10.4 GRAPHS OF RATIONAL FUNCTIONS - mr. …bn002.weebly.com/uploads/5/8/3/3/58339589/rational...10.4 Graphs of Rational Functions (10-25) 553 Suppose f(x) Q P((x x)) is a rational

51. f(x) � (x � 20)2(x � 30)

52. f(x) � (x � 20)2(x � 30)2

550 (10-22) Chapter 10 Polynomial and Rational Functions

53. f(x) � (x � 20)2(x � 30)2x

54. f(x) � (x � 20)2(x � 30)2x2

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 studyfunctions that are defined by rational expressions.

Domain

A rational expression was defined in Chapter 6 as a ratio of two polynomials. If aratio of two polynomials is used to define a function, then the function is called arational function.

Rational Function

If P(x) and Q(x) are polynomials with no common factor and f (x) � �QP(

(xx))

� forQ(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 thatcause the denominator to have a value of 0.

E X A M P L E 1 Domain of a rational functionFind the domain of each rational function.

a) f (x) � �x

x

�

�

31

� b) g(x) � �2x

x2 �

�

43

�

10.4

I n t h i ss e c t i o n

● Domain

● Horizontal and Vertical Asymptotes

● Oblique Asymptotes

● Sketching the Graphs

Solutiona) 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 x2 � 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. ■

Horizontal and Vertical Asymptotes

Consider the simplest rational function f (x) � 1�x. Its domain does not include 0,but 0 is an important number for the graph of this function. The behavior of thegraph of f when x is very close to 0 is what interests us. For this function they-coordinate is the reciprocal of the x-coordinate. When the x-coordinate is close to0, the y-coordinate is far from 0. Consider the following tables of ordered pairs thatsatisfy f (x) � 1�x:

x 0 x � 0

x y x y

0.1 10 �0.1 �100.01 100 �0.01 �100

0.001 1000 �0.001 �10000.0001 10,000 �0.0001 �10,000

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 below0, the values of y are negative but y gets larger and larger. We say that y goes tonegative infinity. The graph of f gets closer and closer to the vertical line x � 0, andso x � 0 is called a vertical asymptote. On the other hand, as x gets larger andlarger, y gets closer and closer to 0. The graph approaches the x-axis as x goes to in-finity, and so the x-axis is a horizontal asymptote for the graph of f. See Fig. 10.5for the graph of f (x) � 1�x.

In general, a rational function has a vertical asymptote for every number ex-cluded from the domain of the function. The horizontal asymptotes are determinedby the behavior of the function when x is large.

E X A M P L E 2 Horizontal and vertical asymptotesFind the horizontal and vertical asymptotes for each rational function.

a) f (x) � �x2 �

31

� b) g(x) � �x2 �

x

4� c) h(x) � �

2x

x

�

�

31

�

Solutiona) The denominator x2 � 1 has a value of 0 if x � �1. So the lines x � 1 and

x � �1 are vertical asymptotes. If x is very large, the value of �x2 �

31

� isapproximately 0. So the x-axis is a horizontal asymptote.

b) The denominator x2 � 4 has a value of 0 if x � �2. So the lines x � 2 and x � �2 are vertical asymptotes. If x is very large, the value of �

x2 �x

4� is

approximately 0. So the x-axis is a horizontal asymptote.

10.4 Graphs of Rational Functions (10-23) 551

1

5432

y

x–1 4 52 31

f(x) = 1x

F I G U R E 1 0 . 5

If the viewing window is toolarge, a rational function willappear to touch its asymp-totes.

c a l c u l a t o r

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8

–8

–8

Because the asymptotes arean important feature of a ratio-nal function, we should draw itso that it approaches but doesnot touch its asymptotes.

c) The denominator x � 3 has a value of 0 if x � �3. So the line x � �3 is avertical 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 expressionby using long division:

2

x � 3�2�x�� 1�2x � 6

�5

Writing the rational expression as quotient � �re

dmivaiisnodrer

�, we get

h(x) � �2xx�

�

31

� � 2 � �x

�

�

53

�. If x is very large, �x

�

�

53

� 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 thedegree of the numerator is less than the degree of the denominator, then the x-axis

is the horizontal asymptote. For example, y � �xx2

�

�

47

� 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

example, y � �23xx

�

�

75

� has y � �23

� as its horizontal asymptote. The remaining case is

when the degree of the numerator is greater than the degree of the denominator. Thiscase is discussed next.

