Chapter 9 Prerequisite Skills

Name: Date: Chapter 9 Prerequisite Skills BLM 9. Consider the function f() 3. a) Show that 3 is a factor of f(). If f() ( 3)g(), what is g()?. Factor each epression fully. a) 30g 4g 6fg 8g c) 6 5 d) 5 e) 6a a 3 f) 0.0 g) 0 y 45 3. Consider the sketch of the function h() 3 5 6. a) What are the zeros of the function? Epress h() in factored form. c) What are the roots of the equation 3 5 6 0? 4. Identify all non-permissible s for each epression. 4a a) 3bc 5 c) d) 3 5 t 6t 9 6 e) f) (3 )( 5) ( a )( a )( a 3) 5. Consider P() 3 5 8. a) Use the Factor Theorem to find a factor of P(). Completely factor P(). c) What is the solution of P() 0? d) Sketch the graph of P(). Eplain the significance of the -intercepts. 6. a) Arrange the equation 4 into the form a b c 0, where a, b, and c are integers. State the solution for this equation. Rewrite the function you developed in part a) in the form y ( h) k, where h and k are real numbers. c) Graph the function. Eplain the relationship of the -intercepts and coordinates of the verte to the functions in parts a) and. 7. Solve each equation algebraically. Epress answers as eact s and identify non-permissible s, where necessary. a) 4 49 0 6 7 3 0 c) 6 7 0 d) 3 4 0 e) 3 4 4 4 f) 4 8. Complete the square. Leave your answers in the form a( h) k 0, where a, h, and k are real numbers. a) 0 4 0 8 3 c) 3 6 d) 4 3 9. Solve each equation by graphing the corresponding function. a) 5 4 0 3 4 7 0 0 0. Simplify each rational epression. State any non-permissible s for the variables. 0k 55k 75 a) 3 0k 0k 50. Simplify each product or quotient. Identify all non-permissible s. 4z 5 z 4 a) z 3z 0 4z 0 7 5 4 9 3 0 6 Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Name: Date: Section 9. Etra Practice BLM 9. Match each function with its graph. 4 4 a) y 3 y 3 4 c) y 3 d) 4 y 3 A D B C 5. Graph the function y using a table of s. Analyse your graph and use a table to summarize the following characteristics: non-permissible (s) behaviour near non-permissible (s) end behaviour domain range equation of vertical asymptote equation of horizontal asymptote 3. Sketch and graph each function. Identify the domain and range, intercepts, and asymptotes. 3 a) y y 6 5 c) y 4 d) y 8 4. Graph each function using technology, and identify any asymptotes and intercepts. 5 a) y 4 3 y Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Name: Date: BLM 9 5. Write the equation of each function in the a form y k. h a) c) d) a 6. The rational function y k passes 5 through points (6, 7) and (4, ). a) Determine the of a and k. Graph the function. 7. Sketch the graph of y and y on the same set of aes. 6 9 Describe how one is a transformation of the other. 8. Use a table of s and a graph to analyse the function y. Then, complete the 7 table. Characteristic Non-permissible Behaviour near non-permissible End behaviour Domain Range Equation of vertical asymptote Equation of horizontal asymptote y 7 9. The distance between two cities is 35 km. a) Write an epression that you can use to calculate the time, t, in hours, that it takes to travel distance, d, in kilometres, at an average speed of s km/h. Use your formula from part a) to determine how long it will take to travel from one city to the other at an average speed of 65 km/h. c) If the trip from one city to the other took 5 h, determine the average speed, s, in kilometres per hour. Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Name: Date: Section 9. Etra Practice BLM 9 3. Eplain the behaviour at each non-permissible in the graph 5 6 of the rational function y. 4 5. Analyse each function and predict the location of any vertical asymptotes, points of discontinuity, and intercepts. Then, graph the function to verify your predictions. 5 7 a) y y 7 0 9 5 4 c) 5 3 y d) y 3 6. Complete the table and compare the behaviour of the two functions near any nonpermissible s. Characteristic y 3 3 9 y 3 3 9. Eplain how the equation of the rational function y can be analysed to 3 determine whether the graph of the function has an asymptote or a point of discontinuity. 3. Complete the table for the given rational function. Characteristic Non-permissible (s) Feature ehibited at each non-permissible Behaviour near each non-permissible Domain Range ( 3)( ) y ( 5)( 3) 4. Create a table of s for each function for s near its non-permissible (s). Eplain how your table shows whether a point of discontinuity or an asymptote occurs in each case. 5 4 a) 5 4 y y 6 8 Non-permissible (s) Feature ehibited at each nonpermissible Behaviour near each nonpermissible 7. Without using technology, match the equation of each rational function with the most appropriate graph. Eplain your reasoning. 4 4 a) y y 3 4 5 4 4 c) y 4 A Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Name: Date: BLM 9 3 B C c) 8. Write the equation for each graphed rational function. a) d) 9. Write the equation of a possible rational function that has an asymptote at, has a point of discontinuity at.5, and passes through (6, 3). Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Name: Date: Section 9.3 Etra Practice BLM 9 4. Solve each equation algebraically. a) 4 5 3 3 5 5 c) d) 8 6 4 8 3 6 9. Solve algebraically. Check your solutions. 3 a) 3 9 c) 5 3 4 4 4 7 d) 3 3. a) Determine the roots of the rational equation 5 60 algebraically. 5 Graph the rational function y 6 and determine the -intercepts. c) Eplain the connection between the roots of the equation and the -intercepts of the graph of the function. 