Investigating Oblique & Non-linear Asymptotes Activity

Oblique and Non-linear Asymptote Activity Name:
Prior Learning Reminder: Rational Functions
In the past we discussed vertical and horizontal asymptotes of the graph of a rational function of the form
n n1
n n1
m m1
m m1
a x a x a0
y
b x b x b
little simpler, let’s rename our rational function
0
where an 0 , b 0 and m and n are nonnegative integers. To make the notation a
m
Nx ( )
y f ( x)
where Nx ( ) and Dx ( ) are the numerator and
Dx ( )
denominator polynomials that define the rational function y f ( x)
. The following chart provides an overview of how we
determine vertical and horizontal asymptotes when dealing with rational functions.
Nx ( )
Let y f ( x)
be a rational function, where Nx ( ) and Dx ( ) have no common factors other
Dx ( )
than constants. The vertical asymptotes of y f ( x)
can be determined as follows:
Finding vertical asymptotes
Procedure Rational Function example Asymptotes of example
Vertical asymptotes occur at
each value of x which makes the
denominator Dx ( ) equal to 0.
15x 10
y f ( x)
2
9x 4
simplifies
to
15x 10
y f ( x)
2
9x4 5(3x 2)
(3x2)(3x2) 5
3x2 Finding horizontal asymptotes
Solving 3x 20yields 2
x as a vertical
3
asymptote.
Note: 3x 20does not
yield a vertical asymptote,
but does result in a hole in
the graph at
2
x .
3
Procedure Rational Function example Asymptote of example
Whenever the degree of the
numerator Nx ( ) is less than the
degree of the denominator Dx ( ) ,
the x-axis, which is the line
y 0 , is a horizontal asymptote.
Whenever the degree of the
numerator is equal to the degree
of the denominator, the line
a
y , where a and b are the
b
leading coefficients of the
numerator and denominator,
respectively, is the horizontal
asymptote.
Whenever the degree of the
numerator is greater than the
degree of the denominator, there
is no horizontal asymptote.
15x 10
y f ( x)
has 1 as the
2
9x4 degree of the numerator and 2 as
the degree of the denominator.
y f ( x)
2
14x 4x11 2
9x4 has 2
as the degree of the numerator
and denominator.
2
4x 4x3 y f ( x)
has 2
x 3
as the degree of the numerator
and 1 as the degree of the
denominator.
Since the degree of the
numerator is less than the
degree of the denominator,
y 0 is the horizontal
asymptote.
Since the degrees of the
numerator and denominator
are the same and the leading
coefficients of the numerator
and denominator are 14 and
14
9, respectively, y is the
9
horizontal asymptote.
Since the degree of the
numerator is greater than the
degree of the denominator,
the function has no horizontal
asymptote.

Investigating Other Types of Asymptotes of Rational Functions
Connection to Prior Learning: Rational Numbers
Given 141
141
r
, use long division to find the quotient q and remainder r. Is equal to q , where d is the divisor? (Recall
8 8 d
that the divisor is the denominator. In this case, d = 8). Verify your answer.
Extending to Rational Functions
1. Let’s investigate a rational function where the degree of the numerator is one greater than the degree of the
2
4x 4x3 denominator. For example, we can examine the rational function y
.
x 3
a) Use either long division or synthetic division in the space provided to determine the quotient and remainder. (Hint:
The remainder should be 45. If you don’t get 45, raise your hand.)
Quotient = Qx ()
__________________
Remainder = Rx ( ) 45
Numerator & Denominator of Rational Function Nx ()
________________
Dx ()
_________
Nx ( )
N( x) R( x)
Rational functions of the form y f ( x)
can be written as y f ( x) Q( x)
Dx ( )
D( x) D( x )
where Qx ( ) and
Rx ( ) are the quotient polynomial and the remainder polynomial, respectively, when we divide Nx ( ) by Dx ( ).
Rewrite
y
2
4x 4x3 x 3
( )
in the form ( ) .
( )
Rx
y Q x
Dx

2
4x 4x3 b) Let’s examine the y values of the rational function y
and the y values of the quotient function
x 3
y4x 16.
To do this put the rational function in and the quotient function in . Next, set up your table by
accessing and set the “Indpnt” variable to “Ask” so that you can input your x values into the table. Finally,
select and enter the x values from the tables below, recording the corresponding y values.
x
What can you say about the values in relative to the values in as x ?
2
4x 4x3 c) Graph the rational function y
x 3
and the quotient function y4x16using your graphing calculator
with the window settings below. Using two different colors, sketch the graphs of both in the box provided.
USE WINDOW
x:[-25,25] scale:5
y:[-150,50] scale:25
– 10
– 20
– 30
x 10 20 30 40 50 100
d) Where do the graphs of the rational function and the quotient function look alike? Where do they look different?
– 40
– 50
-100