Oblique Asymptotes

Each rational function of Example 2 had one horizontal asymptote and a verticalasymptote for each number that caused the denominator to be 0. The horizontalasymptote y � 0 occurs because as x gets larger and larger, the y-coordinate getscloser and closer to 0. Some rational functions have a nonhorizontal line foran asymptote. An asymptote that is neither horizontal nor vertical is called anoblique asymptote or slant asymptote.

E X A M P L E 3 Finding an oblique asymptoteDetermine all of the asymptotes for

g(x) � �2x2 �

x �

3x

2� 5

�.

SolutionIf x � 2 � 0, then x � �2. So the line x � �2 is a vertical asymptote. Use long division to rewrite the function as quotient � �

redmivaiisnodrer

�:

g(x) � � 2x � 1 � �x

�

�

32

�

If x is large, the value of �x

��

32

� is approximately 0. So when x is large, the value

of g(x) is approximately 2x � 1. The line y � 2x � 1 is an oblique asymptote forthe graph of g. ■

We can summarize this discussion of asymptotes with the following strategy forfinding asymptotes for a rational function.

2x2 � 3x � 5��

x � 2


The graph for Example 2(b)should consist of three sepa-rate pieces, but in connectedmode the calculator connectsthe separate pieces. Eventhough the calculator doesnot draw a very good graphof this function, it does sup-port the conclusion that thehorizontal asymptote is the x-axis and the vertical asymp-totes are x � �1 and x � 1.

c a l c u l a t o r

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40

–40

–5

s t u d y t i p

The last couple of weeks ofthe semester is not the time toslack off. This is the time todouble your efforts. Make aschedule and plan every hourof your time. Don’t scheduleanything that isn’t necessary.Get an early start on studyingfor your final exams.


Suppose f (x) � �QP(

(xx))

� is a rational function with the degree of Q(x) at least 1.

1. Solve the equation Q(x) � 0. The graph of f has a vertical asymptote corre-sponding 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 ahorizontal asymptote.

3. If the degree of P(x) is equal to the degree of Q(x), then find the ratio of theleading coefficients. The horizontal line through that ratio is the horizontalasymptote.

4. If the degree of P(x) is greater than the degree of Q(x), then use division torewrite the function as

quotient � �re

dmivaiisnodrer

�.

The equation formed by setting y equal to the quotient gives us an obliqueasymptote.

Strategy for Finding Asymptotes for a Rational Function

M A T H A T W O R K

M A T H E M A T I C SM A C H I N E S

Machines that do computations have been around forthousands of years. The abacus, the slide rule, and thecalculator have simplified computations. However, re-cent calculators have gone way beyond numerical com-putations. Graphing calculators now draw two- andthree-dimensional graphs and even do symbolic compu-tations. Modern calculators are great, but to use one ef-fectively you must still learn the underlying principlesof mathematics.

Consider the fairly simple process of drawing agraph of a function. The graph of a function is a pictureof all ordered pairs of the function. When we graph a function, we typically plot afew ordered pairs and then use our knowledge about the function to draw a graphthat shows all of the important features of the function. A typical graphing calcula-tor plots 96 ordered pairs of a function on a screen that is not much bigger than apostage stamp. The calculator does not generalize and does not make conclusions.We are still responsible for looking at what the calculator shows and making con-clusions. 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 be-lieve the calculator and conclude that the graph has no vertical asymptote? If wegraph y � 1�x 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 � 1�x?

If you have a graphing calculator, use it to help you graph the functions in thischapter and to reinforce your understanding of the properties of functions that weare learning. Do not attempt to use it as a substitute for learning.

Sketching the Graphs

We now use asymptotes and symmetry to help us sketch the graphs of some rationalfunctions.

E X A M P L E 4 Graphing a rational functionSketch the graph of each rational function.

a) f (x) � �x2 �

31

�

b) g(x) � �x2 �

x

4�

Solutiona) From Example 2(a) we know that the lines x � 1 and x � �1 are vertical as-

ymptotes and the x-axis is a horizontal asymptote. Draw the vertical asymptoteson 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, �38

�� are also on the graph.

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 dashedlines. The x-axis is a horizontal asymptote. Because f (�x) � �f (x), the graph

is symmetric about the origin. The ordered pairs (0, 0), �1, ��13

��, (1.9, �4.872),

(2.1, 5.122), �3, �35

��, and �4, �13

�� are on the graph. Use the symmetry to get the

graph shown in Fig. 10.7.


