Question 205136
Please help me graph and label all asymptotes of {{{y=(2x)/(x-4)}}} 
Thank you

{{{y=(2x)/(x-4)}}}
<pre><font size = 4 color = "indigo"><b>
To find the vertical asymptote, set the denominator equal
to zero and solve for x:

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

{{{x=4}}} is the equation of a vertical line 4 units right of
the y-axis, so we draw that vertical asymptote:

{{{drawing(400,400,-7,11,-7,11,
line(4,-8,4,12),
graph(400,400,-7,11,-7,11) 

)}}}

There will be a horizontal asymptote whenever the (degree) largest
exponent of x in the top is not larger than the largest
exponent of x in the bottom.  When the degrees of the top is
smaller, the horizontal asymptote is the line {{{y=0}}} but
when the degrees are the same the horizontal asymptote is

{{{y=(LEADING_COEFFICIENT_OF_THE_TOP)/(LEADING_COEFFICIENT_OF_THE_BOTTOM)}}}

The coefficient of x on the top is 2, and the coefficient of x
on the bottom is 1, so the horizontal asymptote is {{{y=2/1}}} or
{{{y=2}}}. That is a horizontal line 2 units above the x-axis.
Let's draw it:

{{{drawing(400,400,-7,11,-7,11,
line(4,-8,4,12), line(-8,2,12,2), 
graph(400,400,-7,11,-7,11) 

)}}}

Now we need to get some points on both sides of all asymptotes:

(-4,1), (0,0), (2,-2), (3,-6), (5,10), (6,6), (8,4)

Plot those points:

{{{drawing(400,400,-7,11,-7,11,
line(4,-8,4,12), line(-8,2,12,2), 
graph(400,400,-7,11,-7,11), 
line(-4+.2,1,-4-.2,1), line(-4,1+.2,-4,1-.2),
line(0+.2,0,0-.2,0), line(0,0+.2,0,0-.2),
line(2+.2,-2,2-.2,-2), line(2,-2+.2,2,-2-.2),
line(3+.2,-6,3-.2,-6), line(3,-6+.2,3,-6-.2),
line(5+.2,10,5-.2,10), line(5,10+.2,5,10-.2),
line(6+.2,6,6-.2,6), line(6,6+.2,6,6-.2),
line(8+.2,4,8-.2,4), line(8,4+.2,8,4-.2)
)}}}

Draw a smooth graph through the points:

{{{drawing(400,400,-7,11,-7,11,
line(4,-8,4,12), line(-8,2,12,2), 
graph(400,400,-7,11,-7,11,(2x/(x-4))sqrt(4-x)/sqrt(4-x)),
graph(400,400,-7,11,-7,11,(2x/(x-4))sqrt(x-4)/sqrt(x-4)),
line(-4+.2,1,-4-.2,1), line(-4,1+.2,-4,1-.2),
line(0+.2,0,0-.2,0), line(0,0+.2,0,0-.2),
line(2+.2,-2,2-.2,-2), line(2,-2+.2,2,-2-.2),
line(3+.2,-6,3-.2,-6), line(3,-6+.2,3,-6-.2),
line(5+.2,10,5-.2,10), line(5,10+.2,5,10-.2),
line(6+.2,6,6-.2,6), line(6,6+.2,6,6-.2),
line(8+.2,4,8-.2,4), line(8,4+.2,8,4-.2)
)}}}

The vertical asymptote is {{{x=4}}}
The horizontal asymptote is {{{y=2}}}

Edwin</pre>