550
(10-22)
Chapter 10
Polynomial and Rational Functions
51. f(x) (x 20)2(x 30)
53. f(x) (x 20)2(x 30)2x
52. f(x) (x 20)2(x 30)2
54. f(x) (x 20)2(x 30)2x2
10.4 G R A P H S O F R A T I O N A L F U N C T I O N S
We first studied rational expressions in Chapter 6. In this section we will study
functions that are defined by rational expressions.
In this
section
●
Domain
●
Horizontal and Vertical
Asymptotes
●
Oblique Asymptotes
●
Sketching the Graphs
Domain
A rational expression was defined in Chapter 6 as a ratio of two polynomials. If a
ratio of two polynomials is used to define a function, then the function is called a
rational function.
Rational Function
P(x)
for
If P(x) and Q(x) are polynomials with no common factor and f (x) Q(x)
Q(x) 0, then f (x) is called a rational function.
The domain of a rational function is the set of all real numbers except those that
cause the denominator to have a value of 0.
E X A M P L E
1
Domain of a rational function
Find the domain of each rational function.
x3
2x 3
a) f (x) b) g(x) x1
x2 4
10.4
Graphs of Rational Functions
(10-23)
551
Solution
a) Since x 1 0 only for x 1, the domain of f is the set of all real numbers
except 1, x x 1
.
b) Since x 2 4 0 for x 2, the domain of g is the set of all real numbers
■
excluding 2 and 2, x x 2 and x 2 .
calculator
close-up
If the viewing window is too
large, a rational function will
appear to touch its asymptotes.
Horizontal and Vertical Asymptotes
Consider the simplest rational function f (x) 1x. Its domain does not include 0,
but 0 is an important number for the graph of this function. The behavior of the
graph of f when x is very close to 0 is what interests us. For this function the
y-coordinate is the reciprocal of the x-coordinate. When the x-coordinate is close to
0, the y-coordinate is far from 0. Consider the following tables of ordered pairs that
satisfy f (x) 1x:
8
–8
8
x
0
–8
Because the asymptotes are
an important feature of a rational function, we should draw it
so that it approaches but does
not touch its asymptotes.
x0
x
y
x
y
0.1
0.01
0.001
0.0001
10
100
1000
10,000
0.1
0.01
0.001
0.0001
10
100
1000
10,000
y
5
4
3
2
1
–1
f(x) =
1
x
1 2 3 4 5
FIGURE 10.5
E X A M P L E
2
x
As x gets closer and closer to 0 from above 0, the value of y gets larger and larger.
We say that y goes to positive infinity. As x gets closer and closer to 0 from below
0, the values of y are negative but y gets larger and larger. We say that y goes to
negative infinity. The graph of f gets closer and closer to the vertical line x 0, and
so x 0 is called a vertical asymptote. On the other hand, as x gets larger and
larger, y gets closer and closer to 0. The graph approaches the x-axis as x goes to infinity, and so the x-axis is a horizontal asymptote for the graph of f. See Fig. 10.5
for the graph of f (x) 1x.
In general, a rational function has a vertical asymptote for every number excluded from the domain of the function. The horizontal asymptotes are determined
by the behavior of the function when x is large.
Horizontal and vertical asymptotes
Find the horizontal and vertical asymptotes for each rational function.
3
x
2x 1
a) f (x) b) g(x) c) h(x) 2 2 x 1
x 4
x3
Solution
a) The denominator x 2 1 has a value of 0 if x 1. So the lines x 1 and
3
x 1 are vertical asymptotes. If x is very large, the value of is
x2 1
approximately 0. So the x-axis is a horizontal asymptote.
b) The denominator x 2 4 has a value of 0 if x 2. So the lines x 2 and
x
x 2 are vertical asymptotes. If x is very large, the value of is
x2 4
approximately 0. So the x-axis is a horizontal asymptote.
552
(10-24)
Chapter 10
calculator
close-up
The graph for Example 2(b)
should consist of three separate pieces, but in connected
mode the calculator connects
the separate pieces. Even
though the calculator does
not draw a very good graph
of this function, it does support the conclusion that the
horizontal asymptote is the
x-axis and the vertical asymptotes are x 1 and x 1.
40
–5
5
–40
Polynomial and Rational Functions
c) The denominator x 3 has a value of 0 if x 3. So the line x 3 is a
vertical asymptote. If x is very large, the value of h(x) is not approximately 0.
