Asymptotes and Holes Graphing Rational Functions

CHAPTER 2 Polynomial and Rational Functions

University of Houston Department of Mathematics 204

Section 2.3: Rational Functions

Asymptotes and Holes

Graphing Rational Functions

Asymptotes and Holes

Definition of a Rational Function:

Definition of a Vertical Asymptote:

Definition of a Horizontal Asymptote:

SECTION 2.3 Rational Functions

MATH 1330 Precalculus 205

Finding Vertical Asymptotes, Horizontal Asymptotes, and

Holes:

Example:

Solution:



Example:

Solution:



Example:

Solution:



Example:

Solution:



Definition of a Slant Asymptote:



Example:

Solution:



Note: For a review of polynomial long division, please refer to Appendix A.2:

Dividing Polynomials .

Additional Example 1:

Solution:

The numerator and denominator have no common factors.

http://online.math.uh.edu/Math1330/Appendix/sA2/index.html

http://online.math.uh.edu/Math1330/Appendix/sA2/index.html






Solution:




Solution:




Solution:






Solution:



Graphing Rational Functions

A Strategy for Graphing Rational Functions:

Example:

Solution:






Solution:

The numerator and denominator share no common factors.




Solution:







Solution:







Solution:






Solution:



Exercise Set 2.3: Rational Functions


Recall from Section 1.2 that an even function is

symmetric with respect to the y-axis, and an odd function

is symmetric with respect to the origin. This can

sometimes save time in graphing rational functions. If a

function is even or odd, then half of the function can be

graphed, and the rest can be graphed using symmetry.

Determine if the functions below are even, odd, or

neither.

1. 5

( )f xx

2. 3

( )1

f xx

3. 2

4( )

9f x

x

4. 2

4

9 1( )

xf x

x

5. 2

1( )

4

xf x

x

6. 3

7( )f x

x

In each of the graphs below, only half of the graph is

given. Sketch the remainder of the graph, given that the

function is:

(a) Even

(b) Odd

7.

(Notice the asymptotes at 2x and 0y .)

8.

(Notice the asymptotes at 0x and 0y .)

For each of the following graphs:

(j) Identify the location of any hole(s)

(i.e. removable discontinuities)

(k) Identify any x-intercept(s)

(l) Identify any y-intercept(s)

(m) Identify any vertical asymptote(s)

(n) Identify any horizontal asymptote(s)

9.

10.

x

y

x

y

x

y

x

y



For each of the following rational functions:

(a) Find the domain of the function

(b) Identify the location of any hole(s)

(i.e. removable discontinuities)

(c) Identify any x-intercept(s)

(d) Identify any y-intercept(s)

(e) Identify any vertical asymptote(s)

(f) Identify any horizontal asymptote(s)

(g) Identify any slant asymptote(s)

(h) Sketch the graph of the function. Be sure to

include all of the above features on your graph.

11. 5

3)(

xxf

12. 7

4)(

xxf

13. x

xxf

32)(

14. x

xxf

49)(

15. 3

6)(

x

xxf

16. 2

5)(

x

xxf

17. 32

84)(

x

xxf

18. 12

63)(

x

xxf

19. )4)(2(

)3)(2()(

xx

xxxf

20. )3)(2(

)6)(3()(

xx

xxxf

21. 4

20)(

2

x

xxxf

22. 5

103)(

2

x

xxxf

23. 3

2

4( )

1

xf x

x

24. 3

2( )

2 18

xf x

x

25. )2(

)2)(53()(

xx

xxxf

26. )4)(3(

)75)(4()(

xx

xxxf

27. 34

182)(

2

2

xx

xxf

28. 205

168)(

2

2

x

xxxf

29. 4

3

16

2

x

x

30. 3 2

2

2 2( )

4

x x xf x

x

31. 4

8)(

2

xxf

32. 6

12)(

2

xxxf

33. 12

66)(

2

xx

xxf

34. 152

168)(

2

xx

xxf

35. )2)(4)(1(

)4)(2)(3()(

xxx

xxxxf

36. 4595

102)(

23

23

xxx

xxxf

37. )3)(1(

)3)(1)(5()(

xx

xxxxxf

38. )2)(4(

)1)(2)(3)(4()(

xx

xxxxxf

39. 3 2

2

2 9 18( )

x x xf x

x

40. 4 2

3

10 9( )

x xf x

x



Answer the following.

41. In the function 2

2

5 3

3 2 3

x xf x

x x

(a) Use the quadratic formula to find the x-

intercepts of the function, and then use a

calculator to round these answers to the nearest

tenth.

(b) Use the quadratic formula to find the vertical

asymptotes of the function, and then use a


tenth.

42. In the function 2

2

2 7 1

6 4

x xf x

x x

(a) Use the quadratic formula to find the x-

intercepts of the function, and then use a


tenth.

(b) Use the quadratic formula to find the vertical

asymptotes of the function, and then use a


tenth.

The graph of a rational function never intersects a

vertical asymptote, but at times the graph intersects a

horizontal asymptote. For each function f x below,

(a) Find the equation for the horizontal asymptote of

the function.

(b) Find the x-value where f x intersects the

horizontal asymptote.

(c) Find the point of intersection of f x and the


43. 2

2

2 3

3

x xf x

x x

44. 2

2

4 2( )

7

x xf x

x x

45. 2

2

2 3( )

2 6 1

x xf x

x x

46. 2

2

3 5 1

3

x xf x

x x

47. 2

2

4 12 9( )

7

x xf x

x x

48. 2

2

5 1( )

5 10 3

x xf x

x x

Answer the following.

49. The function 12

66)(

2

xx

xxf was graphed in

Exercise 33.

(a) Find the point of intersection of f x and the


(b) Sketch the graph of f x as directed in

Exercise 33, but also label the intersection of

f x and the horizontal asymptote.

50. The function 152

168)(

2

xx

xxf was graphed in

Exercise 34.

(a) Find the point of intersection of f x and the


(b) Sketch the graph of f x as directed in

Exercise 34, but also label the intersection of

f x and the horizontal asymptote.

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Asymptotes and Holes Graphing Rational Functions