SOLUTION: <pre> Find the horizontal asymptote of the rational function {{{f(x) = (3x-12)/(4x-2)}}} Possible answers: {{{y = 1/2}}} {{{y = 3/4}}} {{{y = 2}}} {{{y = 4}}} Thank you

Algebra ->  Graphs -> SOLUTION: <pre> Find the horizontal asymptote of the rational function {{{f(x) = (3x-12)/(4x-2)}}} Possible answers: {{{y = 1/2}}} {{{y = 3/4}}} {{{y = 2}}} {{{y = 4}}} Thank you      Log On


   

Question 71860: 
Find the horizontal asymptote of the rational 
function f%28x%29+=+%283x-12%29%2F%284x-2%29 
Possible answers: 
y+=+1%2F2
y+=+3%2F4
y+=+2
y+=+4 
Thank you. Exactly what is a asymptote? Can anyone 
explain the simplest way to solve this problem?
Found 2 solutions by rcmcc, Edwin McCravy:
Answer by rcmcc(152) About Me  (Show Source):
You can put this solution on YOUR website!
Hello,
Because you have a linear function with a division equation, the asymptote is located where the equation is undefined. this occurs when the function is divisable by 0. because no equation is valid when you divide by 0 that is your asymptote.
to find this simply look at the bottom half of your equation
f%28x%29=%288x-12%29%2F%284x-2%29
the value that makes the bottom half 0 is (1/2) ... 4(1/2)-2=0
so your asymptote is 1/2

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
Find the horizontal asymptote of the rational 
function f%28x%29+=+%283x-12%29%2F%284x-2%29 
Possible answers: 
y+=+1%2F2
y+=+3%2F4
y+=+2
y+=+4 
Thank you. Exactly what is a asymptote? Can anyone 
explain the simplest way to solve this problem?

-------------------------------------------------

I see you reposted with the correction "horizontal"
The correct choice is y=3%2F4. But the answer
does you no good if you don't understand the problem.
So,first I'll show you how to solve the problem by 
graphing. Then I'll show you the simplest 
way.  But you must understand what an asymptote
is before you can know what you are doing.

Let's graph f%28x%29+=+%283x-12%29%2F%284x-2%29
Get some points:

  x | f(x)
----|------
-10 | 1
 -7 | 1.1
 -3 | 1.5
 -2 | 1.8
  0 |  6
  2 | -1
  4 | 0
  8 | .4
 11 |.5 



The graph of the function is in green.  But 
I have drawn two extra lines in, a blue vertical 
line and a red horizontal line. These lines are
called "asymptotes". Look at the red horizontal 
line that goes through 3/4 on the y-axis? The 
green curve approaches that red line but never 
touches it. That line is called the horizontal 
ASYMPTOTE.  I also drew in the vertical 
asymptote in blue.

As you can see, that red horizontal asymptote has 
the equation y+=+3%2F4. and that blue 
vertical asymptote has the equation x+=+1%2F2.

Now here is the easy way to find the equation of that 
red horizontal line, the horizontal asymptote without
graphing the equation:
 
Since numerator and denominator have the same degree 
(same largest exponent of x), we merely get the 
quotient of the coefficients of the largest powers 
of x.  The coefficient of x in the numerator is 3 
and the coefficient of x in the denominator is
4, so we make the fraction3%2F4 and write
y=3%2F4 as the equation of the red horizontal 
asymptote.

You weren't asked to find the blue vertical asymptote's
equation. But let's find it anyway. To find the blue 
vertical asymptote without having to graph:
Set the denominator of f%28x%29+=+%283x-12%29%2F%284x-2%29 equal
to zero, because a function is undefined if x has a value
which causes the function to be undefined.  You know
that having a 0 in the denominator causes a fraction to be
undefined, right?  So we have

4x-2=0
  4x=2
   x=1%2F2

That's all there is to finding the equation of the vertical 
asymptote.

Edwin