Tenn Algebra 2 EOC Practice Workbook

TENNESSEE

EOC Test Preparation Workbook Algebra II

Copyright © by Houghton Mifflin Harcourt Publishing Company

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ISBN 978-0-547-47862-3

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ContentsTo the Student v

Algebra II State Performance Indicators vi

Pre Test 1

Standards Review and Practice

Mathematical Processes

State Performance Indicator 3103.1.1 21




Number & Operations




Algebra















iv

Geometry & Measurement




Data Analysis, Statistics, & Probability









Post Test 97

Answer Grids 117

v

These practice activities are correlated to the state performance indicators for Algebra II and are designed to prepare you to take Tennessee’s high school assessment test. The practice tests reflect the type of wording likely to be encountered on the actual test.

To the Student

Mathematics State Performance Indicators Algebra IIvi

Mathematics State Performance IndicatorsALGEBRA II

Standard 1 — Mathematical Processes

SPI 3103.1.1 Move flexibly between multiple representations (contextual, physical, written, verbal, iconic/pictorial, graphical, tabular, and symbolic) of non-linear and transcendental functions to solve problems, to model mathematical ideas, and to communicate solution strategies.

SPI 3103.1.2 Recognize and describe errors in data collection and analysis as well as identifying representations of data as being accurate or misleading.

SPI 3103.1.3 Use technology tools to identify and describe patterns in data using non-linear and transcendental functions that approximate data as well as using those functions to solve contextual problems.

SPI 3103.1.4 Use mathematical language, symbols, definitions, proofs and counterexamples correctly and precisely to effectively communicate reasoning in the process of solving problems via mathematical modeling with both linear and non-linear functions.

Standard 2 — Number & Operations

State Performance Indicators:

SPI 3103.2.1 Describe any number in the complex number system.

SPI 3103.2.2 Compute with all real and complex numbers.

SPI 3103.2.3 Use the number system, from real to complex, to solve equations and contextual problems.

Standard 3 — Algebra


SPI 3103.3.1 Add, subtract and multiply polynomials; divide a polynomial by a lower degree polynomial.

SPI 3103.3.2 Solve quadratic equations and systems, and determine roots of a higher order polynomial.

SPI 3103.3.3 Add, subtract, multiply, divide and simplify rational expressions including those with rational and negative exponents.

SPI 3103.3.4 Use the formulas for the general term and summation of finite arithmetic and both finite and infinite geometric series.

SPI 3103.3.5 Describe the domain and range of functions and articulate restrictions imposed either by the operations or by the contextual situations which the functions represent.

SPI 3103.3.6 Combine functions (such as polynomial, rational, radical and absolute value expressions) by addition, subtraction, multiplication, division, or by composition and evaluate at speci-fied values of their variables.

SPI 3103.3.7 Identify whether a function has an inverse, whether two functions are inverses of each other, and/or explain why their graphs are reflections over the line y = x.

viiAlgebra II Mathematics State Performance Indicators

SPI 3103.3.8 Solve systems of three linear equations in three variables.

SPI 3103.3.9 Graph the solution set of two or three linear or quadratic inequalities.

SPI 3103.3.10 Identify and/or graph a variety of functions and their transformations.

SPI 3103.3.11 Graph conic sections (circles, parabolas, ellipses and hyperbolas) and understand the relationship between the standard form and the key characteristics of the graph.

SPI 3103.3.12 Interpret graphs that depict real-world phenomena.

SPI 3103.3.13 Solve contextual problems using quadratic, rational, radical and exponential equations, finite geometric series or systems of equations.

SPI 3103.3.14 Solve problems involving the binomial theorem and its connection to Pascal’s Triangle, combinatorics, and probability.

Standard 4 — Geometry & Measurement


SPI 3103.4.1 Exhibit knowledge of unit circle trigonometry.

SPI 3103.4.2 Match graphs of basic trigonometric functions with their equations.

SPI 3103.4.3 Describe and articulate the characteristics and parameters of parent trigonometric functions to solve contextual problems.

Standard 5 — Data Analysis, Statistics, & Probability


SPI 3103.5.1 Compute, compare and explain summary statistics for distributions of data including measures of center and spread.

SPI 3103.5.2 Compare data sets using graphs and summary statistics.

SPI 3103.5.3 Analyze patterns in a scatter-plot and describe relationships in both linear and non- linear data.

SPI 3103.5.4 Apply the characteristics of the normal distribution.

SPI 3103.5.5 Determine differences between randomized experiments and observational studies.

SPI 3103.5.6 Find the regression curve that best fits both linear and non-linear data (using technology such as a graphing calculator) and use it to make predictions.

SPI 3103.5.7 Determine/recognize when the correlation coefficient measures goodness of fit.

SPI 3103.5.8 Apply probability concepts such as conditional probability and independent events to calculate simple probability.

Mathematics State Performance Indicators (continued)

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Pre Test 1Algebra II Tennessee

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Pre Test

1 Which is the equation of the resulting function when the graph of y 5 x2 2 1 is moved down 6 units?

A y 5 26x2 2 1 C y 5 x2 1 5

B y 5 6x2 2 1 D y 5 x2 2 7

2 In the diagram below, what does the set A represent?

WholeNumbers

NaturalNumbers

RationalNumbers

IrrationalNumbers

A

F the complex numbers

G the set of repeating decimals

H the integers

J the real numbers

3 Which of the following is true about the data sets?

Set A {6, 10, 7, 7, 9, 2, 8}

Set B {11, 15, 12, 12, 14, 7, 13}

A The standard deviations are not the same.

B The mean of Set A is 5 less than the mean of Set B.

C The medians are equal.

D The modes are the same.

4 Simplify 5y 1 2 }

xy2 1

2x 2 4 } 4xy

.