To understand the value of h(x), we change the form of the rational expression
by using long division:
2
1
x 32x
2x 6
5
mainder
Writing the rational expression as quotient re
, we get
divisor
2x 1
5
5
h(x) x 3 2 x 3. If x is very large, x 3 is approximately 0, and so
the y-coordinate is approximately 2. The line y 2 is a horizontal asymptote. ■
Example 2 illustrates two important facts about horizontal asymptotes. If the
degree of the numerator is less than the degree of the denominator, then the x-axis
x4
is the horizontal asymptote. For example, y x 2 7 has the x-axis as a horizontal
asymptote. If the degree of the numerator is equal to the degree of the denominator,
then the ratio of the leading coefficients determines the horizontal asymptote. For
2x 7
2
example, y 3x 5 has y 3 as its horizontal asymptote. The remaining case is
when the degree of the numerator is greater than the degree of the denominator. This
case is discussed next.
Oblique Asymptotes
Each rational function of Example 2 had one horizontal asymptote and a vertical
asymptote for each number that caused the denominator to be 0. The horizontal
asymptote y 0 occurs because as x gets larger and larger, the y-coordinate gets
closer and closer to 0. Some rational functions have a nonhorizontal line for
an asymptote. An asymptote that is neither horizontal nor vertical is called an
oblique asymptote or slant asymptote.
E X A M P L E
3
Finding an oblique asymptote
Determine all of the asymptotes for
2x2 3x 5
g(x) .
x2
study
tip
The last couple of weeks of
the semester is not the time to
slack off. This is the time to
double your efforts. Make a
schedule and plan every hour
of your time. Don’t schedule
anything that isn’t necessary.
Get an early start on studying
for your final exams.
Solution
If x 2 0, then x 2. So the line x 2 is a vertical asymptote. Use long
remainder
division to rewrite the function as quotient :
divisor
2x 2 3x 5
3
g(x) 2x 1 x2
x2
3
If x is large, the value of is approximately 0. So when x is large, the value
x2
of g(x) is approximately 2x 1. The line y 2x 1 is an oblique asymptote for
■
the graph of g.
We can summarize this discussion of asymptotes with the following strategy for
finding asymptotes for a rational function.
10.4
Graphs of Rational Functions
(10-25)
553
Strategy for Finding Asymptotes for a Rational Function
P(x)
Suppose f (x) is a rational function with the degree of Q(x) at least 1.
Q(x)
1. Solve the equation Q(x) 0. The graph of f has a vertical asymptote corresponding to each solution to the equation.
2. If the degree of P(x) is less than the degree of Q(x), then the x-axis is a
horizontal asymptote.
3. If the degree of P(x) is equal to the degree of Q(x), then find the ratio of the
leading coefficients. The horizontal line through that ratio is the horizontal
asymptote.
4. If the degree of P(x) is greater than the degree of Q(x), then use division to
rewrite the function as
remainder
quotient .
divisor
The equation formed by setting y equal to the quotient gives us an oblique
asymptote.
M A T H
A T
W O R K
Machines that do computations have been around for
thousands of years. The abacus, the slide rule, and the
calculator have simplified computations. However, recent calculators have gone way beyond numerical computations. Graphing calculators now draw two- and
three-dimensional graphs and even do symbolic compuMATHEMATICS
tations. Modern calculators are great, but to use one efMACHINES
fectively you must still learn the underlying principles
of mathematics.
Consider the fairly simple process of drawing a
graph of a function. The graph of a function is a picture
of all ordered pairs of the function. When we graph a function, we typically plot a
few ordered pairs and then use our knowledge about the function to draw a graph
that shows all of the important features of the function. A typical graphing calculator plots 96 ordered pairs of a function on a screen that is not much bigger than a
postage stamp. The calculator does not generalize and does not make conclusions.
We are still responsible for looking at what the calculator shows and making conclusions. For example, if we graphed y 1(x 100) with the window set for
10 x 10, we would not see the vertical asymptote x 100. Would we believe the calculator and conclude that the graph has no vertical asymptote? If we
graph y 1x with the window set for 500 x 500 and 500 y 500,
the graph will be too close to its horizontal and vertical asymptotes for us to see it.
Would we conclude that there is no graph for y 1x?
If you have a graphing calculator, use it to help you graph the functions in this
chapter and to reinforce your understanding of the properties of functions that we
are learning. Do not attempt to use it as a substitute for learning.
554
(10-26)
Chapter 10
Polynomial and Rational Functions
Sketching the Graphs
We now use asymptotes and symmetry to help us sketch the graphs of some rational
functions.
E X A M P L E
4
calculator
close-up
This calculator graph supports
the graph drawn in Fig. 10.7.
Remember that the calculator graph can be misleading.
The vertical lines drawn by the
calculator are not part of the
graph of the function.