This calculator graph supportsthe graph drawn in Fig. 10.7.Remember that the calcula-tor graph can be misleading.The vertical lines drawn by thecalculator are not part of thegraph of the function.

c a l c u l a t o r

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10

–10

–4

1

5432

y

x–2 –1–2

–3–4–5 4 52 3

f(x) = 3x2 – 1

F I G U R E 1 0 . 6

1

5432

y

x–1

–1

–2–3–4–5

4 53

g(x) = xx2 – 4

F I G U R E 1 0 . 7 ■

E X A M P L E 5 Graphing a rational functionSketch the graph of each rational function.

a) h(x) � �2xx��

31

�

b) g(x) � �2x2 �

x �3x

2� 5

�

Solutiona) Draw the vertical asymptote x � �3 and the horizontal asymptote y � 2 from

Example 2(c) as dashed lines. The points (�2, �3), �0, �13

��, ��12

�, 0�, (7, 1.5),

(�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

from Example 3 as dashed lines. The points (�1, �6), �0, ��52

��, (1, 0), (4, 6.5),

and (�2.5, 0) are on the graph shown in Fig. 10.9.


This calculator graph supportsthe graph drawn in Fig. 10.8.Note that if x is �3, there is noy-coordinate because x � �3is the vertical asymptote.

c a l c u l a t o r

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4

20

–10

–10

1

56789

43

y

x–2 –1–2–3–4–5

–4–5–6–7–8–9 4 521 3

h(x) = 2x + 1x + 3

F I G U R E 1 0 . 8

1

5432

y

x–1

–6

–3–4–5 4 52 3

g(x) = 2x 2 + 3x – 5x + 2

–1

F I G U R E 1 0 . 9 ■

W A R M - U P STrue or false? Explain your answer.

1. The domain of f (x) � �x �

19

� is x � 9. False

2. The domain of f (x) � �x

x

�

�

12

� is �x x � 1 and x � �2. False

3. The domain of f (x) � �x2 �

11

� is �x x � 1 and x � �1. False

4. The line x � 2 is the only vertical asymptote for the graph of f (x) � �x2 �

14

�.False

5. The x-axis is a horizontal asymptote for the graph of f (x) � �x2

x

�3 �

3x

9�

x

5�.

True

6. The x-axis is a horizontal asymptote for the graph of f (x) � �3x

x

�

�

25

�. False

7. The line y � 2x � 5 is an asymptote for the graph of f (x) � 2x � 5 � �1x

�.True

8. The line y � 2x � 5 is an asymptote for the graph of f (x) � 2x � 5 � x2.False

9. The graph of f (x) � �x2

x

�

2

9� is symmetric about the y-axis. True

10. The graph of f (x) � �x2 �

3x

25� is symmetric about the origin. True

Reading and Writing After reading this section, write out theanswers 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), whereP(x) and Q(x) are polynomials with no common factor andQ(x) � 0.

2. What is the domain of a rational function?The domain of a rational function is all real numbers exceptthose that cause the denominator to be 0.

3. What is a vertical asymptote?A vertical asymptote is a vertical line that is approached bythe graph of a rational function.

4. What is a horizontal asymptote?A horizontal asymptote is a horizontal line that is ap-proached by the graph of a rational function.

5. What is an oblique asymptote?An oblique asymptote is a nonhorizontal, nonvertical linethat 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.

7. f (x) � �x �

2

1� �x � x � 1�

8. f (x) � �x

�

�

2

3� �x � x � �3�

9. f (x) � �x2 �

x

1� �x � x � 0�


10. f (x) � ��2x

x2

� 3� �x � x � 0�

11. f (x) � �x2 �

5

16� �x � x � �4 and x � 4�

12. f (x) � �x2

x

�

�

x

1

�

2

6� �x � x � �2 and x � 3�

Determine all asymptotes for the graph of each rationalfunction. See Examples 2 and 3.

