alg2 resources ch 07 toc - MATHEMATICSmathwithjp.weebly.com/uploads/2/0/8/8/20882022/hscc_alg2_rbc_07.pdfdos planes de telefonía celular para un teléfono desbloqueado para los primeros

Copyright © Big Ideas Learning, LLC Algebra 2 All rights reserved. Resources by Chapter

223

Chapter 7 Family and Community Involvement (English) ......................................... 224

Family and Community Involvement (Spanish) ......................................... 225

Section 7.1 ................................................................................................... 226

Section 7.2 ................................................................................................... 231

Section 7.3 ................................................................................................... 236

Section 7.4 ................................................................................................... 241

Section 7.5 ................................................................................................... 246

Cumulative Review ..................................................................................... 251

Algebra 2 Copyright © Big Ideas Learning, LLC Resources by Chapter All rights reserved. 224

Chapter

7 Rational Functions

Name _________________________________________________________ Date _________

Dear Family,

Do you have a cell phone? If so, are you getting the most from your plan? Are you paying too much for your plan?

When choosing a service contract, you should compare different providers and their coverage areas and the different prices for talk, text, and data plans. You want to make sure your plan has what you need, so you don’t end up paying fees for going over what your plan allows. The bigger cell phone companies offer a subsidized plan with a two-year contract in which you pay a smaller amount for a phone and then a monthly charge for service. Another option is an unlocked phone in which you buy the phone outright with a monthly fee for services and no fixed contract.

Use the Internet to compare two different plans for a two-year subsidized contract for the first 24 months of service. Then compare two cell phone plans for an unlocked phone for the first 24 months of service. Remember to include the initial cost of the phone in the total cost for the first month. Then find the average monthly cost for each plan. For a fair comparison of plans, choose the same cell phone for each plan. It may be helpful to use a spreadsheet to organize and calculate your data.

In this chapter, you will solve rational equations for situations in which you want to find the average monthly cost of service. Before you sign a contract with a cell phone provider, researching different plans may end up saving you time and money in the long run.

Have fun talking!

Company A Company B

Month Total cost (in dollars)

Cost per month(in dollars)

Total cost (in dollars)

Cost per month (in dollars)

1

2

24


225

Capítulo

7 Funciones racionales

Nombre _______________________________________________________ Fecha _________

Estimada familia:

¿Tienen teléfono celular? Si lo tienen, ¿aprovechan su plan al máximo? ¿Pagan demasiado por su plan?

Cuando elijan un contrato de servicio, deberían comparar diferentes proveedores y sus áreas de cobertura y los diferentes precios de los planes de datos para hablar y mandar mensajes de texto. Les conviene asegurarse de que su plan tenga que lo necesitan, así no terminan pagando tarifas por pasarse del límite permitido por su plan. Las compañías de telefonía celular más grandes ofrecen un plan subvencionado con un contrato de dos años donde pagan una cantidad menor por un teléfono y luego, un cargo mensual por el servicio. Otra opción es un teléfono desbloqueado donde compran el teléfono en el acto con una tarifa mensual por los servicios sin contrato fijo.

Usen Internet para comparar dos planes diferentes con un contrato de dos años subvencionado para los primeros 24 meses del servicio. Luego, comparen dos planes de telefonía celular para un teléfono desbloqueado para los primeros 24 meses de servicio. Recuerden incluir el costo inicial del teléfono en el costo total del primer mes. Luego, hallen el costo mensual promedio de cada plan. Para hacer una comparación justa de los planes, elijan el mismo teléfono celular para cada plan. Quizás sea útil usar una hoja de cálculo para organizar y calcular sus datos.

En este capítulo, resolverán ecuaciones racionales en situaciones donde quieran hallar el costo mensual promedio de un servicio. Antes de firmar un contrato con un proveedor de telefonía celular, investigar diferentes planes puede terminar ahorrándoles tiempo y dinero a largo plazo.

¡Diviértanse hablando!

Compañía A Compañía B

Mes Costo total (en dólares)

Costo por mes(en dólares)

Costo total(en dólares)

Costo por mes (en dólares)

1

2

24


7.1 Start Thinking

Determine a relationship between x and y in the table. Then create a scatter plot for each set of points. What conclusions can you make about the graphs and the relationship you discovered between x and y?

1. 2. 3.

Use the given values of x and y to find the value of k in the equation.

1. 4, 3

y kxx y

== = −

2. 1 , 52

y kx

x y

=

= =

3. 3.5, 2.8

y kxx y

== − = −

4.

2, 13

kyx

x y

=

= − =

5.

5.2, 5

kyx

x y

=

= = −

6.

3 5, 5 2

kyx

x y

=

= − =

Solve the system.

1. 4 3 76 2 3

x yx y

− − = −+ =

2. 2 43

2 6

y x

y

= − +

− =

3. 2

26y xy x

== −

7.1 Warm Up

7.1 Cumulative Review Warm Up

x y 1 2 3 6 4 8 6 12

x y 1 2 2 1 4 1

2

6 13

x y 1 3 3 1 5 3

5

7 37


227

7.1 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–6, tell whether x and y show direct variation, inverse variation, or neither.

1. 5yx

= 2. 7xy = 3. 6x y=

4. 10yx

= 5. 8x y+ = 6. 2y x=


7. 8.

9. 10.

