Question

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

Answer 1

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

Explanation:

The domain of

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

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.