A Rational Existence Introduction to Rational Functions

Transcription

1 Lesson. Skills Practice Name Date A Rational Eistence Introduction to Rational Functions Vocabular Write the term that best completes each sentence.. A is an function that can be written as the ratio of two polnomials.. A is a vertical line that a function gets closer and closer to, but never intersects. Problem Set Determine whether each function is a rational function or not a rational function. If the function is not rational, eplain wh.. f() 5 6 The function f() is a rational function.. g() q() r() 5 ( 3 ) 5. h() t() 5 5. s() p() 5 ( )( ) Chapter Skills Practice 55

2 Lesson. Skills Practice page Describe the vertical and horizontal asmptotes for each graph, provided the eist. Each figure represents the graph of a rational function. 9.. The vertical asmptote is the -ais or 5. The horizontal asmptote is the -ais or Chapter Skills Practice

3 Lesson. Skills Practice page 3 Name Date Describe the domain and range of each rational function of the form f() 5 a n. Note that a is a non-zero real number and n is an integer greater than or equal to. 5. f() 5 The domain of f() is the set of real numbers ecluding. 4 The range of f() is the set of real numbers greater than. 6. f() 5 3. f() 5 3 Chapter Skills Practice 5

4 Lesson. Skills Practice page Chapter Skills Practice

5 Lesson. Skills Practice page 5 Name Date Describe the end behavior of each rational function of the form f() 5 a n. Note that a is a non-zero real number and n is an integer greater than or equal to.. f() 5 As approaches negative infinit, approaches. 4 As approaches positive infinit, approaches.. f() f() Chapter Skills Practice 59

6 Lesson. Skills Practice page Describe the behavior of each rational function as approaches zero from the left and as approaches zero from the right. Each rational function is in the form f() 5 a n. Note that a is a non-zero real number and n is an integer greater than or equal to.. f() 5 As approaches zero from the left, the values approach positive infinit. 4 As approaches zero from the right, the values approach positive infinit. 8. f() f() Chapter Skills Practice

7 Lesson. Skills Practice page Name Date Chapter Skills Practice 5

8 Lesson. Skills Practice page 8 3. Analze each ke characteristic of a rational function of the form f() 5 n where n is an integer greater than or equal to. Identif whether the given characteristic is modeled b an odd power of n, an even power of n, or both. 33. Range is all real numbers ecluding. This characteristic is modeled b an odd power of n. 34. Domain is all real numbers ecluding. 35. Horizontal asmptote is at Graph onl eists in the first and second quadrants. 3. Graph could be in an of the quadrants. 38. Range is all real numbers greater than. 5 Chapter Skills Practice

9 Lesson. Skills Practice Name Date A Rational Shift in Behavior Translating Rational Functions Problem Set Complete the table. Use our graphing calculator to help. c-value g() 5 c Vertical Asmptote(s) Horizontal Asmptote(s) Domain Range. 3 g() Real Numbers ecept 3 Real Numbers ecept Chapter Skills Practice 53

10 Lesson. Skills Practice page Determine the domain, range, and vertical and horizontal asmptotes of each rational function without using a graphing calculator.. f () 5 3 Domain: All real numbers ecept Range: All real numbers ecept Vertical asmptote at 5 Horizontal asmptote at 5 8. f () f () 5. f () 5 9. f() 5 3. f () 5 Write a rational function for each table, graph, or description provided. Eplain our reasoning. 3. Vertical asmptote at 5 and a horizontal asmptote at 5 Answers will var. f() 5 The denominator cannot be, so cannot equal. There will be a vertical asmptote at 5. The function has a constant in the numerator and variable in the denominator, so the output will approach as increases or decreases, creating a horizontal asmptote at Chapter Skills Practice

11 Lesson. Skills Practice page 3 Name Date 4. Vertical asmptote at 5 5 and a horizontal asmptote at 5 5. f() undefined The domain is all real numbers ecept 5 6. The range is all real numbers ecept 5. Chapter Skills Practice 55

12 Lesson. Skills Practice page 4. Vertical asmptote is at 5 3. The range is all real numbers ecept Sketch each rational function without using a graphing calculator. 9. f() 5. f() 5 56 Chapter Skills Practice

13 Lesson. Skills Practice page 5 Name Date. f() 5 4. f() f() 5 ( 4) 4. f() 5 ( 4) Chapter Skills Practice 5

14 Lesson. Skills Practice page 6 Analze each rational function. Use algebra to determine the vertical asmptotes. Do not use a graphing calculator. 5. f() A vertical asmptote eists at f() 5 ( )( 3). f() f() 5 9. f() 5 3. f() 5 3 ( 3)( ) 58 Chapter Skills Practice