In Exercises 11–13, the variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x 3.=

11. 6, 5x y= = − 12. 1, 7x y= = 13. 233, x y= =

14. The variables x and y vary inversely. Describe and correct the error in writing an equation relating x and y.

15. The number y of songs that can be stored on an MP3 player varies inversely with the average size x of a song. A certain MP3 player can store 3000 songs when the average size of a song is 5 megabytes. Find the number of songs that will fit on the MP3 player when the average size of a song is 4 megabytes.

x 2 4 8 10

y 38 19 9.5 7.6

x 3 5 8 10

y 15 9 6 5.5

x 1.5 4 6.5 10

y 9 24 39 60

x 1.5 4 6 12

y 84 31.5 21 10.5


7.1 Practice B

Name _________________________________________________________ Date _________


1. 12yx

= 2. 15xy = 3. 9x y=

4. 3y x= − 5. 9yx

= 6. 13

xy =


7. 8.

9. 10.

In Exercises 11–13, the variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x 3.=

11. 4, 3x y= = − 12. 23, 5x y= = − 13. 1

510, x y= − = −

14. The variables x and y vary inversely. Describe and correct the error in writing an equation relating x and y.

15. The current y in a certain circuit varies inversely with the resistance x in the circuit. If the current is 8 amperes when the resistance is 20 ohms, what will the current be when the resistance increases to 25 ohms?

x 2.5 4 7.5 9

y 30 48 90 108

x 12 5 2.5 1.5

y 35 84 168 280

x 2.5 3 6 10

y 8 9.6 1.6 6

x 2.5 10 16 21

y 672 168 105 80


229

7.1 Enrichment and Extension

Name _________________________________________________________ Date __________

Inverse Variation Combined variation involves a combination of direct and inverse variation. These equations are a little more complicated, so you will first want to substitute the given values to solve for the constant of variation. Then use the constant variation to find the missing value.

Example: Suppose y varies directly with x and w but varies inversely as the square of z. Find the equation of variation if 100 when 2, 4, and 20.y x w z= = = =

Solution:

( )( )( )

2

2

2

2 4100

20

50005000

axwyz

a

axwy

z

=

=

=

=

2Given the direct variation with , constant , and and are in the numerators. Given the inverse variation with , is in the denominator.

Substitute.

Solve for .N

y a x wy z

aow you can solve for one of the variables when you are given the values

of the other three variables.

In Exercises 1–4, solve the combined variation problems.

1. Suppose x varies directly with y and the square root of z. When 18 and 2,x y= − = 9.z = Find y when 10 and 4.x z= =

2. Suppose w varies inversely with z and the cube root of v, but varies directly with y. When 4, 27, and 2, 5.w v y z= = = = Find w when 3, 64, and 6.y v z= = =

3. The volume V of wood in a tree varies directly with the height h and inversely with the square of the girth g. The volume of a tree is 144 cubic meters when the height is 20 meters and the girth is 1.5 meters. What is the height of a tree with a volume of 100 cubic meters and girth of 2 meters?

4. The pressure P of a gas varies directly with the number of moles n and temperature T of the gas and inversely with volume V. Given the equation of the situation described above, as the pressure of a gas increases, what happens to the volume of the gas? Now write the number of moles as a function of pressure, volume, and temperature.


Puzzle Time

Name _________________________________________________________ Date _________

What Part Of A Computer Keyboard Do Astronauts Like The Best? Write the letter of each answer in the box containing the exercise number.

Tell whether x and y show direct variation, inverse variation, or neither.

1. 3xy =

2. 12x y+ =

3. 6yx

=

The variables x and y vary inversely. Use the given values to write an equation relating x and y.

4. 5, 2x y= = − 5. 8, 2x y= =

6. 1, 62

x y= = 7. 212, 3

x y= − =

8. 1 9, 3 2

x y= − = − 9. 32, 4

x y= = −

10. 2.4, 8.6x y= − =

11. 215, 5

x y= − = −

7.1

Answers

P. 16yx

=

T. inverse variation

S. 10yx

= −

R. 6yx

= −

E. direct variation

A. 3yx

=

A. 20.64yx

= −

C. 8yx

= −

E. 32

yx

=

H. neither

B. 32

yx

= −

1 2 3 4 5 6 7 8 9 10 11


231

7.2 Start Thinking

Complete the table for the function ( ) 1.g xx

= Review the

values in the table to make conclusions about the nature of the graph of g as x-values approach zero and as x-values approach both positive and negative infinity.

1.

2.

Use the graph of f to sketch the transformation.

1. ( ) 2f x + 2. ( )2f x −

3. ( )f x− 4. ( )f x−

5. ( )2 f x 6. ( )3 1f x− + −

Determine the maximum or minimum value of the function.

1. 22 5y x x= − +

2. 23 7 8y x x= − +

3. 23 12 24y x x= − −

7.2 Warm Up


x 1− 12− 1

10− 1100− 1

1001

1012 1

g(x)

x −1000 −100 −10 10 100 1000

g(x)

−4

−8

4 8 x

y

−4−8

4

8f


7.2 Practice A

Name _________________________________________________________ Date _________

In Exercises 1–3, graph the function. Compare the graph with the graph of

( )f xx1 .=

1. ( ) 2h xx

= 2. ( ) 9g xx

= 3. ( ) 4h xx

−=

In Exercises 4–15, graph the function. State the domain and range.