F 20xy2 1 8xy 1 2x 2 4

}} 4xy

G 2x2y2 2 4xy2 1 5y 1 2

}} xy2

H 5x 2 2y 2 2

} xy2 1 4xy

J xy 1 8y 1 4

} 2xy2

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Pre Test (continued)

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5 Which equation represents the hyperbola with its foci labeled in the graph below?

x

y

O

2

2

41)(0,

41)(0, 2

A y2

} 16

1 x2

} 25 5 1

B y2

} 25

1 x2

} 16 5 1

C y2

} 16

2 x2

} 25 5 1

D y2

} 25

2 x2

} 16 5 1

6 Find the range of ƒ(x) 5 x2 2 4.

F ƒ 1 x 2 Þ 22 or 2

G ƒ 1 x 2 $ 24

H x $ 0

J 2` , x , `

7 A newspaper included the following graph in an article that concluded that the value of a collectable doll had increased dramatically since it was released in 1960. Is this an accurate statement?

Valu

e (

do

llars

)

0

270

260

250

Year

Value of Collectable Doll

1960

1970

1980

1990

2000

A No; the broken axis and small increments used for the vertical scale make the graph appear to be steep, implying that the value increased quickly.

B No; the years on the horizontal axis are too spread out.

C No; the value of the collectible doll actually decreased over time.

D Yes, it is an accurate statement.

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10 What are the rational zeros of ƒ(x) 5 2x3 1 x2 1 2x 1 1?

F 2 1 } 2 , 2i, and i

G 21 and 2 1 } 2

H 2 1 } 2

J 21, 2 1 } 2 ,

1 }

2 , and 1

11 What is the solution of the system shown below?

3x 1 2y 2 z 5 21 x 2 y 1 z 5 23 22x 1 4y 2 3z 5 8

A (2, 22, 5)

B (0, 1, 22)

C (21, 0, 22)

D (21, 0, 2)

8 Which is the graph of the function y 5 2(x 2 4)2 2 3?

x

y

2

4

O 4 82426

24

26

28

A

B

C

D

F graph A

G graph B

H graph C

J graph D

9 Which type of trigonometric function is represented by the graph below?

x

y

22

21

1

2

22p 2p p 2pO

A cosecant

B secant

C cotangent

D tangent

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14 A student writes four different models for data in a scatter plot. The correlation coefficient for each model is listed below. Which correlation coefficient indicates the model that has the best fit to the data?

F r 5 0.03

G r 5 0.67

H r 5 20.95

J r 5 20.53

15 Consider the real numbers a, b, and c, such that a > b, b > c, and c < 0. Which of the following statements must be true?

A ac , bc

B a } c .

b } c

C a 1 c . 0

D a 1 c , b 1 c

12 What is the domain of the function shown in this graph?

1

x

y

22

F 22 , x , 5 H 23 , x # 4

G 22 # x # 5 J 23 # x # 4

13 A population of 300 turtles increases at a rate of about 40% per year. Write an exponential growth model giving the population y after t years, and estimate the population after 5 years.

A y 5 300(0.60)t; about 23 turtles

B y 5 300(1.40)t; about 1307 turtles

C y 5 300(1.40)t; about 1613 turtles

D y 5 300 1 (1.40)t; about 305 turtles

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18 An opinion poll is to be conducted on federal funding for education. Which sampling method would give the least biased results?

F calling 2500 randomly chosen households nationwide

G calling 2500 randomly chosen households in California

H calling 2500 randomly chosen households in the Northeast

J calling 2500 randomly chosen members of educational organizations

19 Which statistic has the greatest value for the set of data 4, 7, 5, 6, 3, 8, 3, 7, 9, 6, 5, 7?

A mean

B median

C mode

D range

16 What is the solution to the system of equations shown below? 4x 2 3y 1 z 5 28 22x 1 y 2 3z 5 24 x 2 y 1 2z 5 3

F (1, 22, 4)

G (21, 4, 22)

H (22, 3, 1)

J (22, 1, 3)

17 A set of data has a normal distribution with a mean of 126 and a standard deviation of 16. What value is one standard deviation below the mean?

A 110

B 142

C 122

D 118

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22 Which system of linear inequalities is represented by the graph below?

y

x

22

4

F { x 1 6y $ 4 x 2 y # 3

G { x 1 6y # 4 x 2 y . 3

H { x 1 6y , 4 x 2 y , 3

J { x 1 6y . 4 x 2 y $ 3

20 Simplify 5y 2 15

} 10y 2 5

? 8y 2 4

} 3y 2 9

.

F 4 } 3

G 8 } 3

H y 2 3

} 2y 2 1

J 2y 1 1

} y 1 3

21 Which expression represents f ( g (x)) if f (x) 5 x3 and g(x) 5 2x 2 1?

A (2x 2 1)3

B 2x3 2 1

C 2x4 2 x3

D x3 1 2x 2 1

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24 The table below shows the population of Texas (in millions) for select years since 1950.

Years since 1950

Population (in millions)

0 7.7

10 9.6

20 11.2

30 14.2

40 17.0

50 20.9

Which model approximates the data in the table?

F y 5 20.002x2 1 0.21x 1 7.7

G y 5 7.7(1.02)x

H y 5 0.3x 1 7

J y 5 (7.7 ? 1.02)x

23 A survey is to be conducted on the exercise habits of high school students. Which sampling method would give the least biased results?

A randomly choosing 600 students from the same school and having them complete a questionnaire during their physical education class at school

B randomly choosing 600 students across a state and having them complete a questionnaire during their after-school sports practice

C randomly choosing 600 students across a state and having them complete a questionnaire during their physical education class at school

D randomly choosing 600 students from several city schools and having them complete a questionnaire during their lunch period at school

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27 The height of the tip of the second hand of a wall clock in a classroom can be modeled by a sine curve.

2

4

6

8

1 2 3 4Time (min)

Heig

ht

(ft)

00 x

y

What are the period and amplitude of the function?

A period 5 0.5 min; amplitude 5 0.5 ft

B period 5 0.5 min; amplitude 5 1 ft

C period 5 1 min; amplitude 5 0.5 ft

D period 5 1 min; amplitude 5 1 ft

25 A news service surveys prospective voters by calling them at home between 1 P.M. and 3 P.M. on Monday. Which statement describes the most serious problem with this survey process?

A The time period is not long enough.

B The survey should also be conducted on Tuesday.

C A phone survey does not yield meaningful results.

D Many people are working during the time of the survey.

