Question 77409
If we have a 0 as our denominator, then we have a vertical asymptote, so...


{{{-4+x=0}}}

{{{x=4}}}

So our vertical asymptote is {{{x=4}}}


For our horizontal asymptote, we simply evaluate x for a very large values and see where it ends up. In other words:


{{{(6+2(1000))/(-4+1000)=2.0140562248996}}} Let x=1000


{{{(6+2(10000))/(-4+10000)=2.00140056022409}}} Let x=10,000


{{{(6+2(1000000))/(-4+1000000)=2.000014000056}}} Let x=1,000,000

It looks like as we let x continue on forever, y will slowly approach the value of 2. So our horizontal asymptote is y=2. It turns out that we simply divide 2x by x to get 2

And since the degrees of the numerator and the denominator are the same, we will not have any oblique asymptotes.

So here's our graph:

{{{ graph( 300, 200, -6, 15, -10, 10, (6+2x)/(-4+x), 2) }}} graph of {{{y=(6+2x)/(-4+x)}}} The vertical line is the vertical asymptote (it is not part of the graph) and the horizontal asymptote is the green line (it is not part of the graph also, it is used to show where the asymptote is).