4. ( ) 3 2f xx

= + 5. 5 1yx

= − 6. ( ) 43

g xx

=−

7. 14

yx

=+

8. ( ) 13

h xx

−=+

9. 45

yx

−=−

10. ( ) 32

xf xx

+=−

11. 53

xyx

−=+

12. ( ) 42 6xg xx

+=−

13. 5 23 9

xyx

+=−

14. ( ) 2 33 4

xh xx

− +=+

15. 8 15 1

xyx

−=−

In Exercises 16–21, rewrite the function in the form ( ) ag x kx h

.= +−

Graph the

function. Describe the graph of g as a transformation of the graph of ( ) af xx

.=

16. ( ) 4 51

xg xx

+=+

17. ( ) 6 52

xg xx

+=−

18. ( ) 3 64

xg xx

−=−

19. ( ) 5 122

xg xx

−=+

20. ( ) 155

xg xx

+=−

21. ( ) 39

xg xx

+=−

22. Your choir is taking a trip. The trip has an initial cost of $500, plus $150 for each student.

a. Estimate how many students must go on the trip for the average cost per student to fall to $175.

b. What happens to the average cost as more students go on the trip?

In Exercises 23–25, use a graphing calculator to graph the function. Then determine whether the function is even, odd, or neither.

23. ( ) 25

1f x

x=

− 24. ( )

2

23

4xg x

x=

+ 25. ( )

3

2 42xh x

x x=

+


233

7.2 Practice B

Name _________________________________________________________ Date __________

In Exercises 1–3, graph the function. Compare the graph with the graph of

( )f xx1 .=

1. ( ) 12h xx

= 2. ( ) 8g xx

−= 3. ( ) 0.2h xx

=

In Exercises 4–15, graph the function. State the domain and range.

4. ( ) 5 2f xx

= − 5. ( ) 34

g xx

=+

6. 83

yx

−=−

7. ( ) 15

h xx

−=+

8. 2 31

yx−= ++

9. 9 24

yx

= −−

10. ( ) 54

xf xx

+=−

11. ( ) 32 8xg xx

−=+

12. ( ) 8 35 2

xh xx

− +=+

13. 3 15 1

xyx

−=−

14. 34 1

xyx

−=− −

15. 2 58

xyx

− +=− +

In Exercises 16–21, rewrite the function in the form ( ) ag x kx h

.= +−

Graph the

function. Describe the graph of g as a transformation of the graph of ( ) af xx

.=

16. ( ) 3 72

xg xx

+=+

17. ( ) 4 23

xg xx

−=−

18. ( ) 4 105

xg xx

−=+

19. ( ) 123

xg xx

+=−

20. ( ) 5 304

xg xx

−=+

21. ( ) 7 26

xg xx

−=+

22. You are creating statues made of cement. The mold costs $300. The material for each statue costs $22.

a. Estimate how many statues must be made for the average cost per statue to fall below $30.

b. What happens to the average cost as more statues are created?

23. The concentration c of a certain drug in a patient’s bloodstream t hours after an

injection is given by ( ) 2 .4 1

tc tt

=+

a. Use a graphing calculator to graph the function. Describe a reasonable domain and range.

b. Determine the time at which the concentration is the highest.



Name _________________________________________________________ Date _________

Graphing Rational Functions It is important to take a look at the end behaviors of a function. You will investigate the values of y as the x-values approach a certain value. You will begin by looking at what value y approaches as x approaches and .+∞ −∞

Example: Use a graphing calculator to graph the function ( ) 3 4.2 1xf xx

−=+

Determine

what value y approaches as x approaches and .+∞ −∞

Solution: As the x-values get larger and larger ( ),+∞ the y-values

approach 3.2

As the x-values get smaller and smaller ( )−∞ , the

y-values also approach 3.2

You can write:

As 3, .2

x y→ +∞ → As 3, .2

x y→ −∞ →

In Exercises 1–8 use a graphing calculator to graph the function. Determine the value y approaches as x approaches +∞ and −∞.

1. ( )2

22 8

9x xh x

x− −=

− 2. ( )

3

37 3

2 7 4xq x

x x−=

− +

3. ( ) 2 2xg x

x x=

− − 4. ( )

4

23 5 4

6 2x xk x

x x+ −=

−

5. ( )22 4

7 3xf xx

− +=−

6. ( )3

42 4

6x xj xx

−=+

7. ( ) 4 18

xd xx

− +=−

8. ( )2 9

2xk x

x−=

−

15

−10

−15

10


235

Puzzle Time

Name _________________________________________________________ Date __________

What Has A Head And A Tail But No Body? Write the letter of each answer in the box containing the exercise number.

Match the graph with its function .

1. 2.

3. 4.

5.

Answers

I. 2 4

4y

x= +

−

A. 2 4yx

= +

O. 2

4y

x=

+

N. 4 4

2y

x= −

+

C. 2 4yx

= − +

7.2

−6

4 8 x

y

−4−8

4

−6

4 8 x

y

−4−8

4

−4

−8

−4 4 8 x

y

4

8

−4

−8

−4−8 6 x

y

6

1 2 3 4 5

−4

−8

−4−8 4 8 x

y

4


7.3 Start Thinking

In algebraic expressions, values that make a denominator zero are called excluded values. Determine the excluded values in the expression.

1. 18x +

2. 21

5x x− 3. 2

16 15x x− −

Perform the indicated operation. Simplify your answer and indicate any excluded values of x.

1. 2 57 3

• 2. 9 810 5

•

3. 4 211 3

÷ 4. 1 25 5

x − •

5. 1 1

x xx x

•− +

6. 32 1 1

xx x− ÷

+ +

You deposit $7000 in an account that pays annual interest. Find the balance in the account after 3 years if the interest is compounded as described below at the given rate.

1. quarterly; 1.26% 2. monthly; 3.25%

3. daily; 2.8% 4. continuously; 2.3%

5. annually; 4.125% 6. continuously; 1.86%

7.3 Warm Up



237

7.3 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–6, simplify the expression, if possible.

1. 2

23

5 2x

x x+ 2.

