activity 35: rational functions (section 4.5, pp. 356-369)

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ACTIVITY 35: Rational Functions (Section 4.5, pp. 356-369)

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ACTIVITY 35:

Rational Functions (Section 4.5, pp. 356-369)

Rational Functions and Asymptotes:

A rational function is a function of the form

where P(x) and Q(x) are polynomials (that we assume without common factors). When graphing a rational function we must pay special attention to the behavior of the graph near the x-values which make the denominator equal to zero.

)(

)()(

xQ

xPxr

Example 1:

Find the domain of each of the following rational functions:

4

3)(

2

23

1

x

xxxr

The domain well be all x’s such that x2 – 4 ≠ 0.

042 x

42 x4x 2 ,22,22,

Consequently, the domain is all x’s except -2 and 2. That is,

4

52)(

22

x

xxr The domain well be all x’s such that x2 + 4 ≠ 0.

042 x

42 x

This is impossible! Consequently, the domain is all real numbers.

xxxr

4

5)(

23

The domain well be all x’s such that x2 – 4x ≠ 0.

042 xx 04 xx

0x ,44,00,

Consequently, the domain is all x’s except 0 and 4. That is,

04 x4x

Example 2:

Sketch the graphs of:

xxf

1)(

x y

0 undefined

1 1

2 1/23 1/3

1/2 2

1/3 3

1/4 4

x y

-1 -1

-2 -1/2

-3 -1/3-1/2 -2

-1/3 -3

-1/4 -4

32

1)(

x

xg Hint: observe that g(x) = f(x − 2) + 3

Definition of Vertical and Horizontal Asymptotes:

The line x = a is a vertical asymptote of y = r(x) if y → ∞ or y → −∞ as x → a+ or x → a−

Note: x → a− means that “x approaches a from the left”;x → a+ means that “x approaches a from the right”.

The line y = b is a horizontal asymptote of y = r(x) ify → b as x → ∞ or x → −∞

Asymptotes of Rational Functions:

The vertical asymptotes of r(x) are the lines x = a, where a is a zero of the denominator.

If n < m, then r(x) has horizontal asymptote y = 0.If n = m, then r(x) has horizontal asymptote y = .If n > m, then r(x) has no horizontal asymptote.

Let r(x) be the rational function:

011

1

011

1

...

...)(

bxbxbxb

axaxaxaxr

mm

mm

nn

nn

m

n

b

a

Example 3:

Find the horizontal and vertical asymptotes of each of the following functions:

65

26)(

2

xx

xxr

Vertical Asymptotes

0652 xx

061 xx

01 x 06 x

1x 6x

Horizontal Asymptotes

)26deg( x 1 )65deg( 2 xx 2

Consequently, r(x) has horizontal asymptote y = 0.

xx

xxt

23

3)(

2

Vertical Asymptotes

023 xx

03 x 02 x

3x x2

Horizontal Asymptotes

)3deg( 2x 2

2Consequently, r(x) has horizontal asymptote y =

xx 23 xxx 362 2 xx 62

)6deg( 2 xx

1

3

3y

4

62)(

2

24

x

xxxs

Vertical Asymptotes

042 x

42 x

4x 2

Horizontal Asymptotes

)62deg( 24 xx 4 )4deg( 2x 2

Consequently, r(x) has no horizontal asymptote.

Example 4:

Graph the rational function

65

6)(

2

xxxr

intercept -x

0y

65

60

2

xx

This is impossible so there are NO x-intercepts

intercept -y 0x

6050

6)0( 2 r

6

6

1

So the y-intercept is (0,-1)

Vertical Asymptotes

0652 xx

Horizontal Asymptotes

)6deg( 0 )65deg( 2 xx 2

Consequently, r(x) has horizontal asymptote y = 0.

65

6)(

2

xxxr

061 xx

01 x 06 x

1x 6x

65

6)(

2

xxxr

1. No x-intercepts2. y-intercept is (0,-1)3. Vertical Asymptotes x = – 1 and x = 64. Horizontal Asymptotes y = 0

6252

6)2( 2 r

6104

6

614

6

8

6

43,2

6858

6)8( 2 r

64064

6

624

6

18

6

31,8

Example 5:

Graph the rational function

31

21)(

xx

xxxr

intercept -x

0y

intercept -y 0x

3

2

So the y-intercept is

31

210

xx

xx

210 xx

10 x 20 x

x1 x 2

3010

2010)0(

r

32,0

0,1 0,2

31

21

3

2

Vertical Asymptotes

031 xx

Horizontal Asymptotes

2

Consequently, r(x) has horizontal asymptote y = .

01 x 03 x1x 3x

31

21)(

xx

xxxr

31 xx 332 xxx 322 xx

21 xx 222 xxx 22 xx

21deg xx

31deg xx 2

1

1

1. x-intercepts (1,0) and (-2,0)2. y-intercept is (0,2/3)3. Vertical Asymptotes x = – 1 and x = 34. Horizontal Asymptotes y = 1

31

21)(

xx

xxxr

3212

2212)2(

r 13

41

3

4

3

4,2

3414

2414)4(

r 15

63

5

18

5

18,4