4. Solve each of the following equations by graphing each side of the equation as a separate function. 6 a) 3 5 7 3 5 6 c) 3 5. Solve by rearranging as a single function and then graphing. a) 4 3 4 6. Solve each equation algebraically. Give your answers to the nearest hundredth. a) 4 3 3 c) 5 5 7. Determine the approimate solution(s) to each rational equation graphically, to the nearest hundredth. 3 a) 3 3 3 4 7 9 8. Solve the equation algebraically. 8 n n 9 n 3 9. It takes James 9 h longer to construct a fence than it takes Carmen. If they work together, they can construct the fence in 0 h. How long would it take each of them, working alone, to construct the fence? Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Name: Date: Chapter 9 Study Guide BLM 9 5 This study guide is based on questions from the Chapter 9 Practice Test in the student resource. Question I can Help Needed Refer to # eplain the behaviour of the graph of a rational function for s of the variable near a non-permissible some none # match a function to its graph some none #3 eplain the behaviour of the graph of a rational function some for s of the variable near a non-permissible none #4 describe the relationship between the roots of a rational some equation and the -intercepts of the graph of the none corresponding rational function #5 write equations in equivalent forms some #6 determine if the graph of a rational function will have an asymptote or a point of discontinuity for a nonpermissible #7 determine the solution of a rational equation algebraically and graphically #8 analyse the graph of a rational function to identify characteristics graph a rational function with linear epressions in the numerator and denominator #9 determine, graphically, an approimate solution of a rational equation none some none some none some none some none some none #0 sketch a graph with or without technology some none eplain the behaviour of the graph of a rational function some for s of the variable near a non-permissible none # analyse a rational function to identify characteristics some none # match a set of rational functions to their graphs, and some eplain the reasoning none #3 analyse the graphs of a set of rational functions to some compare characteristics none #4 solve a rational equation algebraically and graphically some none #5 solve problems involving rational equations some none #6 solve problems involving rational equations some none 9. Eample 9. Eample 3 9.3 Eample 9.3 Eample 9.3 Eample 9. Eample 9.3 Eample 9. Eample 9. Eample 3 9.3 Eample 9. Eample 9. Eamples, 9. Eample 3 9. Eample 3 9. Eample 9.3 Eamples, 3 9. Eample 5 9. Eample 5 Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Name: Date: Chapter 9 Test Multiple Choice For # to #8, choose the best answer. k. The -intercept of y is 0.5. What is the of k? A.0 B.5 C.5 D 3.0. Consider the function g ( ). Which statement is false? A g() has two vertical asymptotes. B g() is not defined when 0. C g() has one zero. D g() is a rational function. 3. Consider the functions f(), g(), ( ) and h ( ) f. Which statement is true? g( ) A f(), g(), and h() have the same domain. B The zero of f() is the vertical asymptote of h(). C The non-permissible of h() is the zero of g(). D h() is equivalent to y 0.5 0.5. 4. Consider the following graph of the function f ( ) r. What is the of r? A 3 B C D 3 5. Which of the following is true of the 3 rational function y 6? BLM 9 6 A It has a zero at. B Its range is {yy R}. 6 9 C It is equivalent to y. D It has a vertical asymptote at 6. 6. The graph of which function has a point of discontinuity at? A y B y C y D y 7. Which function has a domain of {, R} and a range of {yy 3, y R}? A y 3 3 3 B y 4 3 3 3 C y D y 8. How many roots does the equation 8 have? 6 4 A 0 B C D 3 Short Answer 9. a) Sketch the graph of the function y. 4 Identify the domain, range, and asymptotes of the function. c) Eplain the behaviour of the function as the of becomes very large. 0. a) Sketch the graph of the function y. 5 State the s of the -intercept and y-intercept. c) Solve 0 algebraically. 5 d) How is your answer to part c) related to your answers to parts a) and? Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Name: Date: BLM 9 6. Select the graph that matches the given function. a) y 5 y 3 ( 3) ( 5) c) y ( 5) A B C 3 3. a) Solve the equation 7 algebraically. Check your answer to part a) graphically. Etended Response 3. a) Graph the functions f( ) and 3 g ( ). Use a table to compare the characteristics of the two graphs. Write g() as a transformation of f(): g() f( a) b. c) Describe the transformation of f() to g(). 4. a) Describe two methods you could use to solve the equation 3 5 ( ) graphically. 4 Use one of the methods from part a) to solve the equation. 5. A rectangle has an area of 6000 cm. a) Write an equation to represent length, l, as a function of the width, w, for this rectangle. Write an equation to represent the change in length, as a function of width, w, when the width is increased by cm. c) Determine the width, w, of the rectangle if the change in length is 0 cm. 6. An emergency patrol boat is patrolling a river. The river has a 5 km/h current. The patrol boat travels 0 km upriver and 0 km back. The total time, t, in hours, for the round trip is given by the function 0v t, where v is the speed of the boat v 5 in kilometres per hour. a) State the domain and range for this function. Sketch the graph over the domain determined in part a). c) Determine the speed of the boat if the round trip took.5 h. Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

