1. VERTICAL ASYMPTOTES TO
Rational Functions BY
TARUNGEHLOTS
Note: It is assumed that you are familiar with long division of polynomials.
Rational functions are functions of the form f(x)=polynomial/polynomial. These
functions are difficult to graph manually. Six aspects of rational functions are worth
looking at but first we need to introduce the concept of a asymptote. A vertical asymptote
is a vertical line to which a graph comes closer and closer but can never cross or touch. A
horizontal asymptote is a horizontal line to which a graph comes closer and closer at the
far left and far right but may sometimes cross it in the vicinity of the origin. A slant (or an
oblique) asymptote is a slanted straight line towards which a graph will come closer and
closer at the far left and far right and may sometimes cross it around the origin. Now, the
first five interesting aspects of a rational function are its y-intercept, the x-intercepts, the
vertical asymptotes, the horizontal or slant asymptote. The sixth aspect is its actual graph
but we will come to that later on. Let us first take a look at the graph of a typical rational
function.
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2. Please note the two vertical asymptotes, x=-2 and x=2, and the horizontal asymptote y=1.
There is no slant asymptote.
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3. Let us see how to get these first five characteristics:
I) As usual, you can find the y-intercept by setting x=0.
II) And you can find the x-intercepts by setting y=0 (in this case f(x)=0). Now please
remember that a fraction has a value of zero only when the numerator equals zero.
So we practically find the x-intercepts of a rational function by setting the
numerator equal to zero and then solving for x.
III) Vertical asymptotes are found by setting the denominator equal to zero and then
solving for x.
IV) Horizontal asymptotes are found by using the theorem for horizontal asymptotes.
This theorem states the following:
a. If the degree of the numerator is less than the degree of the denominator, the
x-axis is the horizontal asymptote.
b. If the degree of the numerator is equal to the degree of the denominator then
the equation of the horizontal asymptote is y = a/b where a is the leading
coefficient of the polynomial in the numerator and b is the leading coefficient
of the polynomial in the denominator. To illustrate this, look at
4x 2 1
f (x) . The leading coefficient in the numerator is 4, in the
3x 2 5 x 2
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denominator 3. Thus the line y is the horizontal asymptote.
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V) Slant asymptotes only occur in cases where the degree of the numerator is one
higher than the degree of the denominator. So we basically may have a horizontal
or a slant asymptote, never both. The equation of the slant asymptote is found by
carrying out the long division of the function. The equation of the slant asymptote
is then y = the quotient or the answer to the division, while the remainder part is
4x 2 6x 5
ignored. To illustrate this, look at f ( x ) . Carrying out the long
2x 3
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division gives us: 2x 6 so that the slant asymptote is just y=2x+6.
2x 3
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4. Let us look at an example and set up the routine to find all the interesting aspects of a
2x 2 8
rational function. Consider f ( x ) .
3x 2 6
I) We find the y-intercept as 0, 8 0, 4
6 3
II) We find the x-intercepts by setting the numerator equal to zero:
2x 2 8 0 x 2 4 0 (x 2)(x 2) 0 so that the x-intercepts are (-2,0)
and (2,0).
III) The vertical asymptotes are found by setting the denominator equal to zero:
3x 2 6 0 x2 2 0 x 2 x 2 0 so that the vertical asymptotes
are x 2.
IV) The horizontal asymptotes are found by using the theorem on horizontal
asymptotes which gives us y 2 since numerator and denominator have the
3
same degree.
V) This function does not have any slant asymptotes since the degree of the
numerator is not one more than the degree of the denominator.
Before we come to the graph there is one more thing to look at, the so-called sign
property of rational functions. This says that the x-intercepts and vertical asymptotes of a
rational function divide the x-axis in intervals such that in every two adjacent intervals
the graph of the rational function alternates between being above or underneath the x-
axis. This holds true only as long as we have x-intercepts of odd multiplicity.
Now we are ready to manually make a graph of the function. Start by plotting the
coordinate system, the intercepts and the asymptotes. Next, take an x-value on the far left,
say x=-5 and calculate the y-value on the graph belonging to x=-5. You will find that
42
f ( 5) which is positive. The sign property says that between -∞ and the first x-
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intercepts, (-2,0) the function is above the x-axis and in the next interval, between (-2,0)
and 2 ,0 is underneath the x-axis etc. We also know that on the far left the graph
must bend towards the horizontal asymptote, pass through (-2,0) and then must go
towards the 'leftmost' vertical asymptote but may never touch it. What happens to the
right of this first 'leftmost' asymptote? We know that according to the sign property the
graph must be above the x-axis until the next asymptote, should pass through the y-
intercept and must approach both asymptotes. By continuing to reason like this we get the
following graph. Taking a couple of x-values like -3,-1, 1 and 3 will definitely improve
the accuracy of the graph.
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5. Of course Winplot can handle this also. Enter (2x^2-8)/(3x^2-6) and the graph will
appear, be it without the asymptotes showing.
x2 4
Let us next consider a graph with a slant asymptote: f ( x ) . The earlier
x 1
established routine gives us the following:
I) The y-intercept is (0,-4).
II) We find the x-intercepts by setting the numerator equal to zero and find (-2,0) and
(2,0).
III) The vertical asymptotes are found by setting the denominator equal to zero. So we
find only one vertical asymptote at x = -1.
IV) Using the theorem on horizontal asymptotes we find that there are none.
V) Since the degree of the numerator is one more than the degree of the denominator
we have a slant asymptote. Carrying out the long division x 2 4 x 1 gives
3
x 1 so that y = x - 1 is the slant asymptote. Following the reasoning of
x 1
the previous example we get the following graph.
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6. When graphing rational functions that have a slant asymptote be aware of the fact that
these graphs often will not appear at once in the display of a graphing calculator or
Winplot. You will have to zoom out to see them completely.
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