D. The Graph of a Rational Function
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CHAPTER 4 Polynomial and Rational Functions
EXAMPLE 4
ᮣ
Graphing Rational Functions
Graph each function given.
x2 Ϫ x Ϫ 6
a. f 1x2 ϭ 2
x ϩxϪ6
Solution
ᮣ
2x2 Ϫ 4x ϩ 2
x2 Ϫ 7
1x ϩ 22 1x Ϫ 32
a. Begin by writing f in factored form: f 1x2 ϭ
.
1x ϩ 32 1x Ϫ 22
1. y-intercept: f 102 ϭ
2.
3.
4.
5.
b. g1x2 ϭ
1221Ϫ32
ϭ 1, so the y-intercept is (0, 1).
1321Ϫ22
x-intercepts: Setting the numerator equal to zero gives 1x ϩ 221x Ϫ 32 ϭ 0,
showing the x-intercepts will be (Ϫ2, 0) and (3, 0).
Vertical asymptote(s): Setting the denominator equal to zero gives
1x ϩ 321x Ϫ 22 ϭ 0, showing there will be vertical asymptotes at
x ϭ Ϫ3 and x ϭ 2.
Horizontal asymptote: Since the degree of the numerator and the degree
x2
of the denominator are equal, y ϭ 2 ϭ 1 is a horizontal asymptote.
x
x2 Ϫ x Ϫ 6
ϭ1
Solving 2
f 1x2 ϭ 1 S horizontal asymptote
x ϩxϪ6
x2 Ϫ x Ϫ 6 ϭ x2 ϩ x Ϫ 6 multiply by x2 ϩ x Ϫ 6
simplify
Ϫ2x ϭ 0
xϭ0
solve
The graph will cross the horizontal asymptote at (0, 1).
The information from steps 1 through 5 is shown in Figure 4.52, and
indicates we have no information about the graph in the interval (Ϫq, Ϫ3).
Since rational functions are defined for all real numbers except the zeroes of
d, we know there must be a “piece” of the graph in this interval.
6. Selecting x ϭ Ϫ4 to compute one additional point, we find
1Ϫ22 1Ϫ72
7
7
f 1Ϫ42 ϭ 1Ϫ12 1Ϫ62 ϭ 14
6 ϭ 3 . The point is (Ϫ4, 3 ).
All factors of f are linear, so function values will alternate sign in the intervals
created by x-intercepts and vertical asymptotes. The y-intercept (0, 1) shows
f (x) is positive in the interval containing 0. To meet all necessary conditions,
we complete the graph as shown in Figure 4.53.
Figure 4.52
Figure 4.53
y xϭ2
x ϭ Ϫ3
y
5
5
Ϫ4, g
yϭ1
pos
n
e
g
pos
Ϫ5
n
e
g
pos
x
Ϫ5
5
Ϫ5
x
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b. Writing g in factored form gives g1x2 ϭ
2.
3.
4.
5.
It’s useful to note that the number
of “pieces” forming a rational graph
will always be one more than the
number of vertical asymptotes. The
3x
graph of f1x2 ϭ 2
(Figure 4.46)
x ϩ2
has no vertical asymptotes and one
piece, y ϭ 1x has one vertical
asymptote and two pieces,
2
Ϫ4
g1x2 ϭ x 2
(Figure 4.47) has two
x Ϫ1
vertical asymptotes and three
pieces, and so on.
21x2 Ϫ 2x ϩ 12
x2 Ϫ 7
ϭ
21Ϫ12 2
21x Ϫ 12 2
1x ϩ 1721x Ϫ 172
.
2
2
ϭ Ϫ . The y-intercept is a 0, Ϫ b.
7
7
1 172 1Ϫ 172
x-intercept(s): Setting the numerator equal to zero gives 21x Ϫ 12 2 ϭ 0,
with x ϭ 1 as a zero of multiplicity 2. The x-intercept is (1, 0).