13. f (x) � �x �

7

4� Vertical: x � �4; horizontal: x-axis

14. f (x) � �x

�

�

89

� Vertical: x � 9; horizontal: x-axis

15. f (x) � �x2 �

1

16�

Vertical: x � 4, x � �4; horizontal: x-axis

16. f (x) � �x2 �

�

5x

2

� 6�

Vertical: x � 2, x � 3; horizontal: x-axis

17. f (x) � �x

5

�

x

7� Vertical: x � 7; horizontal: y � 5

18. f (x) � �3

x

x

�

�

2

8� Vertical: x � 2; horizontal: y � 3

19. f (x) � �x

2�

x2

3� Vertical: x � 3; oblique: y � 2x � 6

20. f (x) � �3x

x2

�

�

12

� Vertical: x � �1; oblique: y � 3x � 3

Match each rational function with its graph a–h.

21. f (x) � ��2

x� c 22. f (x) � ��

x �

1

2� e 23. f (x) � �

x �

x

2� b 24. f (x) � �

x �

x

2� d

25. f (x) � �x2 �

1

2x� g 26. f (x) � �

x2

x

�

2

4� h 27. f (x) � ��

x �

2

4� f 28. f (x) � �

x2 � 2

x

x � 1� a

(a) (b) (c)

–2

–4–3

–5

21

y

x–3–4 –2 1 2–1

–1

–2

–4–3

–5

2345

1

y

x1 3 4–1–3–4 –2 –1

–5

345

y

x–3–4 1 32 4

E X E R C I S E S1 0 . 4

(d) (e) (f)

(g) (h)

–5–4–3–2–1

y

x–5 –4 –3 3

2345

4 5

–5–4–3–2–1

y

x–2 –1–5 –4 –3 1 3 4 5

–5–4–3

–1

32

45

y

x–2 –1–5 1 2 3

1

–5–4–3–2–1

y

x–3 –2 –1–4 1 3

32

45

1

–2

–4–3

–5

23

y

x–3–4 –2 1 43–1

–1


Determine all asymptotes and sketch the graph of each function.See Examples 4 and 5.

29. f (x) � �x �

2

4� x � �4, x-axis

30. f (x) � �x

�

�

3

1� x � 1, x-axis

31. f (x) � �x2 �

x

9�

x � 3, x � �3, x-axis

32. f (x) � �x2 �

�

x

2

� 2�

x � �2, x � 1, x-axis

33. f (x) � �2x

x

�

�

31

� x � �3, y � 2

34. f (x) � �5

x

�

�

2

2

x� x � 2, y � �2

35. f (x) � �x2 � 3

x

x � 1� y-axis, y � x � 3

36. f (x) � �x3

x

�2

1� y-axis, y � x

37. f (x) � �3x

x

2

�

�

12x

� x � 1, y � 3x � 1


38. f (x) � ��x2

x

�

�

5x

3

� 5�

x � 3, y � �x � 2

Find all asymptotes, x-intercepts, and y-intercepts for thegraph of each rational function and sketch the graph of thefunction.

39. f (x) � �x

12�

x � 0, y � 0

40. f (x) � �x2 � 4

2

x � 4�

x � 2, y � 0, �0, �1

2��

41. f (x) � �x2

2

�

x �

x �

3

6�

x � �3, x � 2, y � 0, �0, �1

2��, ��

3

2�, 0�

42. f (x) � �x2 � 4

x

x � 4� x � �2, y � 0, (0, 0)

43. f (x) � �x �

x2

1� x � 0, y � 0, (�1, 0)

44. f (x) � �x �

x2

1� x � 0, y � 0, (1, 0)

45. f (x) � �x

23

x

�

�

9

1

x� x � 0, x � �3, y � 0, ��

1

2�, 0�

46. f (x) � �2

x

x3

2

�

�

x

1� x � 0, x � �1, y � 0


47. f (x) � �x2 �

x

1� x � �1, y � 0, (0, 0)

48. f (x) � �x2 �

x

x � 2� x � �2, x � 1, y � 0, (0, 0)

49. f (x) � �x2 �

2

1� y � 0, (0, 2)

50. f (x) � �x2 �

x

1� y � 0, (0, 0)

51. f (x) � �x �

x2

1� x � �1, y � x � 1, (0, 0)

52. f (x) � �x �

x2

1� x � 1, y � x � 1, (0, 0)

Solve each problem.

