Find the horizontal or oblique asymptotes, if any, of the graph of
Solution
Since the degree of the numerator, 4, is larger than the degree of the denominator, 3, the rational function H is improper. To find any horizontal or oblique asymptotes, we use long division.As a result,
As or as
As or as we have We conclude that the graph of
the rational function H has an oblique asymptote
We verify these results in Tables 15(a) and (b) with and As or the difference in the values between and is indistinguishable.
Y2 Y1 x : q ,
x : - q Y2 = 3x + 3.
Y1 = H1x2 y = 3x + 3.
H1x2 : 3x + 3.
x : q , x : - q
2x2 - 3x - 3 x3 - x2 + 1 L
2x2 x3 =
2 x: 0 x : q ,
x : - q
H1x2 = 3x4 - x2
x3 - x2 + 1 = 3x + 3 + 2x2 - 3x - 3 x3 - x2 + 1 3x + 3
x3 - x2 + 13x4 - x2 3x4 - 3x3 + 3x
3x3 - x2 - 3x 3x3 - 3x2 + 3
2x2 - 3x - 3 H1x2 = 3x4 - x2
x3 - x2 + 1
E X A M P L E 6
f, y = ax + b
f1x2 = ax + b, a Z 0,
y = b f1x2 = b,
Table 15
(a) (b)
Finding Horizontal or Oblique Asymptotes
Find the horizontal or oblique asymptotes, if any, of the graph of
Solution
Since the degree of the numerator, 2, equals the degree of the denominator, 2, the rational function R is improper. To find any horizontal or oblique asymptotes, we use long division.As a result,
Then, as or as
As or as we have We conclude that is a
horizon-tal asymptote of the graph.
✔ CHECK: Verify the results of Example 7 by creating a TABLE with
and
In Example 7, we note that the quotient 2 obtained by long division is the quo-tient of the leading coefficients of the numerator polynomial and the denominator polynomial This means that we can avoid the long division process for rational functions whose numerator and denominator are of the same degree and conclude that the quotient of the leading coefficients will give us the horizontal asymptote.
N O W W O R K P R O B L E M 4 1 .
Finding Horizontal or Oblique Asymptotes
Find the horizontal or oblique asymptotes, if any, of the graph of
Solution
Since the degree of the numerator, 5, is larger than the degree of the denominator, 3, the rational function G is improper. To find any horizontal or oblique asymptotes, we use long division.2x2 - 1
x3 - 12x5 - x3 + 2 2x5 - 2x2
-x3 + 2x2 + 2
-x3 + 1
2x2 + 1 G1x2 = 2x5 - x3 + 2
x3 - 1
E X A M P L E 8
a8 4b.
Y2 = 2.
Y1 = R1x2 y = 2
R1x2 : 2.
x : q , x : - q
-x + 4 4x2 - 1 L
-x 4x2 =
-1 4x : 0 x : q ,
x : - q
R1x2 = 8x2 - x + 2
4x2 - 1 = 2 + -x + 4 4x2 - 1 2
4x2 - 18x2 - x + 2 8x2 - 2 - x + 4 R1x2 = 8x2 - x + 2
4x2 - 1
E X A M P L E 7
As a result,
Then, as or as
As or as we have We conclude that, for large
values of the graph of G approaches the graph of That is, the graph
of G will look like the graph of as or Since
is not a linear function, G has no horizontal or oblique asymptotes. We now summarize the procedure for finding horizontal and oblique asymptotes.
Summary
Finding Horizontal and Oblique Asymptotes of a Rational Function R Consider the rational function
in which the degree of the numerator is n and the degree of the denominator is m.
1. If then R is a proper rational function, and the graph of R will have the horizontal asymptote (the x-axis).
2. If then R is improper. Here long division is used.
(a) If the quotient obtained will be a number and the line is a horizontal asymptote.
(b) If the quotient obtained is of the form (a polynomial of degree 1), and the line is an oblique asymptote.
(c) If the quotient obtained is a polynomial of degree 2 or higher, and R has neither a horizontal nor an oblique asymptote. In this case, for unbounded, the graph of R will behave like the graph of the quotient.
NOTE The graph of a rational function either has one horizontal or one oblique asymptote or else has no horizontal and no oblique
asymptote. ■
‘Are You Prepared?’