26 What is 25 2 (3 1 i ) 2 (8 1 2i ) written as a complex number in standard form?

F 16 1 3i

G 16 2 3i

H 216 1 3i

J 216 2 3i

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29 The scatter plot represents the number of pages and the price for eight novels.

What is the difference between the price of a 300-page novel based on the given trend line and the actual price of a 300-page novel?

A $3

B $5

C $27

D $30

28 Which expression is equivalent to

(2a4b2)3 (27ab9)0

}} 23a10b24 ?

F 14b15 }

3a5

G 22b9 }

a3

H 28a2b10 }

3

J 56a3b19 }

3

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32 What is the sum of the infinite geometric

series 1 1 1 }

3 1

1 }

9 1

1 }

27 1 …?

F 2 } 5

G 2 } 3

H 3 } 2

J Does not exist.

33 The diagram below shows a 60° angle in standard position with its terminal side intersecting the unit circle at (x, y).

x

y

r 5 1

(x, y)

608

Which of the following statements is not true?

A x 5 1 }

2

B x 5 Î

} 3 }

2

C x 5 cos 60°

D y 5 sin 60°

30 What is the solution set of the equation log2 x 1 log2 (x 2 3) 5 2?

F {21, 4}

G {4}

H {21}

J {1, 4}

31 The figure below represents a normal distribution. What are the mean and standard deviation of the data?

34% 34%

27 36 45 54 6313.5%13.5%

2.5%2.5%

A mean: 45; standard deviation: 3

B mean: 45; standard deviation: 36

C mean: 45; standard deviation: 9

D mean: 45; standard deviation: 18

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35 Let f (x) 5 3x 2 2. In finding the inverse of the function, Martin wrote the following “solution” on the chalkboard. His final answer is incorrect.

y 5 3x 2 2

x 5 3y 2 2

x 1 2 5 3y

x 1 2 }

3 5 y

Which statement describes his error?

A He should have solved for x first, then switched x and y.

B He should have multiplied x by 3 before adding 2 to each side.

C He should have divided each term by 3.

D He should have subtracted 2 from each side.

34 The data in the table show the annual average monthly highs y (in °F) for a city, where x 5 1 represents the month of January.

x y

1 37

2 39

3 46

4 56

5 67

6 76

7 82

8 80

9 72

10 62

11 52

12 42

Use a graphing calculator to make a scatter plot of the data. What type of model would best fit the data in the table?

F linear

G quadratic

H exponential

J sinusoidal

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37 The scatter plot below represents the amount of baggage mishandled by an airline compared to the percent of its flights that arrive on time.

x

y

00 4 5 6 7 8

50

60

70

80

90

Mishandled Baggage

(per thousand passengers)

Per

cen

t O

n-T

ime

Arr

ival

s

Which best describes the correlation, if any, illustrated by the scatter plot?

A no correlation

B left correlation

C positive correlation

D negative correlation

36 The scatter plot shows the average cost of a movie ticket in the United States, where x is the number of years since 1975.

x

y

Years since 1975

Average U.S. Ticket Price

5 10 15 20 25 30 35 40 45

Pri

ce (

do

llars

)

0

1

2

3

4

5

6

7

8

9

0

Which type of relationship appears to be the least likely to exist among the data?

F linear H quadratic

G exponential J logarithmic

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40 To which set(s) does the number p belong?

F whole numbers, natural numbers

G rational numbers, integers, whole numbers, natural numbers

H irrational numbers

J imaginary numbers

41 A variable has a normal probability distribution with mean 94 and standard deviation 3. What is the probability that the variable takes a value between 85 and 103?

A 68%

B 95%

C 99.7%

D 99.9%

38 Simplify 2 }

1 2 i .

F 1 2 i

G 1 1 i

H 2 2 2i

J 3

39 The table shows the distance Jenny rides her bike (in kilometers) each week.

Week 1 2 3 4 5

Distance 40 44 48 56 59

Choose the equation that best estimates the regression line for the data, and use it to estimate the distance she rides in week 6.

A y 5 5x 1 34; 64 km

B y 5 4x 1 36; 54 km

C y 5 4x 1 48; 72 km

D y 5 6x 1 34; 70 km

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44 The scatter plot shows Erin’s time for a one-mile run each week for 8 weeks.

x

y

2

9

10

11

12

1210864Week

Time for One-Mile Run

Min

ute

s

Using the line of best fit for the data, which is the best prediction of Erin’s time in week 10?

F 10 min

G 9.5 min

H 9 min

J 8.5 min

42 Monique found the zeros of the function

y 5 6x2 1 19x 1 3 to be 2 1 } 6 and 23.

What are the roots of the equation 6x2 1 19x 5 23?

F 26 and 2 1 } 3 H

1 }

6 and 3

G 6 and 1 }

3 J 2

1 } 6 and 23

43 Solve 3x2 1 14 5 2.

A x 5 4 Î

} 3 }

3

B x 5 2i

C x 5 62i

D x 5 62 Î}

3 i

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47 In a high school, the probability that a student is taking art and music is 0.03. The probability that a student is taking art is 0.15. What is P(music ) art)?

A 0.0045

B 0.18

C 0.2

D 0.5

48 The graphs of the functions f and g are shown below. What is ( f 1 g)(24)?

x

y

O 22428216 212

28

4

8

12

24

f

g

y 5 x

1 2

y 5 2 x 2 4

12

F 24

G 23

H 22

J 0

45 Let f (x) 5 x3 2 7x2 1 2x 1 40 and g(x) 5 x 2 5.

Find f (x)

} g(x)

.

A x2 2 2x 2 8

B x2 1 2x 1 8

C x3 1 7x2 2 35

D x3 2 7x2 1 3x 1 35


Set A {3, 7, 10, 5, 8, 4, 3, 6}

Set B {5, 5, 12, 6, 4, 2, 7, 9}

F The means are equal.

G The medians are equal.

H The modes are equal.

J The standard deviations are equal.

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51 The graph of x2 }

4 1

y2

}

16 5 1 is an ellipse.

Which set of points represents the foci of the ellipse?