Vertical asymptote(s): Setting the denominator equal to zero gives
1x ϩ 172 1x Ϫ 172 ϭ 0, showing there will be asymptotes at x ϭ Ϫ 17
and x ϭ 17.
Horizontal asymptote: The degree of the numerator is equal to the degree
2x2
of denominator, so y ϭ 2 ϭ 2 is a horizontal asymptote.
x
2
2x Ϫ 4x ϩ 2
Solve
g 1x2 ϭ 2 S horizontal asymptote
ϭ2
x2 Ϫ 7
2x2 Ϫ 4x ϩ 2 ϭ 2x2 Ϫ 14 multiply by x 2 Ϫ 7
Ϫ4x ϭ Ϫ16
simplify
xϭ4
solve
1. y-intercept: g102 ϭ
WORTHY OF NOTE
437
Section 4.4 Graphing Rational Functions
The graph will cross its horizontal
asymptote at (4, 2). The information
from steps 1 to 5 is shown in Figure 4.54,
and indicates we have no information
about the graph in the interval
(Ϫq, Ϫ17).
21Ϫ5 Ϫ 12 2
6. Selecting x ϭ Ϫ5, g1Ϫ52 ϭ
1Ϫ52 2 Ϫ 7
21Ϫ62 2
ϭ
25 Ϫ 7
21362
ϭ
18
ϭ4
Figure 4.54
y x ϭ ͙7
x ϭ Ϫ͙7
5
(4, 2)
yϭ2
neg
Ϫ6
(1, 0)
pos
pos
6
x
Ϫ5
The point (Ϫ5, 4) is on the graph
(Figure 4.55).
Figure 4.55
y
Since factors of the denominator have odd
multiplicity, function values will alternate sign
on either side of the asymptotes. The factor in
the numerator has even multiplicity, so the
graph will “bounce off” the x-axis at x ϭ 1
(no change in sign). The y-intercept (0, Ϫ27 )
shows the function is negative in the interval
containing 0. This information and the
completed graph are shown in Figure 4.55.
(Ϫ5, 4)
5
Ϫ6
5
x
Ϫ5
Now try Exercises 31 through 54
ᮣ
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CHAPTER 4 Polynomial and Rational Functions
Examples 3 and 4 demonstrate that graphs of rational functions come in a large variety. Once the components of the graph have been found, completing the graph presents
an intriguing and puzzle-like challenge as we attempt to sketch a graph that meets all
conditions. As we’ve done with other functions, can you reverse this process? That is,
given the graph of a rational function, can you construct its equation?
EXAMPLE 5
ᮣ
Finding the Equation of a Rational Function from Its Graph
Use the graph of f (x) shown to construct its equation.
Solution
ᮣ
The x-intercepts are (Ϫ1, 0) and (4, 0), so the
numerator must contain the factors (x ϩ 1) and
(x Ϫ 4). The vertical asymptotes are x ϭ Ϫ2
and x ϭ 3, so the denominator must have the
factors 1x ϩ 22 and 1x Ϫ 32 . So far we have:
f 1x2 ϭ
y
5
(2, 3)
a1x ϩ 121x Ϫ 42
1x ϩ 221x Ϫ 32
Ϫ5
5
Since (2, 3) is on the graph, we substitute 2 for
x and 3 for f (x) to solve for a:
a12 ϩ 1212 Ϫ 42
substitute 3 for f (x) and 2 for x
3ϭ
12 ϩ 2212 Ϫ 32
Ϫ5
3a
3ϭ
simplify
2
2ϭa
solve
21x ϩ 121x Ϫ 42
2x2 Ϫ 6x Ϫ 8
ϭ 2
The result is f 1x2 ϭ
, with a horizontal
1x ϩ 221x Ϫ 32
x ϪxϪ6
asymptote at y ϭ 2 and a y-intercept of (0, 43), which fit the graph very well.