10
–4
Graphing a rational function
Sketch the graph of each rational function.
3
a) f (x) 2 x 1
x
b) g(x) 2 x 4
Solution
a) From Example 2(a) we know that the lines x 1 and x 1 are vertical asymptotes and the x-axis is a horizontal asymptote. Draw the vertical asymptotes
on the graph with dashed lines. Since all of the powers of x are even,
f (x) f (x), and the graph is symmetric about the y-axis. The ordered pairs
(0, 3), (0.9, 15.789), (1.1, 14.286), (2, 1), and 3, 3 are also on the graph.
8
Use the symmetry to sketch the graph shown in Fig. 10.6.
b) Draw the vertical asymptotes x 2 and x 2 from Example 2(b) as dashed
lines. The x-axis is a horizontal asymptote. Because f (x) f (x), the graph
1
is symmetric about the origin. The ordered pairs (0, 0), 1, 3, (1.9, 4.872),
(2.1, 5.122), 3, 5, and 4,
graph shown in Fig. 10.7.
3
are on the graph. Use the symmetry to get the
1
3
y
y
4
5
4
3
2
1
–10
–5 –4 –3 –2
–1
–2
f(x) =
5
4
3
2
1
3
x2 – 1
2 3 4 5
x
FIGURE 10.6
E X A M P L E
5
Graphing a rational function
Sketch the graph of each rational function.
2x 1
a) h(x) x3
2x 2 3x 5
b) g(x) x2
–1
–1
–2
–3
–4
–5
3 4 5
g(x) =
FIGURE 10.7
x
x
x2 – 4
■
10.4
calculator
close-up
This calculator graph supports
the graph drawn in Fig. 10.8.
Note that if x is 3, there is no
y-coordinate because x 3
is the vertical asymptote.
y
9
8
7
6
5
4
3
4
–10
h(x) = 2x + 1
x+3
y
5
4
3
2
1
1
–9 –8 –7 –6 –5 –4
–2
–1
–2
–3
–4
–5
–5 –4 –3
1 2 3 4 5
x
FIGURE 10.8
WARM-UPS
555
Solution
a) Draw the vertical asymptote x 3 and the horizontal asymptote y 2 from
1
Example 2(c) as dashed lines. The points (2, 3), 0, 1, 2, 0, (7, 1.5),
3
(4, 7), and (13, 2.5) are on the graph shown in Fig. 10.8.
b) Draw the vertical asymptote x 2 and the oblique asymptote y 2x 1
5
from Example 3 as dashed lines. The points (1, 6), 0, 2, (1, 0), (4, 6.5),
and (2.5, 0) are on the graph shown in Fig. 10.9.
20
–10
(10-27)
Graphs of Rational Functions
–1
–1
x
2 3 4 5
–6 g(x) = 2x
2
+ 3x – 5
x+2
■
FIGURE 10.9
True or false? Explain your answer.
1
False
x9
x1
2. The domain of f (x) is x x 1 and x 2
. False
x2
1
3. The domain of f (x) is x x 1 and x 1
. False
x2 1
1. The domain of f (x) is x 9.
1
4. The line x 2 is the only vertical asymptote for the graph of f (x) .
x2 4
False
x 2 3x 5
5. The x-axis is a horizontal asymptote for the graph of f (x) .
x 3 9x
True
3x 5
6. The x-axis is a horizontal asymptote for the graph of f (x) . False
x2
1
7. The line y 2x 5 is an asymptote for the graph of f (x) 2x 5 .
x
True
8. The line y 2x 5 is an asymptote for the graph of f (x) 2x 5 x 2.
False
x2
x 9
3x
10. The graph of f (x) is symmetric about the origin.
x 2 25
True
9. The graph of f (x) 2 is symmetric about the y-axis.
True
556
10. 4
(10-28)
Chapter 10
Polynomial and Rational Functions
EXERCISES
2x 3
10. f (x) x x 0
x2
5
11. f (x) x x 4 and x 4
x2 16
x 12
12. f (x) x x 2 and x 3
2
x x6
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a rational function?
A rational function is of the form f(x) P(x)Q(x), where
P(x) and Q(x) are polynomials with no common factor and
Q(x) 0.
2. What is the domain of a rational function?
The domain of a rational function is all real numbers except
those that cause the denominator to be 0.
Determine all asymptotes for the graph of each rational
function. See Examples 2 and 3.