A (0, 12); (0, 212)

B (12, 0); (212, 0)

C (2 Ï}

3 , 0); (22 Ï}

3 , 0)

D (0, 2 Ï}

3 ); (0, 22 Ï}

3 )

52 Use the unit circle below to determine the exact value of tan 60°.

x

y

608

( , )12

32

!ß

F Ï}

3

G Ï}

3 } 2

H 1 1 Ï}

3 } 2

J 1 }

2

49 A teacher’s Web site requires a password that is 5 digits long with no digits repeated. The teacher uses a computer program to produce 5-digit passwords and distributes them randomly to the students. What is the probability that a student will receive a 5-digit password of consecutive numbers in increasing order?

A about 0.008

B about 0.004

C about 0.002

D about 0.0002

50 What is the inverse of ƒ(x) 5 8 } 27

x3?

F g(x) 5 3 }

2

3 Ï}

x

G g(x) 5 3 }

2 x3

H g(x) 5 3 Î}

2 }

3 x

J g(x) 5 2 }

3

3 Ï}

x

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55 What is the measure of the angle in standard position shown below?

x

y

A 2210°

B 2150°

C 150°

D 210°

56 Use the quadratic formula to solve the equation 4x(x 2 2) 1 1 5 0.

F x 5 22 6 Ï

}

3 } 2

G x 5 2 6 Ï

}

3 } 2

H x 5 3 6 Ï

}

17 } 2

J x 5 1 6 2 Ï}

5

53 Divide 2x2 2 1 qww 6x4 1 4x3 2 2x2 2 2 .

A 3x2 1 2x 2 x2 1 2x 2 2

} 2x2 2 1

B 3x2 1 1 1 2x2 2 1

} x2 1 2x 2 2

C 3x2 1 2x 1 1 }

2 1

2x 2 3 }

2 }

2x2 2 1

D 6x4 1 1 2 x2 2 2

} 2x2 2 1

54 A football is kicked so that the height above the ground, in feet, after t seconds is given by the function y 5 26.4t2 1 32t. What is the maximum height reached by the football?

F 50 ft

G 40 ft

H 38.4 ft

J 32 ft

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59 The first term of an arithmetic sequence is 7 and the 15th term is 49. What is the 20th term of the sequence?

A 57

B 60

C 64

D 67

60 A landscape company is having a tree sale. Five maple trees and 6 ash trees are on sale for $83. Nine maple trees and 11 ash trees are on sale for $151.What is the cost of an ash tree?

F $7 H $14

G $8 J $16

57 Gina has a bag containing 9 red marbles, 12 blue marbles, and 7 green marbles.

She randomly selects a marble from the bag, replaces it, and then randomly selects another marble. What is the probability the two marbles she selects are blue?

A about 0.184

B about 0.168

C about 0.103

D 0.0625

58 Chloe uses the following steps to find the value of x when f (x) 5 0 for the function

f (x) 5 Ï}

3 2 2x 2 5.

Step 1: 0 5 Ï}

3 2 2x 2 5

Step 2: 0 5 3 2 2x 1 25

Step 3: 2x 5 28

Step 4: x 5 14

Is Chloe’s answer correct? If not, in which step did she make her first error?

F no; Step 1

G no; Step 2

H no; Step 3

J yes

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62 Samantha threw a paper airplane into the air. The flight path was part of a parabola, as shown below.

Heig

ht

(ft)

10 2 3 4 95 6 7 8

Distance (ft)

x

y

8

10

12

14

16

18

4

6

2

0

What was the maximum height the plane reached?

F 4 ft

G 9 ft

H 20 ft

J 24 ft

61 Subtract 2x3 1 4x2 2 x 1 8 from x3 2 6x 1 10.

A 22x3 1 4x2 2 5x 1 2

B 2x3 2 4x2 2 5x 1 2

C 2x3 1 4x2 2 7x 1 18

D 2x3 2 5x 1 2

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64 Which polynomial represents (2x2 1 x 2 5)(3x 1 4)?

F 8x3 2 20

G 8x2 1 4x 2 20

H 6x3 1 3x2 2 15x

J 6x3 1 11x2 2 11x 2 20

65 Which function is represented below?

x

y

O

21

1

2p pp2

p2

2

A cos 2x

B 2 cos x

C 1 } 2 cos 2x

D 2 cos 2x

63 Which graph is the solution to the system?

{ y # x 1 2 y , 2x 1 3

A

2

2

2468

68

66

y

x8

B

2

2

2468

68

6

y

x86

C

2

2

2468

68

66

y

x8

D

2

2

2468

68

6

y

x86

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SPI 3103 21Algebra II Tennessee

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SPI 3103.1.1Move fl exibly between multiple representations (contextual, physical, written, verbal, iconic/pictoral, graphical, tabular, and symbolic) of non-linear and transcendental functions to solve problems, to model mathematical ideas, and to communicate solution strategies.

1 Mr. Jackson deposits $5000 in an account that pays 2.25% annual interest. Which equation gives the amount in the account A after t years if interest is compounded annually?

A A 5 5000 1 (1.0225)t

B A 5 5000(1.0225)t

C A 5 5000(1.225)t

D A 5 5000 1 1 1 0.0225

} 12

2 t

2 The length of a rectangular yard is 2 meters longer than twice its width. Which function gives the area of the yard in terms of the length x?

F f (x) 5 0.5(x2 2 x)

G f (x) 5 0.5x2 2 1

H f (x) 5 2(x 2 2)2

J f (x) 5 0.5x2 2 x

In approaching a mathematical problem, remember that you have many types of tools available. Don’t be afraid to use different kinds of descriptions or representations to help you understand and solve a problem.

EXAMPLE

A savings account earns 3% compounded annually. Write an expression for the amount A in the account after t years.

Year Amount

0 A

1 A 1 0.03A 5 A(1.03)

2 A(1.03) 1 A(1.03)(0.03) 5 A(1.03)(1 1 0.03)

5 A(1.03)(1.03)

5 A(1.03)2

3 A(1.03)2 1 A(1.03)2(0.03) 5 A(1.03)2(1 1 0.03)

5 A(1.03)2(1.03)

5 A(1.03)3

: :

t A(1.03)t

To find the amount in an account that earns interest compounded annually after a given number of years, first add the interest rate as a decimal to 1. Calculate the power with this number as the base and the number of years as the exponent. Then multiply the result by the original amount in the account.

