Rational Functions - Precalculus Section 2.3 - Math 1330 - Online ...

CHAPTER 2 Polynomial and Rational Functions

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:

204

University of Houston Department of Mathematics

SECTION 2.3 Rational Functions

Finding Vertical Asymptotes, Horizontal Asymptotes, and Holes:

Example:

Solution:

MATH 1330 Precalculus

205


Example:

Solution:

206



Example:

Solution:


207


Example:

Solution:

208



Definition of a Slant Asymptote:


209

CHAPTER 2 Polynomial and Rational Functions Example:

Solution:

210



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.


211



212


SECTION 2.3 Rational Functions Solution:


213



Solution:

214




Solution:


215



216


SECTION 2.3 Rational Functions Solution:


217


Graphing Rational Functions A Strategy for Graphing Rational Functions:

Example:

Solution:

218




219

CHAPTER 2 Polynomial and Rational Functions Additional Example 1:

Solution: The numerator and denominator share no common factors.

220






221


222






223


224


SECTION 2.3 Rational Functions Additional Example 4:

Solution:


225


226


SECTION 2.3 Rational Functions Additional Example 5:

Solution:


227


228


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.

8.

y







x 

Determine if the functions below are even, odd, or neither. 1.











5 x

f ( x) 





3 x 1

2.

f ( x)  

3.

f ( x) 

4 x2  9

4.

f ( x) 

9x2  1 x4

5.

f ( x) 

x 1 x2  4

6.

f ( x) 

7 x3



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

For each of the following graphs:

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

(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.

 y   

7.



y



x

     











 





x 













y

10. 



 

x



   

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











  


229

Exercise Set 2.3: Rational Functions 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. f ( x)  12. f ( x) 

3 x5 4 x7

24. f ( x) 

x3 2 x 2  18

25. f ( x) 

(3x  5)( x  2) x( x  2)

26. f ( x) 

( x  4)(5 x  7) ( x  3)( x  4)

27. f ( x) 

28. f ( x) 

29.

2 x 2  18 x2  4 x  3 8 x 2  16 x 5 x 2  20

16  x 4 2 x3 x3  2 x 2  x  2 x2  4

2x  3 f ( x)  x

30. f ( x) 

14. f ( x) 

9  4x x

31. f ( x) 

15. f ( x) 

x6 x3

32. f ( x) 

16. f ( x) 

x5 x2

33. f ( x) 

17. f ( x) 

4 x  8 2x  3

34. f ( x) 

18. f ( x) 

3x  6 2x 1

35. f ( x) 

19. f ( x) 

( x  2)( x  3) ( x  2)( x  4)

36. f ( x) 

20. f ( x) 

( x  3)(6  x) ( x  2)( x  3)

37. f ( x) 

21. f ( x) 

x 2  x  20 x4

38. f ( x)  

22. f ( x) 

x 2  3x  10 x5

39. f ( x) 

x3  2 x 2  9 x  18 x2

23. f ( x) 

4 x3 x2  1

40. f ( x) 

x 4  10 x 2  9 x3

13.

230

8 x2  4

12 x  x6 2

6x  6 x  x  12 2

8 x  16 x  2 x  15 2

( x  3)( x  2)( x  4) ( x  1)( x  4)( x  2) 2 x3  10 x 2 x  5 x 2  9 x  45 3

x( x  5)( x  1)( x  3) ( x  1)( x  3) ( x  4)( x  3)( x  2)( x  1) ( x  4)( x  2)


Exercise Set 2.3: Rational Functions Answer the following. 41. In the function f  x  

x2  5x  3 3x 2  2 x  3

(a) Use the quadratic formula to find the xintercepts 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 calculator to round these answers to the nearest tenth. 42. In the function f  x  

2x2  7 x 1 x2  6 x  4

(a) Use the quadratic formula to find the xintercepts 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 calculator to round these answers to the nearest tenth.

47. f ( x) 

4 x 2  12 x  9 x2  x  7

48. f ( x) 

x2  5x  1 5 x 2  10 x  3

Answer the following. 49. The function f ( x) 

6x  6 x  x  12 2

was graphed in

Exercise 33. (a) Find the point of intersection of f  x  and the horizontal asymptote. (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 f ( x) 

8 x  16 x 2  2 x  15

was graphed in

Exercise 34. (a) Find the point of intersection of f  x  and the horizontal asymptote.

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) Sketch the graph of f  x  as directed in Exercise 34, but also label the intersection of f  x  and the horizontal asymptote.

(b) Find the x-value where f  x  intersects the horizontal asymptote. (c) Find the point of intersection of f  x  and the horizontal asymptote. 43. f  x  

x2  2 x  3 x2  x  3

44. f ( x) 

x2  4 x  2 x2  x  7

45. f ( x) 

x2  2 x  3 2 x2  6 x  1

46. f  x  

3x 2  5 x  1 x2  x  3


231