Question

How do you graph `(3x + 3) / (2x + 4)`?

Answer 1

Separate into a polynomial (constant) term and rational expression that tends to zero as `x -> +-oo`. Deduce asymptotic behaviour.

Explanation:

`f(x) = (3x+3)/(2x+4) = (3(x+1))/(2(x+2)) = 3/2*(x+1)/(x+2)`

`=3/2*(x+2-1)/(x+2)`

`=3/2((x+2)/(x+2)-1/(x+2))`

`=3/2(1-1/(x+2))`

`=3/2-3/(2(x+2))`

with exclusion `x != -2`

So as `x->-oo` the term `3/(2(x+2)) ->0_-` so `f(x)->(3/2)_+`

As `x->(-2)_-` the term `3/(2(x+2))->-oo` so `f(x)->oo`

As `x->(-2)_+` the term `3/(2(x+2))->oo` so `f(x)->-oo`

As `x->oo` the term `3/(2(x+2)) -> 0_+` so `f(x)->(3/2)_-`

The intersection with the `x` axis occurs at `(-1, 0)` since the numerator of the original expression is `0` for `x = -1`

The intersection with the `y` axis can be found by substituting `x=0` into the original equation to derive `(0, 3/4)`

graph{ (3x+3)/(2x+4) [-12.29, 7.71, -3.44, 6.56]}