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5 Which function gives the distance from the point at (3, 0) to a point on the line y 5 2x 1 7 as a function of just x?

A f (x) 5 (x 2 3)(2x 1 7)

B f (x) 5 2x 1 7

} x 2 3

C f (x) 5 x 2 3 1 2x 1 7

D f (x) 5 Ï}}

(3 2 x)2 1 (2x 1 7)2

6 There was one cracked cement square in a sidewalk, as indicated by the shaded square. A construction company is adding squares to the sidewalk as shown in the steps below.

Step 3

Step 2

Step 1

Which function will give the ratio of shaded squares to unshaded squares at the nth step of the fi gure? (Hint: In Step 4, the ratio of shaded squares to unshaded squares is 1 to 16.)

F f (n) 5 1 }

2 n H f (n) 5

1 }

2 n2

G f (n) 5 1 1 } 2 2

n J f (n) 5 2n

3 Marlene and Hali are playing catch. They each throw a tennis ball at the same time. The graph shows the fl ight of the 2 balls. Which statement is true about this situation?

Heig

ht

(ft)

010 20 300 40

20

10

Distance (ft)

x

y

MarleneHali

A Marlene’s ball goes higher and farther.

B Marlene’s ball goes lower but farther.

C Hali’s ball goes higher and farther.

D Hali’s ball goes lower but farther.

4 The graph shows the wind speed as a storm develops. To the nearest whole number, what was the average increase in wind speed per minute between 10 and 15 minutes?

Time (min)

Win

d s

peed

(m

i/h

)

0 5 10 15

20

15

10

5

0 x

y

F 2 mi/h H 3 mi/h per minute per minute

G 5 mi/h J 10 mi/h per minute per minute

SPI 3103.1.1 (continued)

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2 A biologist wants to test for germs in lake water. Which sampling method will give the most representative sample?

F collect water samples from 20 random locations from a grid of the lake

G collect water samples from 20 locations along the shoreline

H collect water samples from 20 locations on the northern end of the lake

J collect water samples at 20 different depths from a boat in the middle of the lake

1 The salaries of several people in a company are listed. Which measure will best represent the salary of a “typical” employee?

President $234,560

Vice President $156,090

Manager $95,000

Clerk $30,000

Clerk $27,000

Warehouse clerk $22,000

Warehouse clerk $21,500

Secretary $18,000

Customer Service $17,500

Customer Service $17,500

A mean C mode

B median D range

GO ON

SPI 3103.1.2Recognize and describe errors in data collection and analysis as well as identifying representations of data as being accurate or misleading.

Data should be collected and presented in a way that avoids bias. In reading a report or summary of data, keep in mind the following questions:

Is the data set large enough? Was the data collection technique appropriate? Are graphs properly constructed? Do the conclusions avoid unfounded inferences?

EXAMPLE

A phone survey of 1140 randomly selected households in a small town produced the data summarized in the chart. On this basis, the study concluded that a typical income for this town was about $30,000 per year. Is this a sound conclusion?

If typical refers to the median income, the conclusion is reasonable, because roughly half of households lie on either side of $30,000.

If typical refers to the mean income, the conclusion is unfounded. The mean is almost certainly $35,000 or higher. The unequal intervals from 0 to $60 thousand are a questionable aspect of the graph’s construction.

0220

100

Ho

useh

old

s

200

300

400

20225 25235

Annual Income

($1000)

35260 .60


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3 The graph below shows the sales for a company over a two-month period. Trisha claims that there was a signifi cant decrease in sales from June to July. Is this an accurate statement?

Monthly Sales

6.03

6.01

6.07

6.05

6.09

Sale

s

(th

ou

san

ds o

f d

ollars

)

June JulyMonth

A No; the scale on the vertical axis does not start at 0 and is very small, which makes it appear that sales in July were signifi cantly lower than sales in June.

B No; the monthly sales actually increased from June to July.

C No; the scale on the vertical axis is too large.

D Yes, the statement is accurate.

4 The line graph shows the temperature at the beginning of each hour one morning. Trevor claims that the temperature increased slowly throughout the morning. Is this an accurate statement?

Tem

pera

ture

(8F

)

07 8 9 10 116

80

60

40

20

Time (A.M.)

Temperature Readings

F No; there are not enough data points to make this statement.

G No; the scale on the vertical axis compresses the graph, making it appear that the temperature increased slowly when in fact it increased rapidly.

H No; the temperature slowly decreased throughout the morning.

J Yes, this is an accurate statement.

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SPI 3103.1.3Use technology tools to identify and describe patterns in data using non-linear and transcendental functions that approximate data as well as using those functions to solve contextual problems.

To find an appropriate model for a data set, examine the data and/or its graph for hints of an appropriate type of model. A calculator’s regression feature can then help you find an equation of a model. The closer the value of r2 (the square of the correlation coefficient) is to 1, the better the model.

Once you have a model, you can use it to make predictions. A prediction is most reliable for values of the independent variable that lie between or near the minimum and maximum domain values of the data.

EXAMPLE

Consider the following data:

(3.6, 21.5), (14.1, 493.8), (29.3, 2674.5), (40.5, 5629.3)

a. Find the equation of an appropriate model.

b. Use your model to predict the value of y to the nearest hundred when x 5 21.5.

Solution

a. The data are clearly not linear. A sketch shows a curved, basically parabolic graph. Enter the data into a calculator and try a quadratic regression model:

y < 4.21x2 2 34.5x 1 107

r2 5 0.999845 Very close to 1, indicating a good fit

b. When x 5 21.5, y < 4.21(21.5)2 2 34.5(21.5) 1 107 < 1300.

Experimenting further with a calculator shows that an exponential model is a much poorer fit (r2 < 0.91), but that the power model y < 1.13x2.30 is an even better fit (r2 < 0.999999).

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1 The table shows the number of hours per year a person spent playing video games for the years from 2005 to 2009.

Year Hours Playing Video Games

2005 73

2006 76

2007 85

2008 101

2009 116

If x 5 5 represents 2005, which of the following polynomial functions best approximates the data in the table?