7
13. f (x) Vertical: x 4; horizontal: x-axis
x4
8
14. f (x) Vertical: x 9; horizontal: x-axis
x9
1
15. f (x) x2 16
Vertical: x 4, x 4; horizontal: x-axis
2
16. f (x) x2 5x 6
Vertical: x 2, x 3; horizontal: x-axis
5x
17. f (x) Vertical: x 7; horizontal: y 5
x7
3x 8
18. f (x) Vertical: x 2; horizontal: y 3
x2
3. What is a vertical asymptote?
A vertical asymptote is a vertical line that is approached by
the graph of a rational function.
4. What is a horizontal asymptote?
A horizontal asymptote is a horizontal line that is approached by the graph of a rational function.
5. What is an oblique asymptote?
An oblique asymptote is a nonhorizontal, nonvertical line
that is approached by the graph of a rational function.
6. What is a slant asymptote?
A slant asymptote is the same as an oblique asymptote.
Find the domain of each rational function. See Example 1.
2
7. f (x) x x 1
x1
2
8. f (x) x x 3
x3
x2 1
9. f (x) x x 0
x
2x 2
19. f (x) x3
3x 2 2
20. f (x) x1
Match each rational function with its graph a–h.
2
1
22. f (x) e
21. f (x) c
x
x2
1
x2
26. f (x) 25. f (x) g
h
2
2
x 2x
x 4
(a)
Vertical: x 3; oblique: y 2x 6
Vertical: x 1; oblique: y 3x 3
x
23. f (x) b
x2
x4
27. f (x) f
2
(b)
x2
24. f (x) d
x
x2 2x 1
28. f (x) x
(c)
y
y
y
5
4
3
5
4
3
2
1
2
1
–4 –3
1 2 3 4
–5
x
a
– 4 – 3 – 2 –1
–1
–2
–3
–4
–5
1
3 4
x
– 4 – 3 – 2 –1
–1
–2
–3
–4
–5
1 2
x
10.4
(d)
Graphs of Rational Functions
(e)
y
3
2
1
–4 –3 –2 –1
–1
–2
–3
–4
–5
3 4
(g)
y
5
4
3
2
1
5
4
3
2
1
1
3
x
–5
y
5
4
3
2
– 5 –4 –3 –2 –1
–1
–2
–3
–4
–5
1
3 4 5
x
–5 –4 –3
Determine all asymptotes and sketch the graph of each function.
See Examples 4 and 5.
2
29. f (x) x 4, x-axis
x4
3
30. f (x) x1
x 1, x-axis
–2 –1
–1
–3
–4
–5
(h)
y
557
(f )
y
–4 –3 –2 –1
–1
–2
–3
–4
–5
x
(10-29)
–1
–2
–3
–4
–5
3 4 5
x
x
31. f (x) x2 9
x 3, x 3, x-axis
2
32. f (x) x2 x 2
x 2, x 1, x-axis
1 2 3
x
558
(10-30)
Chapter 10
Polynomial and Rational Functions
2x 1
33. f (x) x3
x 3, y 2
5 2x
34. f (x) x2
x 2, y 2
x 2 5x 5
38. f (x) x3
x 3, y x 2
Find all asymptotes, x-intercepts, and y-intercepts for the
graph of each rational function and sketch the graph of the
function.
1
39. f (x) 2
x
x 0, y 0
x 2 3x 1
35. f (x) x
y-axis, y x 3
2
40. f (x) x 2 4x 4
2
1
x 2, y 0, 0, x3 1
36. f (x) x2
y-axis, y x
2x 3
41. f (x) x2 x 6
2 2, 0
1
3x 2 2x
37. f (x) x1
x 3, x 2, y 0, 0, ,
x 1, y 3x 1
3
10.4
x
42. f (x) 2
x 4x 4
x 2, y 0, (0, 0)
x1
43. f (x) x2
x 0, y 0, (1, 0)
x1
44. f (x) x2
x 0, y 0, (1, 0)
2x 1
45. f (x) x3 9x
x 0, x 3, y 0,
2x 2 1
46. f (x) x3 x
x 0, x 1, y 0
Graphs of Rational Functions
x
47. f (x) 2
x 1
x 1, y 0, (0, 0)
x
48. f (x) x2 x 2
2, 0
x 2, x 1, y 0, (0, 0)
2
49. f (x) 2
x 1
y 0, (0, 2)
x
50. f (x) x2 1
y 0, (0, 0)
1
x2
51. f (x) x1
(10-31)
x 1, y x 1, (0, 0)
559
560
(10-32)
x2
52. f (x) x1
Chapter 10
Polynomial and Rational Functions
www.motortrend.com). If it costs $25,000 to manufacture
each SUV, then the average cost per vehicle in dollars when
x vehicles are manufactured is given by the rational
function
25,000x 700,000,000
A(x) .
x
x 1, y x 1, (0, 0)
Solve each problem.
