orizontal Asymptote:
<em>y</em>
=
0
Vertical Asymptote:
<em>x</em>
=
1
Refer to the graph of
<em>y</em>
=
1
<em>x</em>
when you graph
<em>y</em>
=
4
<em>x</em>
−
1
might help you get some idea of the shape of this function.
graph{4/(x-1) [-10, 10, -5, 5]}
Explanation:
Asymptotes
Find the vertical asymptote of this rational function by setting its denominator to
0
and solving for
<em>x</em>
.
Let
<em>x</em>
−
1
=
0
<em>x</em>
=
1
Which means that there's a vertical asymptote passing through the point
(
1
,
0
)
.
*FYI you can make sure that
<em>x</em>
=
1
does give a vertical asymptote rather than a removable point of discontinuity by evaluating the numerator expression at
<em>x</em>
=
1
. You can confirm the vertical asymptote if the result is a non-zero value. However if you do end up with a zero, you'll need to simplify the function expression, remove the factor in question, for example
(
<em>x</em>
−
1
)
, and repeat those steps. *
You may find the horizontal asymptote (a.k.a "end behavior") by evaluating
lim
<em>x</em>
→
∞
4
<em>x</em>
−
1
and
lim
<em>x</em>
→
−
∞
4
<em>x</em>
−
1
.
If you haven't learned limits yet, you'll still able to find the asymptote by plugging in large values of
<em>x</em>
(e.g., by evaluating the function at
<em>x</em>
=
11
,
<em>x</em>
=
101
, and
<em>x</em>
=
1001
.) You'll likely find that as the value of
<em>x</em>
increase towards positive infinity, the value of
<em>y</em>
getting closer and closer to- but never <em>reaches</em>
0
. So is the case as
<em>x</em>
approaches negative infinity.
By definition , we see that the function has a horizontal asymptote at
<em>y</em>
=
0
Graph
You might have found the expression of
<em>y</em>
=
1
<em>x</em>
, the
<em>x</em>
-reciprocal function similar to that of
<em>y</em>
=
4
<em>x</em>
−
1
. It is possible to graph the latter based on knowledge of the shape of the first one.
Consider what combination of <em>transformations</em> (like stretching and shifting) will convert the first function we are likely familiar with, to the function in question.
We start by converting
<em>y</em>
=
1
<em>x</em>
to
<em>y</em>
=
1
<em>x</em>
−
1
by shifting the graph of the first function to the <em>right</em> by
1
unit. Algebraically, that transformation resembles replacing
<em>x</em>
in the original function with the expression
<em>x</em>
−
1
.

generated with fooplot
Finally we'll vertically stretch the function
<em>y</em>
=
1
<em>x</em>
−
1
by a factor of
4
to obtain the function we're looking for,
<em>y</em>
=
4
<em>x</em>
−
1
. (For rational functions with horizontal asymptotes the stretch would effectively shifts the function outwards.)

generated with footplot