4 3

46

2x x

x− 3.

2

24 57 10

x xx x

− −− +

4. 2

23

5 6x x

x x−

+ + 5.

2

32

8x x

x− −

− 6.

2

33 4

1x x

x− −

+

In Exercises 7–12, find the product.

7. 4 2 3 2

4 5 354

9x y x yy x y

• 8. ( ) ( )( )3

4

2 1 31

x x x xx x

+ − −•

−

9. ( ) ( )( )2

2

5 7 17 4

x x x xx x

− + −•

+ 10.

2 25 4 33

x x x xx x

− + +•+

11. 2 23 5 6

2 4x x x xx x

+ − +•−

12. 2 2

2 24 5 2 66 9 3 2

x x x xx x x x

− − +•+ + + +

13. Compare the function ( ) ( )( )( )

4 1 54 1

x xf x

x+ −

=+

to the function ( ) 5.g x x= −

In Exercises 14–17, find the quotient.

14. 4 9

7 528

2x y yy x

÷ 15. 2

4 3 36 3

3 6 6x x x

x x x− − −÷

+

16. 2

24 12 4

2 3 5 5x x x

x x x+ ÷

+ − − 17. ( )

225 14 4 4

3x x x x

x+ − ÷ − +

+

18. Manufacturers often package products in a way that uses the least amount of material. One measure of the efficiency of a package is the ratio of its surface area to its volume. The smaller the ratio, the more efficient the packaging. A company makes a cylindrical can to hold popcorn. The company is designing a new can with the same height h and twice the radius r of the old can.

a. Write an expression for the efficiency ratio ,SV

where S is the surface area

of the can and V is the volume of the can.

b. Find the efficiency ratio for each can.

c. Did the company make a good decision by creating the new can? Explain.


7.3 Practice B

Name _________________________________________________________ Date _________

In Exercises 1–6, simplify the expression, if possible.

1. 3

34

3 7x

x x+ 2.

2

25 62 3

x xx x

+ ++ −

3. 2

22 5

7 12x x

x x−

+ +

4. 2

320

64x x

x− −

+ 5.

4

3 216

5 3 20 12x

x x x−

− + − 6.

3 2

46 6 5 5

72 50x x x

x− + −

−

In Exercises 7–12, find the product.

7. ( ) ( )( )4

5

4 3 23

x x x xx x

− + −•

+ 8.

2 26 2 84

x x x xx x

+ − −•−

9. 2 22 6 5

5 3x x x x

x x− + +•+

10. 2 2

2 26 3 12

8 16 2 3x x x x

x x x x− − +•

+ + − −

11. ( )2

22

3 28 8 1525

x x x xx+ − • − +

− 12. ( )

22

22 15 12

9x x x x

x+ − • − −

−

In Exercises 13–16, find the quotient.

13. 3 2 2

22 10 2

20 4x x x

x x x+ ÷

+ − − 14. ( )

2210 21 14 49

2x x x x

x− + ÷ − +

+

15. 2 2

2 22 3 4 32 8 6 8

x x x xx x x x

− − + +÷+ − + +

16. 2 2

2 26 5 6

7 12 12x x x x

x x x x+ − − +÷

+ + + −

17. Find the ratio of the perimeter to the area of the square shown.

18. Find the expression that makes the following statement true.

2 23 5

8 12 3 10 6x x

x x x x x+ +÷ =

− + + − −

x + 3

x + 3


239


Name _________________________________________________________ Date __________

Multiplying and Dividing Rational Expressions You can find the horizontal asymptote of a rational function by investigating the coefficients of the leading terms in the numerator and denominator.

( ) ... ,...

n

maxf xbx

+=+

given nth and mth degree polynomials

1. If ,n m< then the x-axis is the horizontal asymptote.

2. If ,n m= then the horizontal asymptote is the line .ayb

=

3. If ,n m> then there is not a horizontal asymptote.

Remember that the function needs to be in the form shown above. In other words, the function must contain a polynomial in the numerator and a polynomial in the denominator. All terms must be combined. Also remember that the polynomial must be in standard form, meaning that the highest exponent should be in the first term of each polynomial. You also may need to factor the polynomials to divide out common factors. Then you will need to expand the remaining monomials or polynomials to analyze the leading term.

In Exercises 1–5, simplify the rational function. Then determine the horizontal asymptote.

1. ( )2 2

2 22 12

3 4 2 35x x x xf xx x x x

− − − −= •− − − −

2. ( )2 2

2 3 22 2 3

9 3 16 48x x x xh x

x x x x+ − + −= ÷

− − − +

3. ( )2 3

2 35 4 2

5 2 7 9x xk x

x x x− +=

+ + −

4. ( )2

2 22 7 6 2 12 3 2 3 5 2

x x xm xx x x x

− + += •− − + −

5. ( )2 23 5 2 2 7 65 4 7

x x x xt xx x+ − + += ÷

− −


Puzzle Time

Name _________________________________________________________ Date _________

What Is Black And White And Red All Over? Write the letter of each answer in the box containing the exercise number.

Simplify the expression, if possible.

1. 4510 15x −

2. 27

3 28x

x x−

− −

3. 22

3 8x

x +

Find the product or quotient.

