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Lesson 13
M3
PRECALCULUS AND ADVANCED TOPICS
Lesson 13: Horizontal and Vertical Asymptotes of Graphs of
Rational Functions
Classwork
Opening Exercise
Determine the end behavior of each rational function below. Graph each function on the graphing calculator and
explain how the graph supports your analysis of the end behavior.
1.
( )=
2.
( )=
3.
( )=
2
3
3
2
3
2 +1
3
2
3
Lesson 13:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Horizontal and Vertical Asymptotes of Graphs of Rational Functions
S.70
Lesson 13
M3
PRECALCULUS AND ADVANCED TOPICS
Example 1
Consider the rational function ( ) =
.
a.
State the domain of .
b.
Determine the end behavior of .
c.
State the horizontal asymptote of the graph of
d.
Graph the function on the graphing calculator and make a sketch on your paper.
Lesson 13:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
= ( ).
Horizontal and Vertical Asymptotes of Graphs of Rational Functions
S.71
Lesson 13
M3
PRECALCULUS AND ADVANCED TOPICS
Exercises 1–9
State the domain and end behavior of each rational function. Identify all horizontal and vertical asymptotes on the
graph of each rational function. Then, verify your answer by graphing the function on the graphing calculator.
1.
( )=
2.
( )=
3.
( )=
+6
2 +3
3
6
3
2
25
Lesson 13:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Horizontal and Vertical Asymptotes of Graphs of Rational Functions
S.72
Lesson 13
M3
PRECALCULUS AND ADVANCED TOPICS
4.
( )=
5.
( )=
6.
( )=
2
2 +2
2
2
3
5 4
+1
5
2 +9
Write an equation for a rational function whose graph has the given characteristic. Graph your function on the graphing
calculator to verify.
7.
A horizontal asymptote of
= 2 and a vertical asymptote of
Lesson 13:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
=
2.
Horizontal and Vertical Asymptotes of Graphs of Rational Functions
S.73
Lesson 13
M3
PRECALCULUS AND ADVANCED TOPICS
= 6 and no horizontal asymptote.
8.
A vertical asymptote of
9.
A horizontal asymptote of
= 6 and no vertical asymptote.
Lesson 13:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Horizontal and Vertical Asymptotes of Graphs of Rational Functions
S.74
Lesson 13
M3
PRECALCULUS AND ADVANCED TOPICS
Lesson Summary
Let be a real number. The line given by
one of the following statements is true.
As
, ( )
As
, ( )
=
.
, ( )
is a horizontal asymptote of the graph of
= ( ) if at
.
, ( )
As
= ( ) if at least
.
Let be a real number. The line given by =
least one of the following statements is true.
As
is a vertical asymptote of the graph of
.
Problem Set
1.
State the domain of each rational function. Identify all horizontal and vertical asymptotes on the graph of each
rational function.
a.
y=
b.
y=
c.
y=
d.
=
e.
( )
f.
( )
2 3 +1
16 9 6
6 4
=
+5
4
= 2
4
3
6
2.
Sketch the graph of each function in Exercise 1 with asymptotes and excluded values from the domain drawn on the
graph.
3.
Factor out the highest power of
nonzero.
a.
=
b.
=
c.
=
d.
=
e.
=
3 +3
3
3
3
4
in each of the following, and cancel common factors if you can. Assume
is
4
2 +2
5
2
6
3 +5 2 +6
2
5
4
3
3 +1
8 1
3
4
2 +1
Lesson 13:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Horizontal and Vertical Asymptotes of Graphs of Rational Functions
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Lesson 13
M3
PRECALCULUS AND ADVANCED TOPICS
4.
Describe the end behavior of each function in Exercise 3.
5.
Using the equations that you wrote in Exercise 3, make some generalizations about how to quickly determine the
end behavior of a rational function.
6.
Describe how you may be able to use the end behavior of the graphs of rational functions, along with the excluded
values from the domain and the equations of any asymptotes, to graph a rational function without technology.
Lesson 13:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Horizontal and Vertical Asymptotes of Graphs of Rational Functions
S.76