15 Lesson. Skills Practice page Name Date Determine two different rational functions with the given characteristics. 3. The rational functions have a vertical asmptote at 5. Answers will var. f() 5 5 or g() 5 ( ) 3. The rational functions have a vertical asmptote at The rational functions have vertical asmptotes at 5 4 and The rational functions have vertical asmptotes at 5 and The rational functions have a vertical asmptote at 5 3 and a -intercept of (, ). 36. The rational functions have a vertical asmptote 5. Also the each have a second vertical asmptote but not the same one. Chapter Skills Practice 59

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17 Lesson.3 Skills Practice Name Date A Rational Approach Eploring Rational Functions Graphicall Problem Set Sketch each function without using a graphing calculator. Indicate the domain, range, vertical and horizontal asmptote(s), and -intercept.. f() 5 3 Domain: All real numbers ecept 3 Range: All real numbers ecept Asmptote(s): Vertical asmptote at 5 3 Horizontal asmptote at 5 -intercept: (, 3 ). f() 5 ( )( 4) Domain: Range: Asmptote(s): -intercept: Chapter Skills Practice 5

18 Lesson.3 Skills Practice page 3. f() 5 3 Domain: Range: Asmptote(s): -intercept: 4. f() 5 6 Domain: Range: Asmptote(s): -intercept: 5 Chapter Skills Practice

19 Lesson.3 Skills Practice page 3 Name Date 5. f() 5 Domain: Range: Asmptote(s): -intercept: 6. f() Domain: Range: Asmptote(s): -intercept: Chapter Skills Practice 53

20 Lesson.3 Skills Practice page 4 The function f() 5 is shown on each coordinate plane. Determine whether the other function shown is the graph of g(), p(), or q(). Eplain our reasoning.. g() 5 3 p() 5 3 q() 5 3 f() 5 Function: g() 5 3 Eplanation: The original function f() 5 has been shifted 3 units to the right. This results from a change in the C value. 8. g() 5 3 p() 5 3 q() 5 3 Function: Eplanation: f() 5 54 Chapter Skills Practice

21 Lesson.3 Skills Practice page 5 Name Date 9. g() 5 4 p() 5 4 q() 5 4 f() 5 Function: Eplanation:. g() 5 4 p() 5 4 q() 5 4 f() 5 Function: Eplanation: Chapter Skills Practice 55

22 Lesson.3 Skills Practice page 6. g() 5 3 p() 5 3 q() 5 3 Function: Eplanation: f() 5. g() 5 3 p() 5 3 q() 5 3 Function: Eplanation: f() 5 56 Chapter Skills Practice

23 Lesson.3 Skills Practice page Name Date Sketch g() on each coordinate plane, given f() g() 5 f ( ) 4. g() 5 f () 4 g() 5. g() 5 f ( 3) 6. g() 5 f () Chapter Skills Practice 5

24 Lesson.3 Skills Practice page 8. g() 5 f ( ) 8. g() 5 f ( ) Write a rational function g() that matches the given characteristic(s). 9. Vertical asmptote at 5 5 Answers will var. g() 5 5. Vertical asmptotes at 5 and 5. Vertical asmptote at 5 4 Horizontal asmptote at 5 3. Vertical asmptotes at 5 3 and 5 5 Horizontal asmptote at 5 3. For f() 5, g() 5 f( ). 4. For f() 5, g() shifts f() left unit and down units. 58 Chapter Skills Practice

25 Lesson.4 Skills Practice Name Date There s a Hole In M Function, Dear Liza Graphical Discontinuities Vocabular Write a definition for the term in our own words.. removable discontinuit Problem Set Determine which function, f() or g(), has a removable discontinuit without using our graphing calculator. Identif the removable discontinuit.. f() 5 g() 5 4 ( 3)( 4) The function g() has a removable discontinuit at f() 5 ( 6) ( ) g() 5 3. f() 5 ( ) g() 5 ( 3)( ) 4. f() 5 3 g() 5 5. f() 5 3 ( 4)( ) g() 5 ( )( )( ) 6. f() 5 ( ) ( 3) g() 5 ( ) Chapter Skills Practice 59

26 Lesson.4 Skills Practice page Simplif each rational epression. List an restrictions on the domain ( 3) ; fi Chapter Skills Practice

27 Lesson.4 Skills Practice page 3 Name Date Determine whether the graph of each rational function has a vertical asmptote, a removable discontinuit, both, or neither. List the discontinuities, if an eist. 5. f() 5 ( 3) 3 The function f() has a removable discontinuit at f() 5. f() f() 5 4 ( )( 4) 9. f() f() Chapter Skills Practice 53

