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Rational Functions

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Graphs of Rational Functions
Prepared for Mth 163: Precalculus 1 Online
By Richard Gill
Through funding provided by a
VCCS LearningWare Grant
A rational function is a function that can be expressed in the form
yпЂЅ
f ( x)
where both f(x) and g(x) are polynomial functions.
g ( x)
Examples of rational functions would be:
yпЂЅ
1
xпЂ­2
f ( x) пЂЅ
2x
3пЂ­ x
x пЂ­4
2
g ( x) пЂЅ
x пЂ­ 2x
2
Over the next few frames we will look at the graphs of each of
the above functions.
First we will look at y пЂЅ
1
xпЂ­2
.
This function has one value of x that is banned from the domain.
What value of x do you think that would be? And why?
If you guessed x = 2, congratulations. This is the value at which
the function is undefined because x = 2 generates 0 in the
denominator.
Consider the graph of the function. What impact do you
think this forbidden point will have on the graph?
Think before you click.
Now just because we cannot use x = 2 in our x-y table, it does not
mean that we cannot use values of x that are close to 2. So before
you click again, fill in the values in the table below.
x
y пЂЅ
1.5
-2
1
xпЂ­2
1.7 -3.33
1.9
-10
2.0
undefined
As we pick values of x that are smaller
than 2 but closer and closer to 2 what do
you think is happening to y?
If you said that y is getting closer and
closer to negative infinity, nice job!
Now fill in the values in the rest of the table.
1
x
y пЂЅ
1.5
-2
1.7
-3.33
1.9
-10
2.0
Und
2.1
10
2.3
3.33
2.5
2
xпЂ­2
What about the behavior of the function
on the other side of x = 2? As we pick
values of x that are larger than 2 but closer
and closer to 2 what do you think is
happening to y?
If you said that y is getting closer and
closer to positive infinity, you are right
on the money!
Let’s see what the points that we have calculated so far would look
like on graph.
y пЂЅ
пЂ±пЂ°пЂ®пЂ°
1
This dotted
vertical line is a
crucial visual aid
for the graph. Do
you know what
the equation of
this dotted line is?
xпЂ­2
y
(2.1, 10)
пЂёпЂ®пЂ°
пЂ¶пЂ®пЂ°
(2.3, 3.33)
пЂґпЂ®пЂ°
пЂІпЂ®пЂ°
(2.5, 2)
пЂІпЂ®пЂ°
пЂ­пЂІпЂ®пЂ°
пЂ­пЂґпЂ®пЂ°
(1.5, -2)
(1.7, -3.33)
пЂ­пЂ¶пЂ®пЂ°
пЂ­пЂёпЂ®пЂ°
пЂ­пЂ±пЂ°пЂ®пЂ°
(1.9, -10)
x
пЂґпЂ®пЂ°
The equation is
x = 2 because
every point on
the line has an
x coordinate of
2.
y пЂЅ
пЂ±пЂ°пЂ®пЂ°
1
Do you know
what this dotted
vertical line is
called?
xпЂ­2
y
(2.1, 10)
пЂёпЂ®пЂ°
пЂ¶пЂ®пЂ°
(2.3, 3.33)
пЂґпЂ®пЂ°
пЂІпЂ®пЂ°
(2.5, 2)
пЂІпЂ®пЂ°
пЂ­пЂІпЂ®пЂ°
пЂ­пЂґпЂ®пЂ°
(1.5, -2)
(1.7, -3.33)
пЂ­пЂ¶пЂ®пЂ°
The line x = 2 is a
vertical asymptote.
пЂ­пЂёпЂ®пЂ°
пЂ­пЂ±пЂ°пЂ®пЂ°
x
пЂґпЂ®пЂ°
Hint: it is one of
the many great
and imaginative
words in
mathematics.
(1.9, -10)
yпЂЅ
пЂ±пЂ°пЂ®пЂ°
Our graph will get
closer and closer
to this vertical
asymptote but
never touch it.
1
xпЂ­2
y
(2.1, 10)
пЂёпЂ®пЂ°
пЂ¶пЂ®пЂ°
(2.3, 3.33)
пЂґпЂ®пЂ°
пЂІпЂ®пЂ°
(2.5, 2)
пЂІпЂ®пЂ°
пЂ­пЂІпЂ®пЂ°
пЂ­пЂґпЂ®пЂ°
(1.5, -2)
(1.7, -3.33)
пЂ­пЂ¶пЂ®пЂ°
пЂ­пЂёпЂ®пЂ°
пЂ­пЂ±пЂ°пЂ®пЂ°
(1.9, -10)
x
пЂґпЂ®пЂ°
If f(x)
approaches
positive or
negative infinity
as x approaches c
from the right or
the left, then the
line x = c is a
vertical
asymptote of the
graph of f.
A horizontal asymptote is a horizontal line that the graph gets
closer and closer to but never touches. The official definition of a
horizontal asymptote:
The line y = c is a horizontal asymptote for the graph of a
function f if f(x) approaches c as x approaches positive or
negative infinity.