4. 21

1 2y x

x y−•

−

5. ( )23 2 11

x x xx

• + ++

6. ( )( )

( )( )( )2

3 1 32 1

x x xx x

+ − +÷

+ −

7. 1 69 3 18

xx x

−÷+ −

8. 2 4

5 23

10 25x yz x y zxy

÷

9. 2

28 4 40

2 48 10x x x

x x x− +•

− − +

10. 2 3 21 9 36

5 5 35x

x x x−÷

−

Answers

E. 4 , 10, 86

x x xx

≠ − ≠+

S. ( )3 1 , 1x x x− ≠ −

A. 92 3x −

P. 3

4 4 , 0z zx y

≠

N. 1 , 74

xx

≠+

W. 1 , 1, 02( 1)

x yx

≠ ≠+

P. 1 , 3, 12

x x xx

− ≠ − ≠+

R. ( )

7 , 0, 79 4

x x xx

− ≠ ≠−

E. 22

3 8x

x +

A. 3 , 69

xx

− ≠+

7.3

1 2 3 4 5 6 7 8 9 10


241

7.4 Start Thinking

To add or subtract rational numbers, you must first find a common denominator. The same is true for rational expressions in algebra. Determine the least common denominator (LCD) for the three rational expressions shown below. Then determine the values of x that make the LCD zero.

2 21 4 5, ,

3 6 3 6x x x x x− + + −

Simplify the expression.

1. 3 27 3

+ 2. 7 810 5

+ 3. 3 128 4

−

4.

2 13 2

35

− 5.

5 38 4

2

− − 6.

5 72 52 43

−

− +

Given ( )f x x x22= − and ( )g x x4 3 ,= − determine the value of the expression.

1. ( )( )f g x+ 2. ( ) ( )f x g x• 3. ( )g xf

4. ( )( )f g x 5. ( )( )2f g− 6. ( )( )3g f

7.4 Warm Up



7.4 Practice A

Name _________________________________________________________ Date _________

In Exercises 1–3, find the sum or difference.

1. 12 35 5x x

+ 2. 2 23

9 9xx x

− 3. 7 32 2

xx x

−− −

In Exercises 4–7, find the least common multiple of the expressions.

4. 23 , 6 18x x − 5. ( )5 , 5 2x x x +

6. 2 9, 3x x− + 7. 2 3 10, 2x x x− − +

8. Describe and correct the error in finding the sum.


9. 27 4

2 3x x− 10. 2 4

1 2x x+

− +

11. 6 54 3

xx x

−+ −

12. 214 67 18 9x x x

++ − +

In Exercises 13 and 14, tell whether the statement is always, sometimes, or never true. Explain.

13. The LCD of two rational functions is the sum of the denominators.

14. The LCD of two rational functions is equal to one of the denominators.

In Exercises 15–18, rewrite the function g in the form ( ) ag x kx h

.= +−

Graph the function. Describe the graph of g as a transformation of the graph

of ( ) af xx

.=

15. ( ) 4 52

xg xx

−=−

16. ( ) 5 34

xg xx

+=+

17. ( ) 103

xg xx

=−

18. ( ) 3 4xg xx+=


243

7.4 Practice B

Name _________________________________________________________ Date __________


1. 2 25

25 25xx x

− 2. 22 86 6

x xx x

++ +

3. 3 124 4

xx x

−− −

In Exercises 4–7, find the least common multiple of the expressions.

4. 2 236 , 9 18x x x− 5. 2 100, 10x x− −

6. 2 225 4, 3 10 8x x x− − − 7. 2 7 18, 9x x x+ − +

8. Describe and correct the error in finding and simplifying the sum.


9. 7 45 1

xx x

+− +

10. 27 3

5 24 8x x x+

− − −

11. 2

23 5

6 16 2x x

x x x− +−

− − + 12. 2 3 6

3 2 1x xx x x

− + +− +

In Exercises 13 and 14, tell whether the statement is always, sometimes, or never true. Explain.

13. The LCD of two rational functions is one of the denominators when the other denominator is a factor.

14. The LCD of two rational functions will have a degree equal to that of the denominator with the higher degree.

In Exercises 15–18, rewrite the function g in the form ( ) ag x kx h

.= +−

Graph the function. Describe the graph of g as a transformation of the graph

of ( ) af xx

.=

15. ( ) 5 34

xg xx

+=+

16. ( ) 912

xg xx

=+

17. ( ) 5 4xg xx−= 18. ( ) 8 13

6xg xx

+=−

( )( )

( )( )

3 3

3 3 43

4 5 74 5 4 357 7 77

x x x xx x x x xx x

++ = + =



Name _________________________________________________________ Date _________

Adding and Subtracting Rational Expressions To add or subtract algebraic fractions, you have seen that you first find a common denominator.

( ) ( )( )( ) ( )( ) 2

3 2 2 13 2 3 6 2 2 5 41 2 1 2 1 2 2

x x x x xx x x x x x x x

− + + − + + −+ = = =+ − + − + − − −

The decomposition of an algebraic fraction into partial fractions is the reverse of this process. The method shown in the example below can be used for fractions in which the degree of the numerator is less than the degree of the denominator, and in which the denominator can be factored into distinct linear factors.

Example: Decompose 25 4

2x

x x−

− − into partial fractions.

Solution: 25 4 5 4Factor the denominator.

( 1)( 2)2x x

x xx x− −=

+ −− −

( )( )5 4Express the fraction as the sum of two fractions,

with the individual factors as denominators and 1 2 1 2the unknown numerators and .

To clear fractions, multiply each side of therational

x A Bx x x x

A B

− = ++ − + −

( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

5 4 2 1 equation by the LCD, 1 2 .

