Question

How do you find the vertical, horizontal and slant asymptotes of: `f(x)=(x+1)/(2x+10)`?

Answer 1

The curve is a rectangular hyperbola.
Horizontal asymptote : `larr y = 1/2 rarr`
Vertical asymptote : `uarr x= -5 darr`

Explanation:

An equation in the form

`(y-m_1x-c_1)(y-m_2x-c_2)=c ne 0` represents a hyperbola

contained between the asymptotes

`(y-m_1x-c_1)(y-m_2x-c_2)=0`

Here, cross multiplying and rearranging,

`(2y-1)(x+5)=6`

The asymptotes at right angles are

#2y-1)(x+5)=0, giving

`x=-5 and y = 1/2`

Now, see the asymptotes-inclusive graph.

graph{((2y-1)(x+5)-6)(2y-1)(x+.000000001y+5)=0 [-15, 5, -7.5, 7.5]}