Chapter 8

Rational Functions and

Relations            

Section  8-­1  Multiplying  and  Dividing  Rational  Expressions  

 • Like  rational  numbers,  rational  expressions  are  closed  under  addition,  subtraction,  

multiplication  and  division  by  a  non-­‐zero  number.    

• A  rational  expression  is  undefined  by  any  value  that  makes  the  denominator  equal  to  zero.  

 • Just  like  rational  numbers,  to  simplify  a  rational  expression  you  divide  the  numerator  

and  denominator  by  their  greatest  common  factor.  (GCF)                      Let’s  practice…….    1.    Simplify  and  state  the  values  at  which  the  expression  is  undefined:  

       

�

3y y2 +10x + 21( )y + 7( ) y2 − 9( )  

 

2.    Multiply  the  expression.    

�

8x21y3

⋅7y2

16x3  

2b.    Multiply  the  expression.    

�

8x21y3

⋅7y2

16x3  

3.    Divide  the  expression  

�

10xy2

21x2y3÷5y2

14x3  

3b.  Divide  the  expression  

�

10xy2

21x2y3÷5y2

14x3  

                   

 When  a  rational  expression  has  more  than  one  term  in  the  numerator  or  denominator  you  may  need  to  factor  before  you  can  simplify.    Let’s  practice…….    4.    Simplify  the  expression,  where  is  this  expression  undefined.    

�

k − 3k +1

⋅1− k2

k2 − 4k + 3  

                             

5.  Simplify  the  expression,  where  is  this  expression  undefined.    

�

2d + 6d2 + d − 2

÷d + 3

d2 + 3d + 2  

                   

 

Complex  Fraction-­          Let’s  practice…….    6.    Simplify  the  complex  fraction.    

           

�

2a2b2

c48ab4

c3

 

                             

7.  Simplify  the  complex  fraction.    

           

�

a2 − b2

3c2 + 6ca + b3c

 

                   

                                 

Section  8-­2  Adding  and  Subtracting  Rational  Expressions  

 Least  Common  Multiple  (LCM)          Let’s  practice:  Find  the  least  common  multiple  of  each  of  the  following  sets.  1.      8,  12,  and  18    

2.      6,  12,  and  15  

3.    12a2b,  15abc,  and  8b3c2   4.    

�

x3 − x2 − 2x    and  

�

x2 − 4x + 4  

 

5.  Simplify:            

�

5a2

6b+

914a2b2

  6.  Simplify    

�

x +103x −15

−3x +156x − 30

 

 

7.    Simplify:  

�

x −1x2 − x − 6

+4

5x +10  

   

 Simplifying  a  complex  fraction  can  be  done  with  different  LCD’s  or  the  same  LCD…    Let’s  Practice….  

7a.  Simplify:      

�

1+2x

3y−4x

  7b.  Simplify:      

�

1+2x

3y−4x

 

   

Section  8-­3  Graphing  Reciprocal  Functions  

 Reciprocal  Function  -­  has  the  equation  of  form      A  reciprocal  function  is  not  defined  for  any  value________________________________________________    The  vertical  asymptotes  show  where  a  function  is  undefined,  while  the  horizontal  asymptotes  show  the  end  behavior  of  a  graph.    Let’s  practice…..  Identify  the  asymptotes,  domain,  and  range  of  each  function.  1.  Identify  where  x  is  not  defined      the  vertical    asymptote  is  ________      the  horizontal  asymptote  is________    The  domain  is  all  real  number  except  ___________      The  range  is  all  real  numbers  except____________        

2.  Identify  where  x  is  not  defined      the  vertical    asymptote  is  ________      the  horizontal  asymptote  is________    The  domain  is  all  real  number  except  ___________      The  range  is  all  real  numbers  except____________    

                                                                                                                                                                                     

   

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

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Chapter 8 20 Glencoe Algebra 2

Skills PracticeGraphing Reciprocal Functions

Identify the asymptotes, domain, and range of each function.

1. 2.

Graph each function. State the domain and range.