Huh?!
Don’t you just love official definitions? At any rate,
rational functions have a tendency to generate
asymptotes, so lets go back to the graph and see if we can
find a horizontal asymptote.
y пЂЅ
пЂ±пЂ°пЂ®пЂ°
1
xпЂ­2
y
(2.1,10)
пЂёпЂ®пЂ°
пЂ¶пЂ®пЂ°
(2.3,3.33)
пЂґпЂ®пЂ°
пЂІпЂ®пЂ°
(2.5,2)
пЂІпЂ®пЂ°
пЂ­пЂІпЂ®пЂ°
пЂ­пЂґпЂ®пЂ°
(1.5,-2)
(1.7,-3.33)
пЂ­пЂ¶пЂ®пЂ°
пЂ­пЂёпЂ®пЂ°
пЂ­пЂ±пЂ°пЂ®пЂ°
Looking at the graph, as
the x values get larger
and larger in the
negative direction, the y
values of the graph
appear to get closer and
closer to what?
(1.9,-10)
x
If you guessed that the
y values appear to get
closer and closer to 0,
you may be onto
something. Let’s look
at a table of values for
confirmation.
Before you click again, take a minute to calculate the y values in
the table below. What is your conclusion about the trend?
x
y пЂЅ
1
xпЂ­2
0
-(1/2)
-5
-(1/7)
-20
-(1/22)
-100
-(1/102)
Conclusion: as the x values get closer and closer to negative
infinity, the y values will get closer and closer to 0.
Question: will the same thing happen as x values get closer to
positive infinity?
How about a guess? What do you think is going to happen to the y
values of our function as the x values get closer to positive infinity?
As x п‚®
п‚Ґ, y пЂЅ
As x п‚® п‚Ґ ,
yпЂЅ
1
xпЂ­2
1
xпЂ­2
п‚® ?
п‚® 0
By looking at the fraction analytically, you can hopefully see that
very large values of x will generate values of y very close to 0. If
you are uneasy about this, expand the table in the previous slide to
include values like x = 10, 100, or 1000.
On the next frame then, is our final graph for this problem
Note how the graph is very much dominated by its asymptotes. You
can think of them as magnets for the graph. This problem was an
exploration but in the future, it will be very important to know where
your asymptotes are before you start plotting points.
y
y = 1/(x-2)
Vertical Asymptote
at x = 2
пЂґпЂ®пЂ°
пЂІпЂ®пЂ°
пЂ­пЂґпЂ®пЂ°
пЂ­пЂІпЂ®пЂ°
пЂІпЂ®пЂ°
пЂґпЂ®пЂ°
пЂ¶пЂ®пЂ°
пЂёпЂ®пЂ°
x
пЂ­пЂІпЂ®пЂ°
пЂ­пЂґпЂ®пЂ°
пЂ­пЂ¶пЂ®пЂ°
Horizontal
Asymptote at y = 0.
Next up is the graph of one of the functions that was mentioned
back in frame #2.
f ( x) пЂЅ
2x
3пЂ­ x
Let’s see if we can pick out the asymptotes analytically before we
start plotting points in an x-y table.
Do we have a vertical asymptote? If so, at what value of x?
We have a vertical asymptote at x = 3 because at that value of x, the
denominator is 0 but the numerator is not. Congratulations if you
picked this out on your own.
The horizontal asymptote is a little more challenging, but go ahead
and take a guess.
Notice though that as values of x get larger and larger, the 3 in the
denominator carries less and less weight in the calculation.
f ( x) пЂЅ
2x
3пЂ­ x
As the 3 “disappears”, the function looks
more and more like…
f ( x) пЂЅ
2x
пЂ­x
which reduces to y = -2.
This means that we should have a horizontal asymptote at y = -2.
We already have evidence of a vertical asymptote at x = 3. So we
are going to set up the x-y table then with a few values to the left
of x = 3 and a few values to the right of x = 3. To confirm the
horizontal asymptote we will also use a few large values of x just
to see if the corresponding values of y will be close to y = -2.
2x
x
y пЂЅ
-5
-10/8 = -1.25
0
0
2.5
5/.5 = 10
3
Undefined
3пЂ­ x
3.5
7/-.5 = -14
5
10/-2 = -5
10 20/-7 = -2.86
50 100/-47= -2.13
Take a few minutes and work out the
y values for this table.
Don’t be lazy now, work them out
yourself.
As expected, y values tend to explode
when they get close to the vertical
asymptote at x = 3.
Also, as x values get large, y values
get closer and closer to the horizontal
asymptote at y = -2.
The graph is a click away.
y
y = 2x/(3-x)
пЂёпЂ®пЂ°
Here is the graph
with most of the
points in our table.
