Lesson 2.3 Exercises, pages

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1 Lesson.3 Eercises, pages A. For the graph of each rational function below: i) Write the equations of an asmptotes. ii) State the domain. a) b) i) There is no vertical asmptote. The degree of the numerator is less than the degree of the denominator, so 0 is the horizontal asmptote. ii) Since all real values of are permissible, the domain is: ç i) The vertical asmptotes have equations and. The horizontal asmptote has equation 1. ii) The domain is: c) d) i) The vertical asmptote has equation 1. The oblique asmptote has slope 1 and -intercept 1, so its equation is 1. ii) The domain is: 1 i) The vertical asmptote has equation. The oblique asmptote has slope and -intercept 1, so its equation is 1. ii) The domain is: P DO NOT COPY..3 Analzing Rational Functions Solutions 9

2 B 5. For the graph of each rational function: i) Write the coordinates of an hole. ii) Write the equations of an vertical asmptotes. a) b) = + - = The function is undefined when 0; that is, when. ( )( ) i) There is a hole at on the line with equation. The coordinates of the hole are: (, ) ii) There is no vertical asmptote. The function is undefined when: 3 0 ( 3)( 1) 0 3 or 1 ( )( 1) i) ( 3)( 1) There is no hole. ii) The vertical asmptotes have equations: 3 and 1 c) d) = = The function is undefined when 0. Since is never equal to 0, the function is defined for all real values of. i) There is no hole. ii) There is no vertical asmptote. The function is undefined when 1 0, or 1. ( )( 1) i) 1 There is a hole at 1 on the line with equation. The coordinates of the hole are: (1, 3) ii) There is no vertical asmptote. 6. For each rational function, determine whether its graph has a horizontal asmptote. If it does, write its equation. a) b) = - 16 = + + The degrees of the numerator and denominator are equal, so there is a horizontal asmptote. The leading coefficients of the numerator and denominator are and 1 respectivel. So, the horizontal asmptote has equation: The degrees of the numerator and denominator are equal, so there is a horizontal asmptote. Both leading coefficients are 1. So, the horizontal asmptote has equation: Analzing Rational Functions Solutions DO NOT COPY. P

3 c) = d) = The degree of the numerator is less than the degree of the denominator, so there is a horizontal asmptote with equation 0. The degrees of the numerator and denominator are equal, so there is a horizontal asmptote. Both leading coefficients are 1. So, the horizontal asmptote has equation: 1 7. For each rational function, determine whether its graph has an oblique asmptote. If it does, write its equation. a) b) = = The numerator does not factor. The degree of the numerator is 1 more than the degree of the denominator, so there is an oblique asmptote. Determine: ( 5) ( 1) The quotient is 1; so the equation of the oblique asmptote is: 1 Factor the numerator: ( 5)( ) 5 After removing a common factor, the equation is:, 5 So, there is no oblique asmptote. c) d) = = more than the degree of the denominator, so there is an oblique asmptote. Determine: ( ) Write: ( ) The quotient is ; so the equation of the oblique asmptote is: 1 more than the degree of the denominator, so there is an oblique asmptote. Determine: ( 3 1) ( ) The quotient is 7; so the equation of the oblique asmptote is: 7 P DO NOT COPY..3 Analzing Rational Functions Solutions 11

4 8. Solve each rational equation b graphing. Give the solutions to the nearest tenth where necessar. a) b) = 0 - = + 3 Graph a related function: f() 1 Use graphing technolog to determine the zero: 6 Graph a related function: f() 3 Use graphing technolog to determine the zeros:.3 or c) d) 3-1 = + 10 = Graph a related function: f() 6 Use graphing technolog to determine the zeros:.6 or. Graph a related function: f() Use graphing technolog to determine the zeros: 1.3 or 0.1 or Match each function to its graph. Justif our choice. i) Graph A ii) Graph B 0 f() 6 iii) Graph C 0 g() iv)graph D 0 h() k() 0 6 v) Graph E vi)graph F 0 p() q() Analzing Rational Functions Solutions DO NOT COPY. P