Now try Exercises 55 through 58
x
ᮣ
As a final note, there are many rational graphs that have x- and y-intercepts and vertical asymptotes, but no horizontal asymptotes. Two examples are shown in Figures 4.56
and 4.57.
Figure 4.56
Figure 4.57
2
x 2 Ϫ 14
x Ϫ4
f1x2 ϭ
g1x2 ϭ
xϩ1
xϪ1
g(x)
f (x)
10
12
8
10
6
8
4
6
2
4
Ϫ10Ϫ8 Ϫ6 Ϫ4 Ϫ2
Ϫ2
D. You’ve just seen how
we can graph general rational
functions
2
2
4
6
8 10
x
Ϫ4
Ϫ4 Ϫ3 Ϫ2 Ϫ1
Ϫ2
Ϫ6
Ϫ4
Ϫ8
Ϫ6
Ϫ10
Ϫ8
1
2
3
4
5
6
x
Without more information regarding nonhorizontal asymptotes, sketching these graphs
requires a substantial number of plotted points. Rational graphs of this type will be
studied in more detail in Section 4.5, enabling us to complete each graph using far
fewer points. See Exercises 59 through 62.
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Section 4.4 Graphing Rational Functions
439
E. Applications of Rational Functions
In many applications of rational functions, the coefficients can be rather large and the
graph should be scaled appropriately.
EXAMPLE 6
ᮣ
Modeling the Rides at an Amusement Park
A popular amusement park wants to add new rides and asks various contractors to
submit ideas. Suppose one ride engineer offers plans for a ride that begins with a
near vertical drop into a dark tunnel, quickly turns and becomes more horizontal,
pops out the tunnel’s end, then coasts up to the exit platform, braking 20 m from
the release point. The height of a rider above ground is modeled by the function
39x2 Ϫ 507x ϩ 468
h1x2 ϭ
, where h(x) is the height in meters at a horizontal
3x2 ϩ 23x ϩ 20
distance of x meters from the release point.
a. Graph the function for x ʦ 3Ϫ1, 20 4 .
b. How high is the release point for this ride?
c. How long is the tunnel from entrance to exit?
d. What is the height of the exit platform?
Solution
ᮣ
468
or 23.4 m. Also, since
20
the degree of the numerator is equal to that of the denominator, the ratio of
39x2
Figure 4.58
leading terms 2 indicates a horizontal
h(x)
3x
32
asymptote at y ϭ 13. Writing h(x) in
391x Ϫ 12 1x Ϫ 122
24
factored form gives h1x2 ϭ
,
13x ϩ 202 1x ϩ 12
16
showing the x-intercepts will be (1, 0) and
(12, 0), with vertical asymptotes at x ϭ Ϫ6.6
8
and x ϭ Ϫ1. Computing midinterval points of
x ϭ 4, 8, and 18 gives (4, Ϫ5.85), (8, Ϫ2.76),
4
8
12
16
20
and (18, 2.83). Graphing the function over the
Ϫ8
specified interval produces the graph shown
in Figure 4.58.
Figure 4.59
b. From the context, the release point is at 23.4 m.
c. The ride enters the tunnel at x ϭ 1 and exits
at x ϭ 12, making the tunnel 11 m long.
d. Since the ride begins braking at a distance of
20 m, the platform must be h1202 Ϸ 3.5 m
high. See Figure 4.59.
a. Here we begin by noting the y-intercept is h102 ϭ
Now try Exercises 65 through 76
E. You’ve just seen how
we can solve applications of
rational functions
x
ᮣ
As a final note, the ride proposed in Example 6 was never approved due to excessive
g-forces on the riders. Example 6 helps to illustrate that when it comes to applications
of rational functions, portions of the graph may be ignored due to the context. In addition, some applications may focus on a specific attribute of the graph, such as the horizontal asymptotes in Exercises 65, 66, and elsewhere, or the vertical asymptotes in
Exercises 67 and 68.