A f (x) 5 11x 1 13, 5 # x # 9

B f (x) 5 2.2x2 2 19x 1 117, 5 # x # 9

C f (x) 5 20.583x3 1 14.46x2 2 103.7x 1 303, 5 # x # 9

D f (x) 5 20.3750x4 2 9.917x3 2 94.13x2 1 387.6x 2 517, 5 # x # 9

2 The table shows the height s in feet above ground of an object t seconds after it is launched.

t h

0 50

1 98.8

2 115.6

3 100.4

4 53.2

Which function models the data in the table?

F s (t) 5 216t2 2 64.8t 1 50

G s (t) 5 16t2 2 64.8t 1 50

H s (t) 5 16t2 1 64.8t 1 50

J s (t) 5 216t2 1 64.8t 1 50

3 The table below shows the number of gallons of bottled water a person drank per year for a select number of years.

Year Gallons of Water

1980 2.7

1985 4.5

1990 8.8

1995 11.6

2000 16.7

2005 25.5

An exponential function that approximates this data is f (x) 5 3(1.09)x, where x 5 0 represents 1980. According to the model, how many gallons of water did a person drink in 2007?

A about 13.0 gal

B about 30.7 gal

C about 72.8 gal

D about 88.3 gal

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2 Arlene believes that the function f (n) 5 n2 1 n 1 17 always produces prime numbers for n, a positive number. Which of the following is a counterexample?

F f (1)

G f (2)

H f (5)

J f (7)

1 Which is the fi rst incorrect step in simplifying (33 p 43)21/3?

Step 1: (33 p 43)21/3 5 1 (3 p 4)3 2 21/3

Step 2: 5 1 (12)3 2 21/3

Step 3: 5 1223

Step 4: 5 1 }

123

A Step 1

B Step 2

C Step 3

D Step 4

GO ON

SPI 3103.1.4Use mathematical language, symbols, defi nitions, proofs and counter-examples correctly and precisely to effectively communicate reasoning in the process of solving problems via mathematical modeling with both linear and non-linear functions.

A counterexample is an example that disproves a hypothesis, proposition, or theorem.

EXAMPLE

Geri believes that the squares of two natural numbers {1, 2, 3, …} cannot add up to the square of another natural number. Give a counterexample.

Solution

Make a table of squares for the natural numbers:

n n2

1 1

2 4

3 9

4 16

5 25

From the table, it is apparent that 9 1 16 5 25, or 32 1 42 5 52. The squares of the natural numbers 3 and 4 add up to the square of the natural number 5. This fact is a counterexample that disproves Geri’s hypothesis.

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6 Which of the following statements is true for 2n $ n2 , where n is a real number?

F 2n $ n2 is always true.

G 2n $ n2 is sometimes true.

H 2n $ n2 is never true.

J 2n $ n2 is only true for n 5 21 .

7 Elena correctly solved a system of equations and got 0 5 5 for a solution. What can she conclude about the graph of the system?

A The equations are the same line.

B The lines intersect at (0, 5).

C The lines are parallel.

D The lines intersect at (5, 0).

8 Assume the statement1 1 2 1 22 1 . . . 1 2n 2 1 5 2n 2 1is true for all positive integers, n.Cory is asked to verify the statementfor n 5 5. His work is shown below:

2n21 5 2n 2 1

2521 0 25 2 1

24 0 32 2 1

16 Þ 31

Based on Cory’s work, which of the following can be concluded?

F The statement is true for all n except 5.

G The statement is false.

H Cory made an error.

J Not enough information is given.

3 Which of the following conclusions is true for the statement below?

logb mn 5 logb m 1 logb n

A The statement is never true.

B The statement is always true.

C The statement is true when m and n are both negative.

D The statement is true when m and n are both positive.

4 Which of the following statements is correct about the expression below?

26 2 3x }

x2 2 6x 1 8

F The expression is defi ned for all real numbers.

G The expression is defi ned for all real numbers except when x 5 2 or 4 .

H The expression is defi ned for all positive real numbers.

J The expression is defi ned for all real numbers except when x 5 4 .

5 Consider the equation loga b 5 0 for positive numbers a. Which statement is valid for real values of b ?

A b 5 0

B b 5 1

C b # 0

D b . 1


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1 To which sets does the number 2 2 } 3

belong?

A real numbers, rational numbers, and negative integers

B rational numbers and real numbers only

C rational numbers, complex numbers, and real numbers

D irrational numbers, complex numbers, and real numbers

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SPI 3103.2.1Describe any number in the complex number system.

Terms to Know Example

A complex number written in standard form is a number in the form a 1 bi where a and b are real numbers. The number a is the real part of the complex number, and the number bi is the imaginary part.

7 2 i and 3 1 8i are complex numbers.

If b Þ 0, then a 1 bi is an imaginary number. 4 1 3i and 5i are imaginary numbers.

A complex plane is a coordinate plane in which each point (a, b) represents a complex number a 1 bi. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

i

1

imaginary

real

2 In the diagram below, what does the set A represent?

Real Numbers

Rational Numbers Irrational Numbers

A

Whole Numbers

Natural Numbers

F complex numbers

G integers

H imaginary numbers

J terminating decimals


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6 Which point on the complex plane below shows the location of 4 2 2i ?

i

1 real

M

N

O

L

imaginary

F L H N

G M J O

5 Which quadrant of the complex plane contains the point 22 1 i ?

A Quadrant I

B Quadrant II

C Quadrant III

D Quadrant IV


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7 If z 5 4 2 3i, what is |z|?

A 1 C 5

B Ï}

7 D 25

8 What is the absolute value of the complex number graphed below?

i

1

imaginary

real

F Ï}

5 H Ï}

13

G Ï}

7 J 13

3 Which one of the following is a true statement?

A The set of irrational numbers is a subset of the set of rational numbers.

B i Ï}

2 is an irrational number.

C 0 is not a whole number.

D Any whole number is also a complex number.

4 Which numbers are irrational numbers in the set below?

H 24. } 7 , 22 Ï}

3 , 22, 0, 1 } 8 , Ï

}

26 , π J F H 22 Ï

} 3 , π J

G H 24. } 7 , 22 Ï}

3 , Ï}

26 , π J H H 24. } 7 , 22 Ï

}

3 J J H 22 Ï

}

3 , π, Ï}

26 J


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1 Simplify 3i(5i).