5 a) b) = - + = ( 3)( ) Rewrite as: ( 3)( ) The graph is a line with a hole at. The function matches Graph C. ( )( ) 1 There is a vertical asmptote at 1. 1 more than the degree of the denominator, so there is an oblique asmptote. The function matches Graph A. 3 c) = d) = ( 1)( 1) There are vertical asmptotes at 1 and 1. 1 more than the degree of the denominator, so there is an oblique asmptote. The function matches Graph F. ( 1)( 1) There are vertical asmptotes at 1 and 1. less than the degree of the denominator, so there is a horizontal asmptote at 0. The function matches Graph B. e) f ) = - 1 = The denominator is alwas positive, so the graph has no hole or vertical asmptote. The degrees of the numerator and denominator are equal, so the graph has a horizontal asmptote that is not the -ais. The function matches Graph D. ( 1)( 1) ( 1)( 1) There are vertical asmptotes at 1 and 1. The degrees of the numerator and denominator are equal, so the graph has a horizontal asmptote that is not the -ais. The function matches Graph E. P DO NOT COPY..3 Analzing Rational Functions Solutions 13

6 10. For the graph of each function: i) Determine the equations of an asmptotes and the coordinates of an hole. ii) Determine the domain. iii) Use graphing technolog to verif the characteristics, and to eplain the behaviour of the graph close to the nonpermissible values. a) = - i) The function is undefined when 0; that is, when 0. There are no common factors, so there is a vertical asmptote with equation 0. The degrees of the numerator and denominator are equal, so there is a horizontal asmptote. The leading coefficients of the numerator and denominator are and 1, respectivel. So, the horizontal asmptote has equation: ii) The domain is: 0 iii) From the calculator screen: as : ˆ, :, which verifies the horizontal asmptote; as : 0, : ˆ, which verifies the vertical asmptote b) = i) The function is undefined when 3 0; that is, when 3. ( 5)( 3) ( 5)( 3), or 3 3 ( 3) is a common factor, so there is a hole at 3. The function is: 5, 3 The coordinates of the hole are: (3, 8) ii) The domain is: 3 iii) From the calculator screen: as : 3, : 8, which verifies the hole c) = - + i) The denominator is alwas positive, so there are no restrictions on, and there is no hole or vertical asmptote. Since the degree of the numerator is less than the degree of the denominator, there is a horizontal asmptote at 0. ii) The domain is: ç iii) From the calculator screen: as : ˆ, : 0, which verifies the horizontal asmptote 1.3 Analzing Rational Functions Solutions DO NOT COPY. P

7 11. The speed of a boat in still water is 10 km/h. It travels 5 km upstream and 5 km downstream in 6 h. This equation models the total time for the journe in terms of the speed of the current, 5 5 v kilometres per hour: 10 - v 10 + v = 6 What is the speed of the current, to the nearest whole number? Write a related function: f(v) v 10 v 6 Use graphing technolog to determine the zeros:.1 or.1 Ignore the negative root because speed cannot be negative. To the nearest whole number, the speed of the current is km/h. C 1. Create an equation for a rational function whose graph has the given characteristics. a) The graph is the line with two holes. For a graph to have holes, the numerator and denominator must have different binomial common factors. Begin with the equation of the line,. Choose two binomial factors, such as ( 3) and ( ). Multipl and divide b these factors. A possible function is ( 3)( ). ( 3)( ) b) The graph has a horizontal asmptote with equation 6, and no vertical asmptotes. For a horizontal asmptote 6, the function must approach 6 as approaches infinit and the degrees of the numerator and denominator must be equal. The leading coefficient of the numerator could be 6, then the leading coefficient of the denominator would be 1. For no vertical asmptotes, the denominator must never be 0. A possible function is 6. 1 c) The graph has an oblique asmptote with equation 1, and a vertical asmptote with equation. For an oblique asmptote, when the denominator is divided into the numerator, the quotient must be 1 and there must be a remainder. For a vertical asmptote with equation, the denominator contains the factor ( ) and the numerator does not. ( 1)( ) 3 A possible function is. P DO NOT COPY..3 Analzing Rational Functions Solutions 15

8 d) The graph has two vertical asmptotes, a horizontal asmptote that is not the -ais, two holes, and: i) the -ais is a line of smmetr ii) the -ais is not a line of smmetr i) For the -ais to be a line of smmetr, the function must have the same value for a and a, so the numerator and denominator are the products of factors of the form ( a)( a), or a. For two vertical asmptotes, the denominator contains two binomial factors, with opposite constant terms, such as ( 3)( 3), or 9. For a horizontal asmptote such as, the leading coefficient of the numerator could be, then the leading coefficient of the denominator would be 1, and the term in the numerator would be. For two holes, the numerator and denominator must have different binomial common factors, with opposite constant terms, such as ( 1)( 1), or 1. A possible function is ( 1). ( 1)( 9) ii) For the graph to have no smmetr about the -ais, replace the factor in the numerator with a factor not of the form ( a ), for eample, ( ). A possible function is ( )( 1). ( 1)( 9) 16.3 Analzing Rational Functions Solutions DO NOT COPY. P