3. f(x) = 1 ! x + 3

- 3 4. f(x) = -1 ! x + 5

- 6

f (x)

x

!2

!4

!6

!2!4!6

2

2

f (x)

x

!2

!4

!6

!2!4!6

2

2

5. f(x) = -1 ! x + 1

+ 3 6. f(x) = 1 ! x + 4

- 2

x

f (x)

!2

!2!4!6

2

4

6

2

x

f (x)

x

!2

!4

!6

!2!4!6

2

2

x

f (x)

!2

!4

!2!4

4

2

2 4

f (x) = 1x - 1

y

x!2

2

4

6

2

-1f (x) = x + 4

asymptotes: x = 1, y = 0D = {x | x " 1}R = {f(x) | f(x) " 0}

asymptotes: x = 0, y = 4D = {x | x " 0}R = {f(x) | f(x) " 4}

D = {x | x " -3}R = {f(x) | f(x) " -3}

D = {x | x " -5}R = {f(x) | f(x) " -6}

D = {x | x " -1}R = {f(x) | f(x) " 3}

D = {x | x " -4}R = {f(x) | f(x) " -2}

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Chapter 8 21 Glencoe Algebra 2

Identify the asymptotes, domain, and range of each function.

1. f(x) = 1 ! x - 1

- 3 2. f(x) = 1 ! x + 1

+3 3. f(x) = -3 ! x - 2

+ 5

x-2

-4

-6

-2

2 4

f (x)

x-2-4

2

4

6

2

f (x)

x-2

2

4

6

2 4 6

f (x)

Graph each function. State the domain and range.

4. f(x) = 1 ! x + 1

- 5 5. f(x) = -1 ! x - 3

- 4 6. f(x) = 3 ! x - 2

+ 4

x-2-4

-4

-6

-2

2

f (x)

x-2

-4

-6

-2

2 4

f (x)

x2

2

4

6

4 6-2

f (x)

7. RACE Kate enters a 120-mile bicycle race. Her basic rate is 10 miles per hour, but Kate will average x miles per hour faster than that. Write and graph an equation relating x (Kate’s speed beyond 10 miles per hour) to the time it would take to complete the race. If she wanted to finish the race in 4 hours instead of 5 hours, how much faster should she travel?

PracticeGraphing Reciprocal Functions

x

t

46

2

8101214

642 10 148 12

x = 1; f(x) = -3 x = -1; f(x) = 3 x = 2; f(x) = 5D = {x | x " 1} D = {x | x " -1} D = {x | x " 2}R = {f(x) | f(x) " -3} R = {f(x) | f(x) " 3} R = {f(x) | f(x) " 5}

D = {x | x " -1} D = {x | x " 3} D = {x | x " 2}R = {f(x) | f(x) " -5} R = {f(x) | f(x) " -4} R = {f(x) | f(x) " 4}

t = 120 # x + 10

;

6 miles per hour

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Chapter 8 22 Glencoe Algebra 2

Word Problem PracticeGraphing Reciprocal Functions

1. VACATION The Porter family takes a trip and rents a car. The rental costs $125 plus $0.30 per mile.

a. Write the equation that relates the cost per mile to the number of miles traveled.

b. Explain any limitations to the range or domain in this situation.

2. PLANES A plane is scheduled to leave Dallas for an 800-mile flight to Chicago’s O’Hare airport at time t = 0. The departure is delayed for two hours. Write two equations that represent the planes’ speed, r, on the vertical axis as a function of travel time, t, on the horizontal axis. Graph the equations below. How do the two curves relate?

3. BIOLOGY A rabbit population follows the function P(t) = 40!

t + 2+ 10, with P(t)

equal to the rabbit population after t months. Eventually, what happens to the rabbit population?

4. COMPUTERS To make computers, a company must pay $5000 for rent and overhead and $435 per computer for parts.

a. Write the equation relating average cost to make a computer to how many computers are being made.

b. Graph the function you found in part a.

c. What is the minimum number of computers the company needs to make so that the average cost is less than $685?t

r

200300

100

400500600700

321 5 74 6

n

C

200300

100

400500600700

302010 50 7040 60

The rabbit population stabilizes at 10 rabbits.

They cannot travel zero miles or a negative number of miles.

C = 125 ! m + 0.3

The second graph is the ! rst graph translated 2 units to the right.

r = 800 ! t - 2

r = 800 ! t ;

C = 5000 ! n + 435

20

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Chapter 8 23 Glencoe Algebra 2

EnrichmentQueue Lengths

An engineer is planning a portion of the interstate highway system, and has to decide how many toll gates to construct at a given place on the road. It is expected that, on the average, one car will pass the given place every 6 seconds. It is also assumed that the average time to collect a toll is 5 seconds.