Chapter 9 BLM Answers BLM 9 Prerequisite Skills. a) 54 3 3 3 3 5 5 5 4 4 0 g() 5 4. a) g(5 g) g(3f 4g) c) ( 5)( ) d) ( )( 5) e) (a 3)(3a ) f) ( 0.)( 0.) g) 5(y 3)(y 3) 3. a),, and 3 h() ( )( )( 3) c),, and 3 5 4. a) b 0, c 0 c), d) t 3 5 e), f) a, a, and a 3 3 5. a) P() () 3 5() () 8 0, therefore, ( ) is a factor. P() ( )( 4)( ) c),, 4 d) 7. a) 7 3, c) 3 6 3 4 d) e), 0, 4 3 f) 6,, 8. a) ( 5) 0 ( 4) 3 0 c) 3( ) 0 d) ( ) 5 0 9. a) 4, 5,, BLM 9 7 The -intercepts are the solution to P() 0. 6. a) 4 0; 6 and y ( ) 6 c) 0. a),, 3 3. a), z 4, 5 k 3 5, k,3 k 3 3 3,,0,5 BLM 9 Section 9. Etra Practice. a) B C c) D d) A. The -intercepts are the solution to the equation. The -coordinate of the verte is the that makes ( h) 0 and the y-coordinate is equal to k. Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

BLM 9 7 Characteristic 5 y Non-permissible Behaviour near nonpermissible As approaches, y becomes very large. End behaviour As becomes very large, y approaches 0. Domain {, R} Range {yy 0, y R} Equation of vertical asymptote Equation of horizontal y 0 asymptote 3. a) d) domain: {, R}; range: {yy 8, y R}; intercepts: (0, 8.5), (.5, 0); asymptotes:, y 8 4. a) domain: {, R}; range: {yy 0, y R}; intercept: (0, 3); asymptotes:, y 0 asymptotes:, y ; intercepts: (.5, 0), (0, 5) domain: { 0, R}; range: {yy 6, y R}; intercept: 3,0 ; asymptotes: 0, y 6 c) asymptotes:, y 4; intercepts: (0,.5), (0.75, 0) 3 4 5. a) y y c) y d) 5 6. a) a 3, k 4 y 4 domain: { 4, R}; range: {yy, y R}; intercepts: (0, 0.75), (.5, 0); asymptotes: 4, y Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

BLM 9 7 7. 9. a) t d s 35 t 5.4, so 5.4 hours or 5 h and 4 min 65 c) 70. km/h The graph of y is the graph of y 6 9 translated 3 units left. 8. y 5 0.9 0.56 0.7 4.33 7 undefined 0 6.33 3 4.7 6 3.44 9 3.08 Characteristic y 7 Non-permissible 7 Behaviour near nonpermissible End behaviour As approaches 7, y becomes very large. As becomes very large, y approaches. Domain { 7, R} Range {yy, y R} Equation of vertical asymptote Equation of horizontal asymptote 7 y BLM 9 3 Section 9. Etra Practice. point of discontinuity at (3, 0 ) vertical asymptote: 7. You can factor the denominator: y ( )( ). Since the factor ( ) appears in the numerator and denominator, the graph will have a point of discontinuity at (, ). The factor ( ) appears in the denominator only, so there will be an asymptote at. 3. Characteristic ( 3)( ) y ( 5)( 3) Non-permissible (s) 5 and 3 Feature ehibited at each non-permissible Behaviour near each non-permissible asymptote at 5; point of discontinuity at (3,.5) As approaches 5, y becomes very large. As approaches 3, y approaches.5. Domain { 3, 5, R} Range 4. a) y 0.9 3. 0.99 3.0 0.999 3.00 0.9999 3.000 undefined.000.9999.00.999.0.99..9 {yy, As approaches, y approaches 3. 5, y R} Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

