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
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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.
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.
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.
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=
In Exercises 7–10, tell whether x and y show direct variation, inverse variation, or neither.
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.
Name _________________________________________________________ Date _________
In Exercises 1–6, tell whether x and y show direct variation, inverse variation, or neither.
1. 12yx
= 2. 15xy = 3. 9x y=
4. 3y x= − 5. 9yx
= 6. 13
xy =
In Exercises 7–10, tell whether x and y show direct variation, inverse variation, or neither.
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?
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.
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.
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 −∞.
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.
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.
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.
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.
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.
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.
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.
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.
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?