пЂ¶пЂ®пЂ°
пЂґпЂ®пЂ°
пЂІпЂ®пЂ°
пЂ­пЂ¶пЂ®пЂ°
пЂ­пЂґпЂ®пЂ°
пЂ­пЂІпЂ®пЂ°
пЂІпЂ®пЂ°
пЂ­пЂІпЂ®пЂ°
пЂґпЂ®пЂ°
пЂ¶пЂ®пЂ°
пЂёпЂ®пЂ°
пЂ±пЂ°пЂ®пЂ°
x
пЂ±пЂІпЂ®пЂ°
Vertical asymptote
at x = 3.
пЂ­пЂґпЂ®пЂ°
пЂ­пЂ¶пЂ®пЂ°
пЂ­пЂёпЂ®пЂ°
пЂ­пЂ±пЂ°пЂ®пЂ°
пЂ­пЂ±пЂІпЂ®пЂ°
пЂ­пЂ±пЂґпЂ®пЂ°
Horizontal
asymptote at y = -2.
Believe it or not, you are now sophisticated enough mathematically
to draw conclusions about the graph three ways:
Analytically:
finding
asymptotes with
algebra!!
Numerically:
supporting and
generating
conclusions
with the x-y
table!!
Graphically: a
visual look at the
behavior of the
function.
If your conclusions from the above areas do not agree, investigate
further to uncover the nature of the problem.
We are going to finish this lesson with an analysis of the third
function that was mentioned in the very beginning:
x пЂ­4
2
g ( x) пЂЅ
x пЂ­ 2x
2
This is a rational function so we have
potential for asymptotes and this is
what we should investigate first. Take
a minute to form your own opinion
before you continue.
Hopefully you began by setting the denominator equal to 0.
x пЂ­ 2x пЂЅ 0
2
xпЂЁx пЂ­ 2 пЂ© пЂЅ 0
x пЂЅ 0, x пЂЅ 2
It appears that we may have vertical
asymptotes at x = 0 and at x = 2. We will
see if the table confirms this suspicion.
x пЂ­4
2
x
yпЂЅ
x пЂ­ 2x
See anything peculiar?
2
-2
0
-1
-1
-.5
-3
-.1
-19
0
Und
1
3
1.5
2.33
1.9
2.05
2
und
Notice that as x values get closer and
closer to 0, the y values get larger and
larger. This is appropriate behavior
near an asymptote.
But as x values get closer and closer
to 2, the y values do not get large. In
fact, the y values seem to get closer
and closer to 2.
Now, if x =2 creates 0 in the
denominator why don’t we have an
asymptote at x = 2?
We don’t get a vertical asymptote at x = 2
because when x = 2 both the numerator and
the denominator are equal to 0. In fact, if we
had thought to reduce the function in the
beginning, we could have saved ourselves a lot
of trouble. Check this out:
x пЂ­4
2
y пЂЅ
x пЂ­ 2x
2
пЂЅ
пЂЁ x пЂ« 2 пЂ©пЂЁ x пЂ­ 2 пЂ©
пЂЅ
xпЂЁx пЂ­ 2 пЂ©
xпЂ«2
x пЂ­4
x
2
Does this mean that
yпЂЅ
x пЂ­ 2x
2
and
yпЂЅ
xпЂ«2
x
are identical functions?
Yes, at every value of x except x = 2 where the former is undefined.
There will be a tiny hole in the graph where x = 2.
x пЂ­4
2
g ( x) пЂЅ
x пЂ­ 2x
2
As was the case with
the previous function,
we concentrate on the
ratio of the term with
the largest power of x
in the numerator to
the term with the
largest power of x in
the denominator. As x
gets large…
x пЂ­4
2
x пЂ­ 2x
2
п‚»
x
2
x
2
пЂЅ1
As we look for horizontal asymptotes, we
look at y values as x approaches plus or
minus infinity. The denominator will get
very large but so will the numerator.
You can verify this in the table.
x пЂ­4
2
x
yпЂЅ
10
1.2
100
1.02
1000
1.002
x пЂ­ 2x
2
So, we have a horizontal asymptote at
y = 1.
To summarize then, we have a vertical asymptote at x = 0, a hole in
the graph at x = 2 and a horizontal asymptote at y = 1. Here is the
graph with a few of the points that we have in our tables.
x пЂ­4
2
g ( x) пЂЅ
Hole in the graph.
x пЂ­ 2 xy
2
пЂґпЂ®пЂ°
Horizontal asymptote
at y = 1.
пЂІпЂ®пЂ°
пЂ­пЂґпЂ®пЂ°
пЂ­пЂІпЂ®пЂ°
пЂІпЂ®пЂ°
пЂ­пЂІпЂ®пЂ°
пЂ­пЂґпЂ®пЂ°
пЂґпЂ®пЂ°
x
пЂ¶пЂ®пЂ°
Vertical asymptote at
x = 0.
Now you will get a chance to practice on exercises that use the
topics that were covered in this lesson:
Finding vertical and horizontal asymptotes in rational functions.
Graphing rational functions with asymptotes.
Good luck and watch out for those asymptotes!
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