A 15i

B 215

C 8i

D 8i 2

2 Simplify 3i 1 (2 2 i).

F 4i

G 6i

H 2 2 2i

J 2 1 2i

SPI 3103.2.2Compute with all real and complex numbers.

Complex Conjugates Example

Two complex numbers of the form a 1 bi and a 2 bi are called complex conjugates. The product of complex conjugates is always a real number.

2 1 3i and 2 2 3i are complex conjugates.

(2 1 3i)(2 2 3i) 5 4 2 6i 1 6i 2 9i2

5 4 1 9

5 13

Complex Number Operations Example

To add (or subtract) two complex numbers, add (or subtract) their real parts and their imaginary parts separately.

Sum of complex numbers:

(a 1 bi) 1 (c 1 di) 5 (a 1 c) 1 (b 1 d)i

Difference of complex numbers:

(a 1 bi) 2 (c 1 di) 5 (a 2 c) 1 (b 2 d)i

Sum:

(2 2 3i) 1 (3 1 4i) 5 (2 1 3) 1 (23 1 4)i

5 5 1 i

Difference:

(4 1 3i) 2 (2 2 4i) 5 (4 2 2) 1 (3 2(24))i

5 2 1 7i

To multiply two complex numbers, use the Distributive Property or the FOIL method just as you do when multiplying real numbers or algebraic expressions.

Distributive Property:

2(2 2 3i) 5 4 2 6i

FOIL Method:

(1 1 4i)(2 2 2i) 5 2 2 2i 1 8i 2 8i2

5 2 1 6i 2 8(21)

5 10 1 6i

To divide two complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of c 1 di is c 2 di.

Use FOIL to multiply a 1 bi

} c 1 di

p c 2 di }

c 2 di .

Then simplify and write in standard form.

2 2 3i

} 3 1 5i

5 2 2 3i

} 3 1 5i

p 3 2 5i }

3 2 5i

5 29 2 19i

} 9 2 25i2

5 29 2 19i

} 34

5 2 9 } 34 2

19 }

34 i


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3 Simplify (26 2 7i) 2 (1 1 3i).

A 25 2 10i

B 7 1 10i

C 27 2 10i

D 27 2 4i

4 Which is the sum of 2(5 2 3i) and 4 1 3i?

F 21

G 1 1 6i

H 21 1 6i

J 9 1 6i

5 Write the expression i(4 2 5i)(1 1 2i) as a complex number in standard form.

A 23 1 14i

B 3 1 14i

C 213 2 6i

D 13 1 6i

6 Write the expression 28 1 (2 1 2i) 2 (1 1 5i) as a complex number in standard form.

F 7 1 3i

G 7 2 3i

H 27 1 3i

J 27 2 3i

7 Multiply: (5 2 2i)(1 1 3i)

A 13 1 11i

B 21 1 13i

C 11 1 13i

D 13 2 i

8 What is the equivalent form of 3i }

5 2 2i ?

F 21

} 29

1 15

} 29

i

G 2 21

} 29

1 15

} 29

i

H 6 }

29 1

15 }

29 i

J 2 6 }

29 1

15 }

29 i

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11 Simplify (1 2 6i) 1 2i 2 (21 1 3i).

A 2i

B 27i

C 2 2 7i

D 2 2 11i

12 Simplify (22 1 5i) 2 .

F 24 1 10i

G 221 2 20i

H 221 1 20i

J 220 2 21i

9 Write the product of Ï}

24 and Ï}

29 in standard form.

A 26

B 6

C 26i

D 6i

10 Subtract 25 1 Ï}

212 from 27 1 Ï}

23 .

F 212 2 3 Ï}

3 i

G 22 1 Ï}

3 i

H 22 2 Ï}

3 i

J 2 1 Ï}

3 i


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15 Simplify 1 1 3i

} 1 2 3i

.

A 24 1 3i } 5

B 4 2 3i } 5

C 28 1 6i

D 10

16 Simplify 1 2 Ï

}

24 }

3 1 Ï}

21 .

F 5 1 5i } 8

G 1 2 7i } 10

H 1 1 i } 2

J 5 2 7i } 10

13 Simplify (4 2 7i)(4 1 7i).

A 33 2 56i

B 233 2 56i

C 233 1 56i

D 65

14 Simplify 7 2 5i

} 22i

.

F 5 2 7i } 2

G 5 1 7i } 2

H 25 2 7i } 2

J 25 1 7i } 2


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35Algebra II Tennessee SPI 3103

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3 Solve log2 1 x2 2 1 2 5 3.

A 23

B 3

C 22 and 2

D 2 3 and 3

4 Solve Ï}}

x2 2 5x 1 1 5 Ï}

x 2 4 .

F 5 1 6

G 5 5 6

H 5 21, 25 6

J 5 1, 5 6

1 Find all solutions to x2 1 6x 5 212 .

A x 5 23 6 i Ï}

3

B x 5 3 6 i Ï}

3

C x 5 23 6 2i Ï}

3

D x 5 3 6 2i Ï}

3

2 Solve (x 2 5)2 1 24 5 0 over the complex number system.

F 5 1 2 Ï}

6 i

G 5 1 2 Ï}

6 i and 5 2 2 Ï}

6 i

H 5 1 4 Ï}

6 i and 5 2 4 Ï}

6 i

J 5 2 2 Ï}

6

SPI 3103.2.3Use the number system, from real to complex, to solve equations and contextual problems.

A solution of a linear equation with real number coefficients will be a real number. Once you begin solving polynomial equations of degree two and higher, however, the solution set may include complex numbers. For example, a quadratic equation ax2 1 bx 1 c 5 0 has complex solutions when the discriminant, b2 2 4ac, is negative.

EXAMPLE

Solve: x2 1 4x 1 5 5 0

Solution

x2 1 4x 1 5 5 0 Write the original equation.

x 5 2b 6 Ï

}

b2 2 4ac }}

2a Use the quadratic formula.

x 5 24 6 Ï

}}

(4)2 2 4(1)(5) }}

2(1) Substitute a 5 1, b 5 4, and c 5 5.

x 5 24 6 Ï

}

24 }

2 Simplify.

x 5 24 6 2i } 2 Rewrite using the imaginary unit i.

x 5 22 6 i Simplify.

