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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