To determine and , first eliminate by 5 1 4 1 2 1 1letting 1. 5 4 3

3

Then eliminate by letting 2. 5 2 4 2 2 2 110 4 3

2

Use the values of

x A x B xx x

A B B A Bx A

A

A x A BB

B

− = − + ++ −

− − = − − + − += − − − = −

=

= − = − + +− =

=

( )( )5 4 3 2 and to express the original

fraction as the sum of two partial fractions. 1 2 1 2xA B

x x x x− = +

+ − + −

In Exercises 1–6, express the given fraction as the sum of partial fractions.

1. 23 18

5 4x

x x+

+ + 2.

2

36 2 4

4x x

x x+ −

− 3.

2

3 26 10 4

2 9 18x x

x x x+ −

+ − −

4. 22 12

3 28x

x x+

− − 5.

2

32 7 4

9x x

x x+ −

− 6.

2

3 27 18 19

2 5 6x x

x x x+ −

+ − −


245

Puzzle Time

Name _________________________________________________________ Date __________

What Do You Get When You Cross Cinderella With A Barber? Write the letter of each answer in the box containing the exercise number.

Find the sum or difference.

1. 10 32 2x x

− 2. 5 2 31 1x x

x x− +++ +

3. 52 4 1

xx x

−− +

4. 2 23

x xx x

− +

5. 7 28 1x x

−+ −

6. 3 84 3

xx x

+− −

7. 21 43 10 2x x x

−+ − −

Simplify the complex fraction.

8.

2

254

x

x

9.

2

4

4

x

xx +

10. 2

4 23

41 1

x xx

x x

+

−+ +

11. 2

4 45

x

x−

12.

2 19 4

6

x

x

+ 13. 2

1 52 4

52 2

x

x

+−

−

Answers

R. 24 9

216x

x+ L. 2

4 193 10

xx x

− −+ −

S. 25 23

7 8x

x x−

+ − L. 2 7

1xx

++

P. 2

3 24 10 6, 1

3 12x x x

x x+ + ≠ −

−

S. 7 23

xx− I.

3, 0

100x x ≠

G. 72x

P. 2 4 , 4, 0

4x x x x+ ≠ − ≠

C. 2

23 32

7 12x x

x x− −

− +

A. 2

29 20

2 2 4x xx x

− +− −

E. 35 , 0

4 20x x

x≠

−

S. 23

2 10x

x− −

−

7.4

1 2 3 4 5 6 7 8 9 10 11 12 13


7.5 Start Thinking

A racquetball club offers a $200 annual membership that includes unlimited free court rental. They also offer a $120 membership with a $10 fee for each court rental. Write a rational equation that represents the cost per court rental for each membership. Then use a graphing calculator to graph both rational functions and determine the point of intersection. What does this point of intersection represent in the context of the problem? Which membership is a better deal?

Solve the equation.

1. ( ) ( )3 1 4 1x x− − = + 2. ( ) 22 1x x x− = −

3. ( )34 2 32

x x + = −

4. ( ) ( )( )2 1 8 1x x x− + = − −

5. ( ) ( )3 2 8 1x x x= + 6. ( ) ( ) ( )3 4 4 3 4x x x x x x− + − = −

Perform the indicated operation. Write your answer in standard form.

1. ( ) ( )3 2 5i i+ − + 2. ( )2 11 7i i− +

3. ( )25 3i− + 4. ( )( )4 9 5 6i i− + +

5. ( )5 23 4 5 8i i i− + 6. ( ) ( )7 65 3 11 2i i− − +

7.5 Warm Up



247

7.5 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–3, solve the equation by cross multiplying. Check your solution(s).

1. 3 14 2x x

=−

2. 4 62 2x x

=+ −

3. 3 51 5

xx x

− −=+ −

4. So far in baseball practice, you have pitched 47 strikes out of 61 pitches. Solve the

equation 80 47100 61

xx

+=+

to find the number x of consecutive strikes you need to

pitch to raise your strike percentage to 80%.

In Exercises 5 and 6, identify the least common denominator of the equation.

5. 2 52

xx x x

+ =−

6. 3 8 25

xx x x

− =+

In Exercises 7–12, solve the equation by using the LCD. Check your solution(s).

7. 4 2 43 x

+ = 8. 5 1 92 4 2x x

+ =

9. 2 233

x xx x

− + =−

10. 4 1 15 5

xx x x

−+ =− −

11. 8 834

xx x

++ =−

12. 212 3 3

2 2x x x x− =

− −

13. Describe and correct the error in the first step of solving the equation.

14. You can clean the gutters of your house in 5 hours. Working together, you and your friend can clean the gutters in 3 hours. Let t be the time (in hours) your friend would take to clean the gutters when working alone. Write and solve an equation to find how long your friend would take to clean the gutters when working alone.

( ) ( ) ( )( )Work done Work rate TimeHint: = ×


7.5 Practice B

Name _________________________________________________________ Date _________

In Exercises 1–3, solve the equation by cross multiplying. Check your solution(s).

1. 3 52 2x x

=+ −

2. 2 34 1

xx x

−=− −

3. 25 5

4 4x x

x− −=

+

4. So far in soccer practice, you have made 10 out of 32 goal attempts. Solve the

equation 100.4532

xx

+=+

to find the number x of consecutive goals you need to

make to raise your goal average to 0.45.

In Exercises 5 and 6, identify the least common denominator of the equation.

5. 6 43 2 5

xx x

+ =+ +

6. 6 2 98 3 2 4

xx x

− =− −

In Exercises 7–12, solve the equation by using the LCD. Check your solution(s).

