y = x+ 6. These are the asymptotes of the hyperbola represented by the equation. See illustrative graph.

#### Explanation:

Cross multiply and reorganize to the form

$x \left(y - x\right) - 2 y - 4 x + 2 = 0$. This suggest the form

(x+a)(y-x+b=c and we have

$\left(x - 2\right) \left(x - y + 6\right) = - 14$ that represents the hyperbola with asymptotes

$x - 2 = 0 \mathmr{and} x - y + 6 = 0$.

Now, look for the asymptotes in the graph.

graph{y(x-2)-x^2-4x+2=0 [-79.2, 79.2, -39.6, 39.6]}