Let U = average service time

!!! average time between customer arrivals

.

Thus, in this case, U = 5 ! 6 .

A line of waiting customers is called a “queue.” A queue length of one means one person is being served; a queue length of two means one person is being served and one person is waiting to be served; and so on.

According to queue theory, the average queue length (assuming random arrival and service time) is U !

(1 - U) .

1. What is the average queue length if only one toll gate is installed in the situation described above? 5

2. If two toll gates are installed, the average time between customer arrivals at each gate should double. Find the new value for U and the new average queue length. U = 5 !

12 ; queue length: 5 !

7

3. If three toll gates are installed, find the values for U and the average queue length. U = 5 !

18 ; queue length: 5 !

13

4. If n toll gates are installed, write expressions for U and the average queue length. U = 5 !

6n ; queue

length: 5 ! 6n - 5

5. In a different part of the expressway, the average time between customer arrivals is predicted to be 2 seconds while the average service time remains 5 seconds. If 4 toll gates were built, what would be the average queue length? 5 !

3

6. What will happen on the expressway if the value of U is greater than or equal to one? The queue length would increase without bound.

7. In a third part of the expressway, it is expected that one car will pass the proposed toll gate site each second while the average service time remains 5 seconds. Find the number of toll gates needed for an average queue length of less than 2. 8 or more gates

8. What other things can be done to reduce queue length besides adding toll gates? Can you think of other applications for queue theory? Decrease service time with automatic tellers. Grocery store checkout lines, ticket lines for concerts or sports events

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Section  8-­5  Variation  Functions  

Read  pages  562  –  565.    Study  the  examples.    Direct  Variation:      Constant  of  variation:      Example:    If  y  varies  as  x  and  y  =  15  when  x  =  5,  find  y  when  x  =  7.          What  is  the  constant  of  variation?    Joint  variation:      Example:    It  is  stated  that  y  varies  jointly  as  x  and  z.    If  y  =  12  when  z  =  8  and  x  =  3  

using  this  same  joint  variation  find  y  when  x  =  10  and  z  =  5.        

What  is  the  constant  of  variation?    Inverse  variation:          Example:    If  r  varies  inversely  as  t  and  r  =  6  when  t  =  2,  find  r  when  t  =7.    

   Example:  The  intensity,  I,  of  light  (in  ft-­‐candles)  received  from  a  source  varies  inversely  as  the  square  of  the  distance  d  (in  ft),  from  the  source.    If  the  light  intensity  is  2  ft-­‐candles  at  13  feet,    find  the  light  intensity  at  19  feet,  to  the  nearest  hundredth.  

 Combined  variations  –      Example:    Suppose  f  varies  directly  as  g,  and  f  varies  inversely  as  h.    Find  g  when  f  =  6  and  h  =  5,  if  g  =  18  when  h  =  3  and  f  =  5.  

Section  8-­6  Solving  Rational  Equations  and  Inequalities  

 Rational  Equation:    

1. Find  the  least  common  denominator  (LCD)  2. Multiply  each  term  by  the  common  denominator  or  make  common  denominators.  3. Simplify  the  fractions.  4. Solve  the  equation.  5. Check  your  answer.  

 Let’s  practice….    

1.    

�

5y − 2

+ 2 =176   2.    

�

524

+23− x

=14  

3.  

�

7n3n + 3

−5

4n + 4=12  

           

4.  

�

x − 2x + 2

+1

x − 2>x − 4x − 2

 

1. You  have  16  ounces  of  a  10%  brine  (salt  and  water)  solution.    How  much  of  an  80%  salt  solution  will  you  need  to  have  to  make  your  brine  solution  30%  brine?  

                       

2. Jason’s  takes  5  hours  to  travel  24  kilometers  downstream  and  the  same  distance  back  in  his  boat.    If  the  current  in  the  river  is  2  kilometers  per  hour,  what  is  the  speed  of  Jason’s  boat  in  still  water?  

                       3. Willy  and  Lou  mow  lawns  together.    Willy  can  mow  all  the  lawns  by  himself  and  

complete  the  job  in  20  hours.    If  Lou  does  the  lawns  himself  he  can  complete  the  job  alone  in  15  hours.    How  long  will  it  take  them  to  complete  the  job  when  they  work  together?