7. 3 1 74 8 4x x

+ = 8. 5 1 16 6

xx x x

−+ =− −

9. 4 455

x xx x

− + =−

10. 216 8 4

4 4x x x x− =

− −

11. 1 1 2 12 2

x xx x x

+ ++ =+ +

12. 4 412x x

− =+

13. Describe and correct the error in the first step of solving the equation.

14. You can kayak around a certain island in 3 hours. Kayaking together, you and your friend can kayak around the island in 1.4 hours. Let t be the time (in hours) your friend would take to kayak around the island when kayaking alone. Write and solve an equation to find how long your friend would take to kayak around the island when kayaking alone.

( ) ( ) ( )( )Work done Work rate TimeHint: = ×


249


Name _________________________________________________________ Date __________

Solving Rational Equations

In Exercises 1–12, you are given the function ( ) xf xx

2 9.2

+=

+

1. What are three points that lie on the graph of 1?f − Explain your reasoning.

2. Does the point 11, 23

lie on the graph of 1?f − Explain your reasoning.

3. Algebraically find ( )1 .f x−

4. Prove that they are inverse functions by algebraically showing that( )( ) ( )( )1 1 .f f x f f x x− −= =

5. Graph both f and 1f − in the same coordinate plane.

6. Write the domain and range of .f

7. Write the domain and range of 1.f − How do the domain and range of 1f − compare with the domain and range of ?f

8. What do you notice about the vertical asymptotes and horizontal asymptotes of the graphs of f and 1?f −

9. What is significant about where the graphs of f and 1f − intersect?

10. Is there a relationship between the asymptote of the graph of a function and the zero of its inverse? Explain.

11. Can you find the inverse of ( ) 7 ?2 1xg xx

−=+

If so, find ( )1 .g x−

12. Do all rational functions have an inverse? Explain.


Puzzle Time

Name _________________________________________________________ Date _________

What Is Green And Sings? Write the letter of each answer in the box containing the exercise number.

Solve the equation by cross multiplying.

1. 5 415 5

x − = 2. 6 2 23 15

x x− − −=

3. 3 151 7

xx x

+ =+ +

4. 5 41

xx

=−

5. 2 2 42 1

x xx x

+ +=− +

6. 3 25x x

=−

Solve the equation by using the LCD.

7. 5 1 13x

+ = 8. 2 1 45 3 15x x

+ =

9. 1 432

xx x

− =+

10. 23 1 10

1 1 1x x x+ =

− + −

11. 212 3 1

4 4x x x x+ =

+ +

12. 28 2 2

2 2x

x x x x−− =

+ +

Answers

L. 2x =

S. 2, 5x x= − =

A. 152x =

E. 17x =

P. 3x =

S. 27 , 1x x= = −

E. 12x = −

I. 4x = −

R. 25x = −

Y. 25x =

L. 4x =

V. 3, 2x x= =

7.5

1 2 3 4 5 6 7 8 9 10 11 12


251

Chapter

7 Cumulative Review

Name _________________________________________________________ Date __________

In Exercises 1–18, factor the polynomial.

1. 2 4 21x x− − 2. 2 8 48x x+ − 3. 2 9 8x x+ +

4. 2 9 20x x− + 5. 2 6 27x x− − 6. 2 12 32x x− +

7. 22 5 3x x+ − 8. 22 2 24x x− − 9. 23 23 36x x+ −

10. 25 23 12x x+ + 11. 23 25 42x x− + 12. 24 7 3x x− +

13. 24 30 56x x+ + 14. 26 7 5x x− − 15. 28 2 3x x− −

16. 25 2 7x x− − 17. 28 22 6x x+ − 18. 212 18 12x x− −

In Exercises 19–34, write the prime factorization of the number. If the number is prime, then write prime.

19. 12 20. 201 21. 88 22. 56

23. 75 24. 300 25. 2 26. 99

27. 188 28. 199 29. 85 30. 41

31. 17 32. 169 33. 131 34. 65

In Exercises 35–46, solve the equation. Check your solution.

35. 1 562 2

x x+ = 36. 3 1174 4

x x− = 37. 5 247 5

x − =

38. 3 29 75 5

x x+ = − 39. 1 55 14 4

x x+ = − 40. 1 3 82

x x− =

41. 1 353 5

x x+ = 42. 2 537 8

x x− = 43. 7 2 332 5 4

x x− = −

44. 4 2 1 25 7 4 3

x x− = + 45. 5 1 7 29 4 8 5

x x− = − 46. 3 2 3 75 7 4 8

x x+ = −

47. The expression for the area of a rectangle is 2 5 24.x x− − What are the expressions for the length and the width?

48. The expression for the area of a rectangle is 22 3 20.x x+ − What are the expressions for the length and the width?

49. The expression for the area of a rectangle is 26 15.x x+ − What are the expressions for the length and the width?


Chapter

7 Cumulative Review (continued)

Name _________________________________________________________ Date _________

In Exercises 50–63, find the zero(s) of the function.

50. ( ) 2 2 35f x x x= + − 51. ( ) 2 3 18h x x x= + −

52. ( ) 2 2 99f x x x= + − 53. ( ) 2 2 48h x x x= − −

54. ( ) 2 8 15h x x x= − + 55. ( ) 2 4 45g x x x= − −

56. ( ) 2 5 4g x x x= + + 57. ( ) 2 25f x x= −

58. ( ) 2 4 32h x x x= − − 59. ( ) 2 3h x x x= +

60. ( ) 22 17 8f x x x= + + 61. ( ) 23 26 35g x x x= − +

62. ( ) 22 7 15f x x x= + − 63. ( ) 24 14f x x x= −

In Exercises 64–75, find the discriminant of the quadratic equation and describe the number and type of solutions of the equation.