53. Oscillating modulators. The number of oscillating modu-lators produced by a factory in t hours is given by the poly-nomial function n(t) � t2 � 6t for t � 1. The cost in dol-lars of operating the factory for t hours is given by thefunction c(t) � 36t � 500 for t � 1. The average cost per

modulator is given by the rational function f (t) � �36

t

t2

�

�

5

6

0

t

0�

for t � 1. Graph the function f. What is the average cost permodulator at time t � 20 and time t � 30? What can youconclude about the average cost per modulator after a longperiod of time?f (20) � $2.35, f (30) � $1.46, average approaches 0

54. Nonoscillating modulators. The number of nonoscillatingmodulators produced by a factory in t hours is given by thepolynomial function n(t) � 16t for t � 1. The cost in dol-lars of operating the factory for t hours is given by the func-tion c(t) � 64t � 500 for t � 1. The average cost per mod-

ulator is given by the rational function f(t) � �64t

1

�

6t

500� for

t � 1. Graph the function f. What is the average cost permodulator at time t � 10 and t � 20? What can youconclude about the average cost per modulator after a longperiod 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 mil-lion to design its new 1998 SUV (Motor Trend,


www.motortrend.com). If it costs $25,000 to manufactureeach SUV, then the average cost per vehicle in dollars whenx vehicles are manufactured is given by the rationalfunction

A(x) � .

a) What is the horizontal asymptote for the graph of thisfunction?

b) What is the average cost per vehicle when 50,000 vehi-cles 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

56. Average cost of a pill. Assuming Pfizer spent a typical$350 million to develop its latest miracle drug and $0.10each to make the pills, then the average cost per pill in dol-lars when x pills are made is given by the rational function

A(x) � .0.10x � 350,000,000��

x

25,000x � 700,000,000��

x

F I G U R E F O R E X E R C I S E 5 6

a) What is the horizontal asymptote for the graph of thisfunction?

b) What is the average cost per pill when 100 million pillsare 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

10.5 Transformations of Graphs (10-33) 561

GR APHING CALCULATOR EXERCISES

Sketch the graph of each pair of functions in the same coordi-nate system. What do you observe in each case?

57. f (x) � x2, g(x) � x2 � 1�xThe graph of f (x) is an asymptote for the graph of g(x).

58. f (x) � x2, g(x) � x2 � 1�x2

The graph of f (x) is an asymptote for the graph of g(x).

59. f (x) � � x �, g(x) � � x � � 1�xThe graph of f (x) is an asymptote for the graph of g(x).

60. f (x) � � x �, g(x) � � x � � 1�x2


61. f (x) � x, g(x) � x � 1�xThe graph of f (x) is an asymptote for the graph of g(x).

62. f (x) � x3, g(x) � x3 � 1�x2


T R A N S F O R M A T I O N S O F G R A P H S

We can discover what the graph of almost any function looks like if we plot enoughpoints. However, it is helpful to know something about a graph so that we do nothave to plot very many points. For example, it is important to know that the graphof a function is symmetric about the y-axis before we begin calculating orderedpairs. In this section we will learn how one graph can be transformed into anotherby modifying the formula that defines the function.

Reflecting

Consider the graphs of f (x) � x2 and g(x) � �x2

shown in Fig. 10.10. Notice that the graph of gis a mirror image of the graph of f. For anyvalue of x we compute the y-coordinate of anordered pair of f by squaring x. For an orderedpair of g we square first and then find the oppo-site because of the order of operations. Thisgives a correspondence between the orderedpairs of f and the ordered pairs of g. For everyordered pair on the graph of f there is a corre-sponding ordered pair directly below it on thegraph of g, and these ordered pairs are the samedistance from the x-axis. We say that the graphof g is obtained by reflecting the graph of f in the x-axis or that g is a reflection ofthe graph of f.

Reflection

The graph of y � �f (x) is a reflection in the x-axis of the graph of y � f (x).

10.5I n t h i s

s e c t i o n

● Reflecting

● Translating

● Stretching and Shrinking

With a graphing calculator,you can quickly see the resultof modifying the formula for afunction. If you have a graph-ing calculator, use it to graphthe functions in the examples.Experimenting with it willhelp you to understand theideas in this section.

c a l c u l a t o r

c l o s e - u p

10

10

–10

–10

F I G U R E 1 0 . 1 0

–5–4–3–2

y

x–5 –4 –3 –2 32

21

345

4 5

f (x) = x2

g(x) = –x2

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10.4 GRAPHS OF RATIONAL FUNCTIONS - mr. …bn002.weebly.com/uploads/5/8/3/3/58339589/rational...10.4 Graphs of Rational Functions (10-25) 553 Suppose f(x) Q P((x x)) is a rational