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
3.3 Assess Your Understanding
ƒxƒ
n 7 m + 1, y = ax + b
ax + b n = m + 1,
y = an bm an
bm, n = m,
n Ú m,
y = 0 n 6 m,
R1x2 = p1x2 q1x2 =
anxn + an - 1xn - 1 + Á + a1x + a0 bmxm + bm - 1xm - 1 + Á + b1x + b0 y = 2x2 - 1
x : q . x : - q
y = 2x2 - 1
y = 2x2 - 1.
ƒxƒ,
G1x2 : 2x2 - 1.
x : q , x : - q
2x2 + 1 x3 - 1 L
2x2 x3 =
2 x: 0 x : q ,
x : - q
G1x2 = 2x5 - x3 + 2
x3 - 1 = 2x2 - 1 + 2x2 + 1 x3 - 1
1. True or False: The quotient of two polynomial expressions is a rational expression.(p. 971)
2. What is the quotient and remainder when
is divided by x2 - 1?(pp. 977–979) 3x3 - 6x2 + 3x - 4
3. Graph (p. 21)
4. Graph using transformations.
(pp. 118–126)
y = 21x + 122 - 3 y = 1
x.
Concepts and Vocabulary
5. The line _____ is a horizontal asymptote of
6. The line _____ is a vertical asymptote of
7. For a rational function R, if the degree of the numerator is less than the degree of the denominator, then R is _____.
R1x2 = x - 1 x + 1. R1x2 = x3 - 1
x3 + 1. 8. True or False: The domain of every rational function is the set of all real numbers.
9. True or False: If an asymptote is neither horizontal nor vertical, it is called oblique.
10. True or False: If the degree of the numerator of a rational function equals the degree of the denominator, then the ratio of the leading coefficients gives rise to the horizontal asymptote.
Skill Building
In Problems 11–22, find the domain of each rational function.
11. 12. 13. 14.
15. 16. 17. 18.
19. 20. 21. 22.
In Problems 23–28, use the graph shown to find:
(a) The domain and range of each function (b) The intercepts, if any (c) Horizontal asymptotes, if any (d) Vertical asymptotes, if any (e) Oblique asymptotes, if any
23. 24. 25.
26. 27. 28.
In Problems 29–40, graph each rational function using transformations. Verify your result using a graphing utility.
29. 30. 31. 32. R1x2 = 3
33. 34. 35. 36.
37. 38. 39. 40.
In Problems 41–52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function.
41. 42. 43. 44.
53. Gravity In physics, it is established that the acceleration due to gravity, g (in ), at a height h meters above sea level is given by
where is the radius of Earth in meters.
(a) What is the acceleration due to gravity at sea level?
(b) The Sears Tower in Chicago, Illinois, is 443 meters tall.
What is the acceleration due to gravity at the top of the Sears Tower?
(c) The peak of Mount Everest is 8848 meters above sea level. What is the acceleration due to gravity on the peak of Mount Everest?
(d) Find the horizontal asymptote of
(e) Solve g1h2 = 0.How do you interpret your answer?g1h2.
6.374 * 106
g1h2 = 3.99 * 1014 16.374 * 106 + h22
meters>sec2 54. Population Model A rare species of insect was discovered in the Amazon Rain Forest. To protect the species, environ-mentalists declare the insect endangered and transplant the insect into a protected area. The population P of the insect t months after being transplanted is given below.
(a) How many insects were discovered? In other words, what was the population when
(b) What will the population be after 5 years?
(c) Determine the horizontal asymptote of What is the largest population that the protected area can sustain?
55. If the graph of a rational function R has the vertical asymp-tote then the factor must be present in the de-nominator of R. Explain why.
56. If the graph of a rational function R has the horizontal asymptote then the degree of the numerator of R equals the degree of the denominator of R. Explain why.
y = 2,
x - 4 x = 4,
57. Can the graph of a rational function have both a horizontal and an oblique asymptote? Explain.
58. Make up a rational function that has as an oblique asymptote. Explain the methodology that you used.
y = 2x + 1
‘Are You Prepared?’ Answers
1. True 2. Quotient: Remainder: 3. 4.
x
3.4 The Graph of a Rational Function; Inverse and Joint Variation
PREPARING FOR THIS SECTION Before getting started, review the following:
• Intercepts (Section 1.2, pp. 15–17) • Even and Odd Functions (Section 2.3, pp. 80–82) Now work the ‘Are You Prepared?’ problems on page 207.
OBJECTIVES 1 Analyze the Graph of a Rational Function
2 Solve Applied Problems Involving Rational Functions
3 Construct a Model Using Inverse Variation
4 Construct a Model Using Joint or Combined Variation