64. 2 4 12 0x x+ − = 65. 2 5 10 0x x+ + = 66. 2 8 16 0x x+ + =

67. 2 9 24 0x x+ − = 68. 2 6 32 0x x− − = 69. 2 11 150 0x x+ + =

70. 2 42 3x x+ = 71. 2 14 49x x+ = − 72. 22 4 3x x− =

73. 23 6 8x x= − + 74. 24 8 6x x+ = − 75. 2 25 10x x+ =

In Exercises 76–87, solve the equation using the Quadratic Formula.

76. 22 4 2 0x x− + = 77. 23 5 2 0x x− + = 78. 25 4 16 0x x+ − =

79. 23 2 18x x− + = 80. 2 9 15x x+ = 81. 24 5 2x x+ =

82. 26 11 8x x+ = 83. 23 24 7x x− − = 84. 25 36 2x x+ = −

85. 24 12 64x x− + = − 86. 28 7 5x x− = − 87. 2 4x x= −

88. The expression for the area of a square is 2 16 64.x x+ + What is the expression for the length of one side?

89. The expression for the area of a square is 24 20 25.x x+ + What is the expression for the length of one side?

90. The expression for the area of a square is 216 8 1.x x− + What is the expression for the length of one side?


253

Chapter


Name _________________________________________________________ Date __________

In Exercises 91–102, solve the system by substitution.

91. 912

y xy

= −=

92. 143

y xx

= += −

93. 2 94

y x xx

= + −= −

94. 2 86

y xy

= += −

95. 2 1713

y xy

= −=

96. 2 711

y xy

= += −

97. 4 126

y xy x

= +=

98. 4 59

y xy x

= +=

99. 3 85

y xy x

= −= −

100. 2 5 61

y x xy x

= + −= −

101. 2 2 153

y x xy x

= − −= +

102. 2 10 162

y x xy x

= + += +

In Exercises 103–112, solve the system by elimination.

103. 2 88 4x y

x y− =

− + = 104. 3 8 20

3 4 8x y

x y+ =

− − =

105. 3 4 153 3 13x y

x y− =

− + = − 106. 2 6 14 0

5 6 10x y

x y− + − =

− =

107. 2 4 134 12 22x y

x y− = −

− + = 108. 3 4 7

2 8 10x y

x y− + = −

− = −

109. 2

2

6 8 12 14

5 7 14 12

x x y

x x y

− + + =

− + − − = −

110. 2

2

3 2 7 12 8

4 5 8 2 14

x y x

x x y

− + + − = −

− + − =

111. 2

2

3 7 8 3

3 5 9 2

x x y

x y x

+ − =

+ + + = −

112. 2

2

9 14 3 8 4 18

6 4 7 21 5 23

x x x y

x y x x

− − + − =

− + − + + =

113. Your friend has a swimming pool that is in the shape of a rectangular prism. It measures 18 feet by 34 feet, and is 4 feet deep. What is the volume of the swimming pool?

114. A cube has an edge length of 3 inches.

a. Find the surface area of the cube.

b. Find the volume of the cube.

115. A sphere has a radius of 7 centimeters.

a. Find the surface area of the sphere.

b. Find the volume of the sphere.


Chapter


Name _________________________________________________________ Date _________

In Exercises 116–122, evaluate the function for the given value of x.

116. ( ) 3 24 2 6 14; 3f x x x x x= − + − + = −

117. ( ) 5 3 24 3 7 1; 4h x x x x x x= − + + + =

118. ( ) 4 33 2 11 36; 5g x x x x x= − + − + = −

119. ( ) 6 4 3 22 7 3 12 24 38; 2f x x x x x x x= − + − + + = −

120. ( ) 4 3 26 3 2 14 12; 3g x x x x x x= − + − + − =

121. ( ) 3 2 123 14 21 37; h x x x x x= − + − =

122. ( ) 3 2 142 8 12 52; g x x x x x= − + − + = −

In Exercises 123–129, find the sum.

123. ( ) ( )3 2 3 25 4 7 3 12 9x x x x− + + − + −

124. ( ) ( )4 2 4 23 7 12 9 7 5 14 8x x x x x x− + − + − − +

125. ( ) ( )5 2 4 26 12 7 5 4 2 9 12x x x x x x− − + − + − + − +

126. ( ) ( )4 3 3 29 7 3 11 5 8 15x x x x x x− + − + − − +

127. ( ) ( )4 4 3 24 2 9 8 3 9 7 10x x x x x x− + − + − − + +

128. ( ) ( )5 4 3 2 4 28 7 5 10 3 1 3 3 5 9x x x x x x x x− + − − + + − − +

129. ( ) ( )5 3 2 5 4 3 27 3 4 8 12 4 12 8 7 13 10x x x x x x x x x− + − − + − + + − + +

In Exercises 130–135, find the difference.

130. ( ) ( )3 311 10 1 5 8 9x x x x− + − − − − +

131. ( ) ( )2 212 2 1 7 8 11x x x x− + − − + +

132. ( ) ( )4 3 412 4 6 9 12 10 2x x x x x− + + − − − +

133. ( ) ( )4 3 2 33 3 12 7 3 5 9 10x x x x x x− + + − − − + +

134. ( ) ( )5 2 5 4 3 28 4 2 1 9 2 7 4 12 6x x x x x x x x− + + − − − + + + −

135. ( ) ( )5 2 3 26 4 5 8 12 10 5x x x x x x− + − + − − − −

Documents

alg2 resources ch 07 toc - MATHEMATICSmathwithjp.weebly.com/uploads/2/0/8/8/20882022/hscc_alg2_rbc_07.pdfdos planes de telefonía celular para un teléfono desbloqueado para los primeros