How do you graph y=(6x-1)/(3x-1)y=6x13x1 using asymptotes, intercepts, end behavior?

1 Answer
Nov 30, 2016

graph{(6x-1)/(3x-1) [-10, 10, -5, 5]}

Explanation:

The domain of

y(x) = frac (6x-1) (3x-1)y(x)=6x13x1

is all RR except the point x=1/3 where the denominator vanishes.

Look at the behavior of the function at the limits of the intervals of definition:

lim_(x->-oo) y(x) = 2

lim_(x->+oo) y(x) = 2

The limits are finite, so y(x) has an horizontal asymptote y=2 on both sides.

lim_(x->(1/3)^-) y(x) = -oo

lim_(x->(1/3)^+) y(x) = +oo

There is a vertical asymptote at x=1/3

The intercepts with the axis can be found as:

y(x=0) = 1

0=frac (6x-1) (3x-1) at x=1/6

The derivative is:

y'(x)=frac (6(3x-1)-3(6x-1))((3x-1)^2)=-3/(3x-1)^2

and is always negative, so the function is strictly decreasing in the whole domain.

Summing it up, as x goes from -oo to +oo, y(x) starts from below the horizontal line y=2, decreases constantly crossing the y axis at y=1 and the x axis at x=1/6, then goes negative without bound until the vertical line x=1/3.
To the right of this line, y(x) starts from +oo and decreases approaching to the horizontal line y=2 as x grows.