53. Oscillating modulators. The number of oscillating modulators produced by a factory in t hours is given by the polynomial function n(t) t2 6t for t 1. The cost in dollars of operating the factory for t hours is given by the
function c(t) 36t 500 for t 1. The average cost per
a) What is the horizontal asymptote for the graph of this
function?
b) What is the average cost per vehicle when 50,000 vehicles are made?
c) For what number of vehicles is the average cost
$30,000?
d) Graph this function for x ranging from 0 to 100,000.
a) y 25,000 b) $39,000 c) 140,000
36t 500
t 6t
modulator is given by the rational function f (t) 2
for t 1. Graph the function f. What is the average cost per
modulator at time t 20 and time t 30? What can you
conclude about the average cost per modulator after a long
period of time?
f (20) $2.35, f (30) $1.46, average approaches 0
56. Average cost of a pill. Assuming Pfizer spent a typical
$350 million to develop its latest miracle drug and $0.10
each to make the pills, then the average cost per pill in dollars when x pills are made is given by the rational function
0.10x 350,000,000
A(x) .
x
54. Nonoscillating modulators. The number of nonoscillating
modulators produced by a factory in t hours is given by the
polynomial function n(t) 16t for t 1. The cost in dollars of operating the factory for t hours is given by the function c(t) 64t 500 for t 1. The average cost per modulator is given by the rational function f(t) 64t 500
16t
for
t 1. Graph the function f. What is the average cost per
modulator at time t 10 and t 20? What can you
conclude about the average cost per modulator after a long
period of time?
f (10) $7.13, f (20) $5.56, average cost approaches
$4.00
55. Average cost of an SUV. Mercedes-Benz spent $700 million to design its new 1998 SUV (Motor Trend,
FIGURE FOR EXERCISE 56
10.5
a) What is the horizontal asymptote for the graph of this
function?
b) What is the average cost per pill when 100 million pills
are made?
c) For what number of pills is the average cost per pill $2?
d) Graph this function for x ranging from 0 to 100 million.
a) y 0.10 b) $3.60 c) 184,210,526
Transformations of Graphs
(10-33)
561
G R A P H I N G C ALC U L ATO R
EXERCISES
Sketch the graph of each pair of functions in the same coordinate system. What do you observe in each case?
57. f (x) x 2, g(x) x 2 1x
The graph of f (x) is an asymptote for the graph of g(x).
58. f (x) x 2, g(x) x 2 1x2
The graph of f (x) is an asymptote for the graph of g(x).
59. f (x) x , g(x) x 1x
The graph of f (x) is an asymptote for the graph of g(x).
60. f (x) x , g(x) x 1x 2
The graph of f (x) is an asymptote for the graph of g(x).
61. f (x) x
, g(x) x
1x
The graph of f (x) is an asymptote for the graph of g(x).
62. f (x) x 3, g(x) x 3 1x 2
The graph of f (x) is an asymptote for the graph of g(x).
10.5 T R A N S F O R M A T I O N S O F G R A P H S
In this
section
●
Reflecting
●
Translating
●
Stretching and Shrinking
Reflecting
calculator
close-up
With a graphing calculator,
you can quickly see the result
of modifying the formula for a
function. If you have a graphing calculator, use it to graph
the functions in the examples.
Experimenting with it will
help you to understand the
ideas in this section.
10
–10
We can discover what the graph of almost any function looks like if we plot enough
points. However, it is helpful to know something about a graph so that we do not
have to plot very many points. For example, it is important to know that the graph
of a function is symmetric about the y-axis before we begin calculating ordered
pairs. In this section we will learn how one graph can be transformed into another
by modifying the formula that defines the function.
y
Consider the graphs of f (x) x2 and g(x) x2
shown in Fig. 10.10. Notice that the graph of g
5
f (x) = x 2
4
is a mirror image of the graph of f. For any
3
value of x we compute the y-coordinate of an
2
ordered pair of f by squaring x. For an ordered
1
pair of g we square first and then find the oppo–5 –4 –3 –2
2 3 4 5 x
site because of the order of operations. This
–
2
gives a correspondence between the ordered
–3
pairs of f and the ordered pairs of g. For every
–4
ordered pair on the graph of f there is a corre–5
g(x) = –x 2
sponding ordered pair directly below it on the
graph of g, and these ordered pairs are the same
FIGURE 10.10
distance from the x-axis. We say that the graph
of g is obtained by reflecting the graph of f in the x-axis or that g is a reflection of
the graph of f.
10
Reflection
–10
The graph of y f (x) is a reflection in the x-axis of the graph of y f (x).