Asymptotes and Holes Graphing Rational Functions · PDF fileAsymptotes and Holes Graphing...
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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:
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 206
Example:
Solution:
SECTION 2.3 Rational Functions
MATH 1330 Precalculus 207
Example:
Solution:
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 208
Example:
Solution:
SECTION 2.3 Rational Functions
MATH 1330 Precalculus 209
Definition of a Slant Asymptote:
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 210
Example:
Solution:
SECTION 2.3 Rational Functions
MATH 1330 Precalculus 211
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.
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 212
Additional Example 2:
SECTION 2.3 Rational Functions
MATH 1330 Precalculus 213
Solution:
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 214
Additional Example 3:
Solution:
SECTION 2.3 Rational Functions
MATH 1330 Precalculus 215
Additional Example 4:
Solution:
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 216
Additional Example 5:
SECTION 2.3 Rational Functions
MATH 1330 Precalculus 217
Solution:
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 218
Graphing Rational Functions
A Strategy for Graphing Rational Functions:
Example:
Solution:
SECTION 2.3 Rational Functions
MATH 1330 Precalculus 219
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 220
Additional Example 1:
Solution:
The numerator and denominator share no common factors.
SECTION 2.3 Rational Functions
MATH 1330 Precalculus 221
Additional Example 2:
Solution:
The numerator and denominator share no common factors.
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 222
SECTION 2.3 Rational Functions
MATH 1330 Precalculus 223
Additional Example 3:
Solution:
The numerator and denominator share no common factors.
CHAPTER 2 Polynomial and Rational Functions
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SECTION 2.3 Rational Functions
MATH 1330 Precalculus 225
Additional Example 4:
Solution:
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 226
SECTION 2.3 Rational Functions
MATH 1330 Precalculus 227
Additional Example 5:
Solution:
CHAPTER 2 Polynomial and Rational Functions
University of Houston Department of Mathematics 228
Exercise Set 2.3: Rational Functions
MATH 1330 Precalculus 229
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
Exercise Set 2.3: Rational Functions
University of Houston Department of Mathematics 230
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
Exercise Set 2.3: Rational Functions
MATH 1330 Precalculus 231
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
calculator to round these answers to the nearest
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
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.
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
horizontal asymptote.
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
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 152
168)(
2
xx
xxf was graphed in
Exercise 34.
(a) Find the point of intersection of f x and the
horizontal asymptote.
(b) Sketch the graph of f x as directed in
Exercise 34, but also label the intersection of
f x and the horizontal asymptote.