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CHAPTER 4 Polynomial and Rational Functions
4.4 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
1. Write the following in direction/approach notation.
As x becomes an infinitely large negative number, y
approaches 2.
2. For any constant k, the notation as ͿxͿ S ϩq,
y S k is an indication of a
asymptote,
while x S k, ͿyͿ S ϩq indicates a
asymptote.
3. Vertical asymptotes are found by setting the
equal to zero. The x-intercepts are found
by setting the
equal to zero.
4. If the degree of the numerator is equal to the
degree of the denominator, a horizontal asymptote
occurs at y ϭ ab , where ab represents the ratio of
the
.
3x2 Ϫ 2x
and a table of
2x2 Ϫ 3
values to discuss the concept of horizontal
asymptotes. At what positive value of x is the graph
of g within 0.01 of its horizontal asymptote?
5. Use the function g1x2 ϭ
ᮣ
6. Name all of the “tools” at your disposal that play a
role in the graphing of rational functions. Which
tools are indispensable and always used? Which are
used only as the situation merits?
DEVELOPING YOUR SKILLS
Give the location of the vertical asymptote(s) if they
exist, and state the function’s domain.
7. f 1x2 ϭ
xϩ2
xϪ3
9. g1x2 ϭ
3x2
x2 Ϫ 9
8. F1x2 ϭ
4x
2x Ϫ 3
10. G1x2 ϭ
xϩ1
9x2 Ϫ 4
11. h1x2 ϭ
x2 Ϫ 1
xϪ5
12. H1x2 ϭ 2
2x ϩ 3x Ϫ 5
2x Ϫ x Ϫ 3
13. p1x2 ϭ
2x ϩ 3
2
x ϩxϩ1
2
14. q1x2 ϭ
2x3
x2 ϩ 4
Give the location of the vertical asymptote(s) if they
exist, and state whether function values will change sign
(positive to negative or negative to positive) from one
side of the asymptote to the other.
15. Y1 ϭ
xϩ1
x ϪxϪ6
2
17. r1x2 ϭ
16. Y2 ϭ
2x ϩ 3
x Ϫ x Ϫ 20
2
x2 ϩ 3x Ϫ 10
x2 Ϫ 2x Ϫ 15
18.
R1x2
ϭ
x2 Ϫ 6x ϩ 9
x2 Ϫ 4x ϩ 4
19. Y1 ϭ
x
x ϩ 2x Ϫ 4x Ϫ 8
20. Y2 ϭ
Ϫ2x
x3 ϩ x2 Ϫ x Ϫ 1
3
2
For the functions given, (a) determine if a horizontal
asymptote exists and (b) determine if the graph will
cross the asymptote, and if so, where it crosses.
21. Y1 ϭ
2x Ϫ 3
x2 ϩ 1
22. Y2 ϭ
4x ϩ 3
2x2 ϩ 5
23. r1x2 ϭ
4x2 Ϫ 9
2x2 Ϫ x Ϫ 10
24.
R1x2
ϭ
x2 Ϫ 3x Ϫ 18
x2 ϩ 5
25. p1x2 ϭ
3x2 Ϫ 5
x2 Ϫ 1
26. P1x2 ϭ
3x2 Ϫ 5x Ϫ 2
x2 Ϫ 4
Apply long division to find the quotient and remainder
for each function. Use this information to determine the
equation of the horizontal asymptote, and whether the
graph will cross this asymptote. Verify answers by
graphing the functions on a graphing calculator and
locating points of intersection.
27. v1x2 ϭ
8x
x ϩ1
28. f 1x2 ϭ
4x ϩ 8
x2 ϩ 1
29. g1x2 ϭ
2x2 Ϫ 8x
x2 Ϫ 4
30. h1x2 ϭ
x2 Ϫ x Ϫ 6
x2 Ϫ 1
2