Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
1
TASK 5.3.1: RATIONAL FUNCTIONS AND THEIR RECIPROCALS
Solutions
Rational functions appear frequently in business, science, engineering, and medical
applications. This activity explores some aspects of rational functions.
1.
()
Graph the function f x = x using your graphing calculator. Use a ZOOM 4
window. Sketch your graph below:
(a) What is the domain of this function?
D: all real numbers, !" < x < " or !","
(
(b) What is the range of this function?
R: all real numbers, !" < x < " or !","
(
)
)
(c) Is the function increasing or decreasing, and over what interval(s)?
Increasing over all real numbers, !" < x < " or !","
(
)
Introduce the use of interval notation as a shortcut means to describe the domain
and range of a function. It is likely that many Algebra II teachers will not be
familiar with interval notation, as it is generally not introduced until Precalculus.
Closed interval - !" a,b #$ for a % x % b
Open interval - a,b for a < x < b
( )
)
Semi or half intervals - !" a,b for a # x < b
a,b $% for a < x # b
(
Note : ! and -! are always enclosed using parentheses.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
2.
2
()
Graph the function f x = x using your graphing calculator. Use a ZOOM 4
window. Sketch your graph below:
(a) What is the domain of this function?
D: all real numbers, !","
(
)
(b) What is the range of this function?
R: all positive real numbers, "#0,!
)
(c) Is the function increasing or decreasing, and over what interval(s)?
Decreasing over D: !",0
Increasing over D:
3.
( )
(0,! )
()
Graph the function f x = x 3 + 2x 2 + 1 using your graphing calculator. Use a
ZOOM 4 window. Sketch your graph below:
(a)
What is the domain of this function?
D: all real numbers, !","
(
)
(b) What is the range of this function?
R: all real numbers, !","
(
)
(c) Is the function increasing or decreasing, and over what interval(s)?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
3
#
4&
Increasing over D: % !",! (
3'
$
" 4 %
Decreasing over D: $ ! ,0'
# 3 &
( )
Increasing over D: 0,!
Ask participants how they found the relative maximum. Ask for volunteers to
demonstrate their method on the overhead calculator or on the board.
It can be approximated on the TI-83+ by using TRACE, or it can be calculated
using the maximum feature under the CALC menu (2nd ,TRACE, 4:maximum). It
can also be found by differentiating the function and setting the derivative equal to
zero. Demonstrate all three methods.
d 3
x + 2x 2 + 1 = 3x 2 + 4x
dx
3x 2 + 4x = 0
x(3x + 4) = 0
(
)
x = 0 or
3x + 4 = 0
3x = ! 4
4
x=!
3
4.
Graph the following functions along with their reciprocals. (Hint: Put the function
in Y1=, then make Y2= 1/Y1). Graph using the ZOOM 4 window. Sketch your
graphs on the following page.
()
(a) f x = x
()
(b) g x = x
()
(c) h x = x 2
()
(d) p x = x 2 + 1
()
(e) q x = 1 ! x
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
4
Compare the graph of each function with the graph of its reciprocal. Compare the
domain and range of each function with the domain and range of its reciprocal.
Compare the intervals over which each is increasing or decreasing. Note any points
where either the function or its reciprocal does not exist.
4.
(a)
( )
: ( !"," )
: ( !",0 ) # ( 0," )
: ( !",0 ) # ( 0," )
D f : !","
Rf
D1
R1
f
f
f is increasing over D: ( !", " ) .
( !",0) # (0," ) .
1
is decreasing over D:
f
1
does not exist at x = 0.
f
(b)
(
D f : !","
)
)
R f : #$0,"
D1 f : !",0 % 0,"
R1
f
( ) ( )
: ( 0," )
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
5
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
(
)
( )
f is decreasing over D: !",0 and increasing over D: 0,! .
(
)
( )
increasing over D: !",0 and decreasing over D: 0,! .
1
is
f
1
does not
f
exist at x = 0.
(c)
(
D f : !","
)
)
R f : #$0,"
D1 f : !",0 % 0,"
R1
f
( ) ( )
: ( 0," )
(
)
( )
f is decreasing over D: !",0 and increasing over D: 0,! .
(
)
( )
increasing over D: !",0 and decreasing over D: 0,! .
1
is
f
1
does not
f
exist at x = 0.
(d)
(
D f : !","
)
)
R f : #$1,"
D1 f : !","
R1
f
( )
: ( 0,1%&
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
6
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
(
)
( )
increasing over D: ( !",0 ) and decreasing over D: ( 0,! ) .
f is decreasing over D: !",0 and increasing over D: 0,! .
1
is
f
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
7
(e)
( )
: ( !"," )
: ( !",1) # (1," )
: ( !",0 ) # ( 0," )
D f : !","
Rf
D1
R1
f
f
(
)
f is decreasing over D: !"," .
( !",1) # (1," ) .
1
is increasing over D:
f
1
does not exist at x = 1.
f
Write any generalizations that you observe about the relationship between a
function and its reciprocal:
•
•
•
•
When a function is increasing, its reciprocal must be decreasing.
When the function is decreasing, its reciprocal must be increasing.
The existence of a function at all points on an interval does not guarantee
the existence of its reciprocal at all points on the interval.
When the value of a function is zero, its reciprocal does not exist.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
8
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
5.
Without using your graphing calculator, sketch the graph of the reciprocal of each
of the given functions. After you have sketched all three, then use your graphing
calculator to check your sketches. Correct any errors.