2
4x 4x3 e) Notice that as the graph of y
approaches the graph of the quotient function y4x16. x 3
Since the quotient function’s degree is one, its graph is a line. When the graph of a rational function approaches a
non-horizontal line as x , that line is called an oblique asymptote of the rational function. For our rational
2
4x 4x3 function, y = 4x 16 is an equation of the oblique asymptote. Knowing that y
can be expressed as
x 3
2
45
4x 4x3 y4x16 , explain why the graph of y
must approach the line y4x16as .
x 3
x 3
(Hint: Think about what happens to 45
x
x
as x
.)
x 3
f) In part e above you explained why the graphs are close to each other as x . Give one possible reason why the
graphs are not close to each other near 3
x ?
2
4x 4x3 g) In part e above you learned that the rational function y
has an oblique asymptote that was obtained by
x 3
dividing the numerator of the rational function by the denominator of the rational function. Also, you learned that an
oblique asymptote of a rational function is a non-horizontal line, so it is defined by a degree 1 polynomial. What must
the relationship between the degrees of the numerator and the denominator of a rational function be in order for the
rational function to have an oblique asymptote? Explain.

3 2
2x 7x 13x 18
2. Now let’s consider the rational function y
where the degree of the numerator is 2 greater than
x 1
the degree of the denominator.
a) Use long division or synthetic division to determine the quotient polynomial Qx () and the remainder polynomial
Rx () . Then write the function in the form of
Qx ()
___________________
( )
( ) .
( )
Rx
y Q x
Dx
Rx ()
y Q() x
Dx () ________________________________
Rx ( ) 22
How does the degree of this quotient polynomial differ from the degree of the quotient polynomial in problem 1 a)?
What could be the reason for this difference?
b) As in 1b) examine the y values of the rational function and the quotient function y Q( x ) . Remember to let
represent the rational function and the quotient function.
x
– 2
– 10
– 18
x 2 10 18 26 34 42
What can you say about the values of the rational function relative to the values of the quotient function as x ?
– 26
– 34
– 42

c) Graph the rational function and the quotient function using your graphing calculator with the settings below. Using
two different colors, sketch the graphs of both in the box provided.
USE WINDOW
x:[-10,10] scale:1
y:[-40,100] scale:20
3 2
2x 7x 13x 18
d) Notice that as x the graph of y
x 1
approaches the graph of the quotient function
y Q( x ) . This quotient function is a non-linear asymptote of our rational function. This asymptote is non-linear
because the quotient polynomial’s degree is not 1. Knowing that
y
3 2
22
2x 7x 13x 18
y Q( x)
, explain why the graph of y
x 1
x 1
x
. (Hint: If needed, refer to the hint in 1e.)
3. a) Given the rational function
y
Qx ()
__________________________
5 3
x 5x4x 2
x x12 Rx ()
y Q() x
Dx () ______________________________
find the following:
3 2
2x 7x 13x 18
x 1
can be expressed as
must approach the graph of y Q( x) as
R( x) 120x 240
The degree of the numerator is how much greater than the degree of the denominator? ___________
What is the degree of the quotient polynomial? ___________
Your answers to the previous two questions should be the same. Why should they be the same?

) Graph the rational function and the quotient function using your graphing calculator with the settings below. Sketch
the graphs of both in the box provided.
USE WINDOW
x:[-8,8] scale:1
y:[-500,500] scale:100
c) What do the graphs in part b) suggest to you about the rational function and its corresponding quotient function?
Explain.
4. Given the rational function
a) Use the above to determine the asymptote of the graph of
5 3
2x 58x 200x 4 3 2
144
y 2x 2x 56x 56x 144 .
x1 x1
(Note that the division has already been done for you)
5 3
2x 58x 200x
y
x 1
as x .
b) Verify your answer by graphing the rational function and your asymptote in the box provided.
USE WINDOW
x:[-6,6] scale:1
y:[-200,400] scale:50
5. Find the equation of the asymptote as x for each of the following rational functions. Graph each rational
function and asymptote in a suitable window to verify your results.
a)
y
4 3 2
x x x x
x 2
2
b)
y
2
x x
x 2
c)
y
3 2
x x x
2 4 9
2x3