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

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Answer 1 · 2015-07-06T20:03:43

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

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

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

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

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

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

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]}

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