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:
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SECTION 2.3 Rational Functions
Finding Vertical Asymptotes, Horizontal Asymptotes, and Holes:
Example:
Solution:
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Example:
Solution:
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Example:
Solution:
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Example:
Solution:
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SECTION 2.3 Rational Functions
Definition of a Slant Asymptote:
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CHAPTER 2 Polynomial and Rational Functions Example:
Solution:
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SECTION 2.3 Rational Functions
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.
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Additional Example 2:
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SECTION 2.3 Rational Functions Solution:
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Additional Example 3:
Solution:
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Additional Example 4:
Solution:
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Additional Example 5:
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SECTION 2.3 Rational Functions Solution:
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Graphing Rational Functions A Strategy for Graphing Rational Functions:
Example:
Solution:
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CHAPTER 2 Polynomial and Rational Functions Additional Example 1:
Solution: The numerator and denominator share no common factors.
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Additional Example 2:
Solution: The numerator and denominator share no common factors.
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Additional Example 3:
Solution: The numerator and denominator share no common factors.
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SECTION 2.3 Rational Functions Additional Example 4:
Solution:
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SECTION 2.3 Rational Functions Additional Example 5:
Solution:
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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 .)
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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 x5 4 x7
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)
x6 x3
32. f ( x)
16. f ( x)
x5 x2
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 x4
38. f ( x)
22. f ( x)
x 2 3x 10 x5
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.
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8 x2 4
12 x x6 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)
University of Houston Department of Mathematics
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
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