1
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.3
TASK 5.3.3: RATIONAL FUNCTIONS AND ASYMPTOTES
Solutions
1.
Graph the following functions and their quotient
f
.
g
Note: Graphs below represent only the quotients.
(a) f x = 3x + 1 and g x = x + 1
()
()
()
()
(b) f x = !2(x + 2)(x ! 1) and g x = (x + 1)(x ! 1)
()
()
(c) f x = 4x 2 + 1 and g x = ! 2x 2 ! 3
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.
2
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.3
()
()
(d) f x = !3(x ! 1)2 (x + 2) and g x = (x + 1)2 (x ! 1)
Compare the graph of the quotient with the graphs of the numerator and
denominator. Compare the graphs with the function values found in the function
table. Next, turn off graphs Y1 and Y2. This will allow you to compare X and Y3
in adjacent columns in the table. Compare the graph of the quotient with the
function values in the table. Note what happens to the function value as x ! " .
What observations can you make?
The function value approaches the leading coefficient as x ! " .
Lead participants to discover that a horizontal asymptote occurs when the leading
a
term powers are the same. The asymptote will be located at y = , where a is the
b
leading coefficient of the numerator and b is the leading coefficient of the
denominator.
Ask, “Under what circumstances would a rational function have the x-axis as a
horizontal asymptote? Give an example.”
A rational function will approach the x-axis as a horizontal asymptote when the
degree of the numerator is less than the degree of the denominator.
Ask, “It was not possible for a function to cross a vertical asymptote. Is it
possible for a function to cross or touch a horizontal asymptote? Why, or why
not?”
The function may cross or touch a horizontal asymptote because the asymptote
describes the behavior of the function in the long run. Another way to say this is
that the asymptote is the limit of the function as x approaches ± ∞.
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.
3
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.3
Point out that (a), (b), and (d) have vertical asymptotes. Note that (b) and (d) also
have removable discontinuities which are clearly revealed in the table. Point out
that (c) has a horizontal asymptote, but no vertical asymptotes or removable
discontinuities.
2.
()
Graph the function f x =
window shown below:
2x 3 + x 2 ! 1
using your graphing calculator. Use the
x2
(a) Use long division to divide the numerator by the denominator. Write the result
in the form q(x) + r(x) , where q(x) is the quotient and r(x) is the remainder.
f (x) =
2x 3 + x 2 ! 1
1
1
=
2x
+
1
!
,
where
q(x)
=
2x
+
1
and
r(x)
=
!
x2
x2
x2
(b) Look closely at the function in this form. What happens to the function as x
gets large?
As x gets large, r(x) vanishes and f(x) tends to 2x + 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.
4
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.3
(c) Enter the quotient, q(x) , in Y2=, and graph simultaneously with the original
function, f(x). How does the graph of the quotient compare with the graph of
the function near the origin? How do they compare for large absolute values of
x?
The graph of f(x), tends to the graph of q(x) as x→ ± ∞.
3.
2x 3 + x 2 ! 1
Let g(x) =
.
x2 ! 1
(a) Divide the numerator by the denominator and write the result as q(x) + r(x) .
2x 3 + x 2 ! 1
1
1
g(x) =
= 2x + 1 ! 2
, where q(x) = 2x + 1 and r(x) = ! 2
2
x !1
x !1
x !1
(b) Enter q(x) in Y1= and g(x) in Y2=. Change the graph style for Y2 to show the
path. Set the window as shown below, then graph.
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 5.3.3
(c) Compare the graphs of g(x) and q(x) . What observations can you make?
As x gets large, r(x) vanishes and g(x) tends to 2x + 1.
4.
Let h(x) =
2x 3 + x 2 ! 1
.
x2 + 1
(a) Divide the numerator by the denominator and write the result as q(x) + r(x) .
q(x) = 2x + 1, r(x) =
-2x-2
x2 + 1
(b) Enter q(x) in Y1= and h(x) in Y2=. Change the graph style for Y2 to show the
path. Set the window as shown below, then graph.
(c)
Compare the graphs of h(x) and q(x) . What observations can you make?
As x gets large, r(x) vanishes and h(x) tends to 2x + 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 5.3.3
TASK 5.3.3: RATIONAL FUNCTIONS AND ASYMPTOTES
1.
Graph the following functions and their quotient
()
f
.
g
()
(a) f x = 3x + 1 and g x = x + 1
()
()
(b) f x = !2(x + 2)(x ! 1) and g x = (x + 1)(x ! 1)
()
()
(c) f x = 4x 2 + 1 and g x = ! 2x 2 ! 3
()
()
(d) f x = !3(x ! 1)2 (x + 2) and g x = (x + 1)2 (x ! 1)
Compare the graph of the quotient with the graphs of the numerator and
denominator. Compare the graphs with the function values found in the function
table. Next, turn off graphs Y1 and Y2. This will allow you to compare X and Y3
in adjacent columns in the table. Compare the graph of the quotient with the
function values in the table. Note what happens to the function value as x ! " .
What observations 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.
7
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.3
2.
()
Graph the function f x =
window shown below:
2x 3 + x 2 ! 1
using your graphing calculator. Use the
x2
(a) Use long division to divide the numerator by the denominator. Write the result
in the form q(x) + r(x) , where q(x) is the quotient and r(x) is the remainder.
(b) Look closely at the function in this form. What happens to the function as x
gets large?
(c) Enter the quotient, q(x) , in Y2=, and graph simultaneously with the original
function, f(x). How does the graph of the quotient compare with the graph of
the function near the origin? How do they compare for large absolute values of
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.
8
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.3
3.
Let g(x) =
2x 3 + x 2 ! 1
.
x2 ! 1
(a) Divide the numerator by the denominator and write the result as q(x) + r(x) .
(b) Enter q(x) in Y1= and g(x) in Y2=. Change the graph style for Y2 to show the
path. Set the window as shown below, then graph.
(c) Compare the graphs of g(x) and q(x) . What observations can you make?
4.
Let h(x) =
2x 3 + x 2 ! 1
.
x2 + 1
(a) Divide the numerator by the denominator and write the result as q(x) + r(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.
9
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.3
(b) Enter q(x) in Y1= and h(x) in Y2=. Change the graph style for Y2 to show the
path. Set the window as shown below, then graph.
(c) Compare the graphs of h(x) and q(x) . What observations 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.