(
(
)(
)(
B. f (x) = x ! 3 x + 2
A. f (x) = x ! 3
)(
)
)
C. f (x) = x ! 3 x + 2 x ! 1
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
6.
9
Graph the following functions and their sum. (Hint: Put Function 1 in Y1=,
Function 2 in Y2=, then make Y3= Y1+Y2. Change the graph style for Y3 to show
the path (“football” icon -0). This will make it easier to view the graph of the sum).
Use the window shown.
()
()
()
()
(a) f x = x and g x =
(b) f x = x 2 and g x =
1
x
1
x
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
()
()
(c) f x = 2x + 1 and g x =
()
()
()
()
(d) f x = x ! 1 and g x =
(e) f x = x ! 1 and g x =
10
1
x2
(
1
x !1
(
1
x +1
)
)
Compare the graph of the sum of the two functions with the graph of each
individual function. What observations can you make about the sum of two
functions?
•
When x ! " , the fractional term in the sum goes to zero; therefore, the graph
•
of the sum approaches the graph of the non-fractional term.
When x ! a , where a is a zero of the fractional term, the graph of the sum
approaches the graph of the fractional term.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
7.
11
Graph the following functions. Compare each graph with the corresponding graph
from Problem 6. Turn off the graph for Y1 and Y2. Change the graph style for Y3
back to line, then make Y4 = the new function. Change the graph style for Y4 to
path. Use the window shown.
()
x2 + 1
x
()
x3 + 1
x
()
2x 3 + x 2 + 1
x2
()
x 2 ! 2x + 2
x !1
()
x2
x +1
(a) f x =
(b) g x =
(c) h x =
(d) p x =
(e) q x =
Compare the two graphs. What generalizations can you make?
The graphs are the same.
Rational functions are not always written as algebraic fractions. They can also be
expressed as the sum of a polynomial and a fraction.
Ask participants, “How are the sums in Problem 6 related to the rational
functions in Problem 7?”
Give them some think time. If no one offers a suggestion, give them a hint by
suggesting that they use long division to divide each rational function in Problem 7.
Have them write their answers in the form q(x) + r(x), where q(x) is the quotient
and r(x) is the remainder. Ask them to compare their results with f(x) and g(x) given
in Problem 6. Lead them to see that f(x) and g(x) are q(x) and r(x), respectively.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
12
TASK 5.3.1: RATIONAL FUNCTIONS AND THEIR RECIPROCALS
Rational functions appear frequently in business, science, engineering, and medical
applications. This activity explores some aspects of rational functions.
1.
()
Graph the function f x = x using your graphing calculator. Use a ZOOM 4
window. Sketch your graph below:
(a) What is the domain of this function?
(b) What is the range of this function?
(c) Is the function increasing or decreasing, and over what interval(s)?
2.
()
Graph the function f x = x using your graphing calculator. Use a ZOOM 4
window. Sketch your graph below:
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
13
(a) What is the domain of this function?
(b) What is the range of this function?
(c) Is the function increasing or decreasing, and over what interval(s)?
3.
()
Graph the function f x = x 3 + 2x 2 + 1 using your graphing calculator. Use a
ZOOM 4 window. Sketch your graph below:
(a) What is the domain of this function?
(b) What is the range of this function?
(c) Is the function increasing or decreasing, and over what interval(s)?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
4.
14
Graph the following functions along with their reciprocals. (Hint: Put the function
in Y1=, then make Y2= 1/Y1). Graph using the ZOOM 4 window. Sketch your
graphs on the following page.
()
(a) f x = x
()
(b) g x = x
()
(c) h x = x 2
()
(d) p x = x 2 + 1
()
(e) q x = 1 ! x
Compare the graph of each function with the graph of its reciprocal. Compare the
domain and range of each function with the domain and range of its reciprocal.
Compare the intervals over which each is increasing or decreasing. Note any points
where either the function or its reciprocal does not exist.
Write any generalizations that you observe about the relationship between a
function and its reciprocal:
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
15
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
5.
16
Without using your graphing calculator, sketch the graph of the reciprocal of each
of the given functions. After you have sketched all three, then use your graphing
calculator to check your sketches. Correct any errors.
A. f (x) = x ! 3
B. f (x) = (x ! 3)(x + 2)
C. f (x) = (x ! 3)(x + 2)(x ! 1)
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
6.
17
Graph the following functions and their sum. (Hint: Put Function 1 in Y1=,
Function 2 in Y2=, then make Y3= Y1+Y2. Change the graph style for Y3 to show
the path (“football” icon -0). This will make it easier to view the graph of the sum).
Use the window shown.
()
()
()
()
(a) f x = x and g x =
1
x
(b) f x = x 2 and g x =
()
1
x
()
(c) f x = 2x + 1 and g x =
()
()
()
()
(d) f x = x ! 1 and g x =
(e) f x = x ! 1 and g x =
1
x2
(
1
x !1
(
1
x +1
)
)
Compare the graph of the sum of the two functions with the graph of each
individual function. What observations can you make about the sum of two
functions?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.1
7.
18
Graph the following functions. Compare each graph with the corresponding graph
from Problem 6. Turn off the graph for Y1 and Y2. Change the graph style for Y3
back to line, then make Y4 = the new function. Change the graph style for Y4 to
path. Use the window shown.
()
x2 + 1
x
()
x3 + 1
x
()
2x 3 + x 2 + 1
x2
()
x 2 ! 2x + 2
x !1
()
x2
x +1
(a) f x =
(b) g x =
(c) h x =
(d) p x =
(e) q x =
Compare the two graphs. What generalizations can you make?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.