BLM 9 7 y.9 4.38 095 4.99 4.47 636 8.999 4.497 5 37.9999 4.499 75 0 undefined.000 4.500 75 0.00 4.50 75 38.0 4.57 638 9. 4.789 473 68 y 3.9 09 3.99 099 3.999 0 999 3.9999 09 999 4 undefined 4.000 0 00 4.00 00 4.0 0 4. As approaches, y approaches 4.5, and as approaches 4, y becomes very large, approaching negative infinity or positive infinity. 5. a) vertical asymptote: ; point of discontinuity at (5, 5 3 ); -intercept: (0, 0); y-intercept: (0, 0) vertical asymptote: 3; point of discontinuity at (3, ); 6 4 -intercept: (4, 0); y-intercept: (0, ) 3 c) no vertical asymptote; point of discontinuity at (, 3); -intercept: (4, 0); y-intercept: (0, 4) d) no vertical asymptote; point of discontinuity at (3, 7); -intercept: (0.5, 0); y-intercept: (0, ) 6. Characteristic Non-permissible (s) Feature ehibited at each nonpermissible Behaviour near each nonpermissible y 3 3 9 y 3 3 point of discontinuity As approaches 3, y approaches. 3 3 9 asymptote As approaches 3, y becomes very large. 7. a) C; Eample: In factored form, the rational function has two non-permissible s in the denominator, which do not appear in the numerator. Therefore, the graph with two asymptotes is the most appropriate choice. B; Eample: In factored form, the rational function has one non-permissible that appears in both the numerator and denominator, and another nonpermissible that is only in the denominator. Therefore, the graph with one asymptote and one point of discontinuity is the most appropriate choice. c) A; Eample: In factored form, one non-permissible appears in the numerator and denominator. Therefore, the graph has a point of discontinuity, but no asymptote. Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

BLM 9 7 3 or y 6 3 3 or y 4 y 4 or y 4 c) y 4 4 6 5 or y 5 d) y 3 5 8 5 8. a) y 9. Eample: y ( 5) ( )( 5) BLM 9 4 Section 9.3 Etra Practice 3. a) 5 and 6 c) 0.5 and 5 c) 4 d) 4. a) 0 and 4 7 and c) 0 and 3 d) 3 and 3. a) 5 and c) The of the function is 0 when the of is or 5. The -intercepts of the graph of the function are the same as the roots of the corresponding equation. 4. a) 0 and 3.5 5. a) 0 8 and 6 Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

BLM 9 7 6 y BLM 9 6 Chapter 9 Test. D. B 3. C 4. D 5. C 6. A 7. C 8. B 9. a) 3 6. a) 0.76 and 5.4.79 and.79 c) 0.53 and 4.87 7. a) 0.63 domain: {, R}; range: {yy 0,, y R}; 4 vertical asymptote: ; horizontal asymptote: y 0 c) As becomes very large, y approaches 0. 0. a) -intercept 3; y-intercept 0.6 c) 3 d) The root of the equation is the same as the -intercept.. a) B A c) C. a) 0.85 and 6.5 8. The solution n 3 is a non-permissible, so there is no solution. 9. Carmen: 36 h; James: 45 h Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4

BLM 9 7 3. a) g() = f ( ) + 3 g ( ) 3 c) a vertical translation 3 units up and a horizontal translation unit right 4. a) Eample: 3 5 Method : Graph y ( ), and 4 determine the -intercepts of the graph. 3 5 Method : Graph y and y ( ), 4 and determine the -coordinates of the points where the two graphs intersect. Characteristic Nonpermissible Behaviour near nonpermissible End behaviour Domain Range Equation of vertical asymptote Equation of horizontal asymptote f( ) 0 As approaches 0, y becomes very large. As becomes very large, y approaches 0. { 0, R} {y y 0, y R} 0 y 0 y 3 3 g ( ) As approaches, y becomes very large. As becomes very large, y approaches 3. {, R} {y y 3, y R} and 3 6000 5. a) l w 6000 6000 change in l w w 6000 w w c) w 4 cm 6. a) domain: {vv > 5, v R}; range: {tt > 0, t R} c) v 5 km/h Copyright 0, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073887-4