Section 2.3: Rational Functions
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1 CHAPTER Polynomial and Rational Functions Section.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: 04 University of Houston Department of Mathematics
2 SECTION.3 Rational Functions Finding Vertical Asymptotes, Horizontal Asymptotes, and Holes: Eample: MATH 1330 Precalculus 05
3 CHAPTER Polynomial and Rational Functions Eample: 06 University of Houston Department of Mathematics
4 SECTION.3 Rational Functions Eample: MATH 1330 Precalculus 07
5 CHAPTER Polynomial and Rational Functions Eample: 08 University of Houston Department of Mathematics
6 SECTION.3 Rational Functions Definition of a Slant Asymptote: MATH 1330 Precalculus 09
7 CHAPTER Polynomial and Rational Functions Eample: 10 University of Houston Department of Mathematics
8 SECTION.3 Rational Functions Note: For a review of polynomial long division, please refer to Appendi A.: Dividing Polynomials. Additional Eample 1: The numerator and denominator have no common factors. MATH 1330 Precalculus 11
9 CHAPTER Polynomial and Rational Functions Additional Eample : 1 University of Houston Department of Mathematics
10 SECTION.3 Rational Functions MATH 1330 Precalculus 13
11 CHAPTER Polynomial and Rational Functions Additional Eample 3: 14 University of Houston Department of Mathematics
12 SECTION.3 Rational Functions Additional Eample 4: MATH 1330 Precalculus 15
13 CHAPTER Polynomial and Rational Functions Additional Eample 5: 16 University of Houston Department of Mathematics
14 SECTION.3 Rational Functions MATH 1330 Precalculus 17
15 CHAPTER Polynomial and Rational Functions Graphing Rational Functions A Strategy for Graphing Rational Functions: Eample: 18 University of Houston Department of Mathematics
16 SECTION.3 Rational Functions MATH 1330 Precalculus 19
17 CHAPTER Polynomial and Rational Functions Additional Eample 1: The numerator and denominator share no common factors. 0 University of Houston Department of Mathematics
18 SECTION.3 Rational Functions Additional Eample : The numerator and denominator share no common factors. MATH 1330 Precalculus 1
19 CHAPTER Polynomial and Rational Functions University of Houston Department of Mathematics
20 SECTION.3 Rational Functions Additional Eample 3: The numerator and denominator share no common factors. MATH 1330 Precalculus 3
21 CHAPTER Polynomial and Rational Functions 4 University of Houston Department of Mathematics
22 SECTION.3 Rational Functions Additional Eample 4: MATH 1330 Precalculus 5
23 CHAPTER Polynomial and Rational Functions 6 University of Houston Department of Mathematics
24 SECTION.3 Rational Functions Additional Eample 5: MATH 1330 Precalculus 7
25 CHAPTER Polynomial and Rational Functions 8 University of Houston Department of Mathematics
26 Eercise Set.3: Rational Functions Recall from Section 1. that an even function is symmetric with respect to the y-ais, 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 f( ) 8. y. 3 f( ) f( ) f( ) f( ) f( ) 3 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. y (Notice the asymptotes at 0 and y 0.) For each of the following graphs: (j) Identify the location of any hole(s) (i.e. removable discontinuities) (k) Identify any -intercept(s) (l) Identify any y-intercept(s) (m) Identify any vertical asymptote(s) (n) Identify any horizontal asymptote(s) 9. y (Notice the asymptotes at and y 0.) 10. y MATH 1330 Precalculus 9
27 Eercise Set.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 -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 f( ) 18 (3 5)( ) ( ) ( 4)(5 7) ( 3)( 4) f( ) ( 3)( )( 4) ( 1)( 4)( ) 19. ( )( 3) ( )( 4) ( 3)(6 ) ( )( 3) 37. ( 5)( 1)( 3) ( 1)( 3) ( 4)( 3)( )( 1) ( 4)( ) f( ) f( ) f( ) University of Houston Department of Mathematics
28 Eercise Set.3: Rational Functions Answer the following. 41. In the function f (a) Use the quadratic formula to find the - 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. 4. In the function f (a) Use the quadratic formula to find the - 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 f below, horizontal asymptote. For each function (a) Find the equation for the horizontal asymptote of the function. (b) Find the -value where f intersects the horizontal asymptote. (c) Find the point of intersection of f and the horizontal asymptote f( ) 7 51 f( ) Answer the following. 49. The function Eercise was graphed in 1 (a) Find the point of intersection of f horizontal asymptote. (b) Sketch the graph of f as directed in and the Eercise 33, but also label the intersection of f and the horizontal asymptote. 50. The function Eercise was graphed in 15 (a) Find the point of intersection of f horizontal asymptote. (b) Sketch the graph of f as directed in and the Eercise 34, but also label the intersection of f and the horizontal asymptote. 43. f f( ) f( ) f MATH 1330 Precalculus 31
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