28 Lesson.4 Skills Practice page 4. f() 5 3. f() 5 4 Write an eample of a rational function that models each of the given characteristics. 3. A vertical asmptote at 5 Answers will var. f() 5 4. A removable discontinuit at A vertical asmptote at 5 A removable discontinuit at 5 6. A vertical asmptote at 5 3 and 5 5 A removable discontinuit at 5. A vertical asmptote at 5 3 A removable discontinuit at 5 and No vertical asmptote A removable discontinuit at Chapter Skills Practice

29 Lesson.4 Skills Practice page 5 Name Date Sketch each rational function without using a graphing calculator. Identif an restrictions. 9. f() 5 (, ) f() 5 ( ) 5 5 ; fi 3. f() 5 4 Chapter Skills Practice 533

30 Lesson.4 Skills Practice page 6 3. f() f() Chapter Skills Practice

31 Lesson.4 Skills Practice page Name Date 33. f() f() Chapter Skills Practice 535

32 Lesson.4 Skills Practice page f() f() Chapter Skills Practice

33 Lesson.5 Skills Practice Name Date The Breaking Point Using Rational Functions to Solve Problems Problem Set Solve each problem. Eplain our reasoning.. Courtne plas softball. Her goal for the season is to have an overall batting average of.3 or better. Currentl she has 45 base hits in 5 at bats. How man consecutive base hits must she get to reach her goal? Courtne must get at least 8 consecutive hits to reach her goal. The ratio of her hits to her at bats is 4. Additional hits increases the numerator as well as the denominator, represented b the 5 ratio 4. The result of graphing the functions 5 4 and 5.3 and finding their 5 5 point of intersection provides the solution.. Tito is miing green and red paint. Currentl his miture is 3 parts green to 5 parts red. What is the least amount of red paint Tito needs to add so that the miture is in the ratio part green to 6 parts red? 3. Talk-Tell is a cellular service provider. The advertise that ou can bu a monthl plan for as low as $5 per month as long as ou bu a cell phone costing $. If ou bu the monthl plan along with the phone, how man months will it take for our average cost of owning the phone and the plan to be less than $9? Chapter Skills Practice 53

34 Lesson.5 Skills Practice page 4. Mr. Motle is tweaking the final eam he intends to give to his Algebra students. The current test has multiple choice and free response questions. He would like that ratio of multiple choice to free response questions to be :3. How man free response questions does Mr. Motle need to add to his test to achieve the desired ratio? 5. Ms. Greener owns a lawn service compan. She placed the following fler in the mailboes of everone living in the town of Stork. Greener Lawn Service One-time fee $5 Monthl application fee $5 If ou purchase Ms. Greener s service, how man months will it take for our average monthl cost for the service to be less than $5? 6. Conro is a budding entomologist, that means that he likes to stud insects. In fact, Conro has an insect collection that currentl contains 3 insects that fl and 45 insects that crawl. He would like his collection to contain enough insects so that the ratio of the number of insects that fl to the number of insects that crawl is :. How man insects that crawl should Conro add to his collection to achieve the desired ratio? 538 Chapter Skills Practice

35 Lesson.5 Skills Practice page 3 Name Date. Upon entering Sl s Long Term Rentals one is greeted b the following two signs. Big Deal 6 HD Television $3 Deposit $8 Monthl Rental Fee Not So Big Deal 6 HD Television $ Deposit $ Monthl Rental Fee Determine the number of months for which the average monthl cost for the Not So Big Deal is better than the average monthl cost for the Big Deal. 8. Milton and Tor both work at Widget Kingdom, a compan that produces widgets. Milton is paid $ a da plus $. for each widget he produces; while Tor is paid $96 a da plus $.5 for each widget she produces. If Milton and Tor consistentl produce the same number of widgets, at what point will the average cost of a widget produced b Tor be greater than the average cost of a widget produced b Milton? Chapter Skills Practice 539

36 Lesson.5 Skills Practice page 4 9. Cod and Melissa pla on the same basketball team. Up to this point in the season Cod has made 9 out of 3 free throw attempts for an average of about 63.33%; while Melissa has made out of 5 free throw attempts for an average of 4%. Suppose that during the balance of the basketball season Melissa shoots twice as man free throws as Cod and both Melissa and Cod make all of them. If at the end of the season the both end up with the same free throw average, what is the least number of free throws made b Cod during the balance of the season?. Condoleezza works from home 6 hours per da 5 das a week. The compan she works for pas her $ per hour plus $5 for ever prospective customer she contacts who signs up for the service provided b the compan. The compan will continue to emplo Condoleezza provided she can maintain an average weekl cost per new customer that is less than or equal to $. How man new customers per week must Condoleezza sign up for the compan s service to be able to keep her job? 54 Chapter Skills Practice

37 Lesson.5 Skills Practice page 5 Name Date Sketch a graph to solve each equation. Do not use a graphing calculator (, ) Chapter Skills Practice 54

38 Lesson.5 Skills Practice page Chapter Skills Practice