The solutions are 22 1 i and 22 2 i.


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8 The number of bacteria in a culture increases exponentially with time. For a particular culture, the equation y 5 10(2) t/20 gives the number of bacteria present in the culture after t minutes. How many bacteria are present in the culture after one hour?

F 10 bacteria

G 80 bacteria

H 20 bacteria

J 1000 bacteria

7 A manufacturer determines the number of defective items d produced at one factory to be d 5 5x 1 37, where x is the number of months after the factory began production. Meanwhile, the number of workers n employed at the factory grows according to the function n 5 0.2x2 1 52. Therefore, the average number of defective items per worker y produced in

month x is y 5 d } n 5

5x 1 37 }

0.2x2 1 52 . The largest

value for y occurs 10 months after beginning production.

What is the value of y after 10 months rounded to the nearest tenth?

A 1.0 defective items/worker

B 1.1 defective items/worker

C 1.2 defective items/worker

D 1.3 defective items/worker

5 Jason drops a rubber ball from a bridge 80 meters above a gorge. The height h of the ball above the ground as it falls can be found using the equation h 5 29.8t2 + 80, where t is the time in seconds. Estimate the height of the ball above the ground when t 5 2 seconds.

A about 20 m

B about 40 m

C about 60 m

D about 120 m

6 A population of 80 red-eyed tree frogs increases at a rate of about 8% per year. The equation y 5 80(1 1 0.08) t models the growth of the frog population over t years. What will the frog population be in 3 years?

F about 100

G about 110

H about 120

J about 130

GO ON


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1 Add: (3x2 2 2) 1 (x3 2 4x 1 5)

A x3 1 3x2 2 4x 1 3

B 4x3 2 6x 1 5

C 4x2 1 2x 1 3

D 6x3 1 3x2

2 What is the product of 24x2y and x2 2 xy 1 y2?

F 24x4y 1 4x3y2 2 4x2y3

G 24x4y 1 3x2y 2 3x2y2

H 24x4 1 16x3y2 2 4y3

J 24x4y 2 4x3y2 2 4x2y3

SPI 3103.3.1Add, subtract, and multiply polynomials; divide a polynomial by a lower degree polynomial.

like terms

EXAMPLE

a. (3x2 2 2x 1 7) 2 (x2 2 5x 1 2)

b. (5x2y)(2x2 2 3y 1 1)

c. (x2 1 3x 2 12) 4 (x 1 6)

Solution

a. (3x2 2 2x 1 7) 2 (x2 2 5x 1 2) 5 3x2 2 2x 1 7 2 x2 1 5x 2

5 (3x2 2 x2) 1 (22x 1 5x) 1 (7 2

5 2x2 1 3x 1

b. (5x2y)(2x2 2 3y 1 1) 5 (5x2y)(2x2) 1 (5x2 y)(23y) 1 (5x2y

5 10x4y 2 15x2y2 1 5x2y

c. x 1 6 qww x2 1 3x 2

x2 1 6xx2

} x 5 x

23x 2

23x 2 1823x

} x 5 2

(x2 1 3x 2 12) 4 (x 1 6) 5 x 2 3 1 6 }

x 1 6

x 2 3

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3 x2 25x 1 2) 2 (x2 2 3x 1 4)

A 2x2 2 2x 2 2

B 2x2 2 8x 1 6

C 3x2 2 2x 2 2

D 3x2 2 8x 1 6

5 x4 2 6x2 2 x 1 4 x4 2 3x3 2 x 1 7?

A 27x4 1 3x3 2 6x2 2 3

B 7x4 1 3x3 2 3

C 7x4 2 3x3 1 6x2 1 3

D 13x4 2 3x3 1 6x2 1 2x 1 11

4 (5y 2 3)(5y 1 3)?

F 25y2 2 9

G 25y3 2 9

H 25y2 2 15y2 1 15y 2 9

J 25y3 1 15y2 2 15y 2 9

6 x2 2 11x 1 15

x 2

F 2x 2

G x 2

H x 1

J 2x 1

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9 x3 1 2x2 2 2) and (x3 2 3x 1 5)?

A 3x3 1 5x2 2 7

B 4x3 2 x2 1 3

C 5x3 2 x2 1 3

D 5x3 1 2x2 2 3x 1 3

10 x2 1 x 1 7 is x3 1 4x2 1 9?

F 11x2 2 x 1 2

G 11x3 1 3x2 2 x 1 2

H 14x3 1 x2 2 x 1 2

J 14x3 1 7x2 1 x 1 16

7 3x4 1 6x3y 2 3x2

A (x 2 y)2 (3x 1 2y)

B (x 2 y)(3x2 1 y2)

C (3x2)(x2 1 2xy 2 1)

D (3x 2 1)2(x 2 1)2

8 x5 2 x3 1 x2 2 6x 2 x2 2 3?

F x3 2 2x 2 1

G x3 1 2x 1 1

H x4 2 2x 2 1

J x4 1 x3 2 2x 1 1

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14 x2 1 5 qww 6x3 1 2x2 1 22x

F 3x 1 1 2 7x 2 5

} 2x2 1 5

G 3x 1 1 1 7x 2 5

} 2x2 1 5

H 2x 1 1 1 7x 2 5

} 2x2 1 5

J 2x 1 1 2 7x 2 5

} 2x2 1 5

15 a2 2 3ab 1 b2)(a2 2 b2)

A a4 2 b4

B a4 2 3a3b 1 3ab3 2 b4

C a4 2 3a3b 1 2a2b2 1 3ab3 2 b4

D a4 1 6a3b3 2 b4

16 y2 2 3)3

F 8y6 2 27

G 8y6 2 24y4 1 18y2 2 27

H 8y6 2 12y4 2 18y2 2 27

J 8y6 2 36y4 1 54y2 2 27

11 x2 2 2x 1 2 qww 2x3 1 6x 1 5

A x 1 6 1 2x 1 4 } 10x 2 3

B x 1 6 2 2x 1 4 } 10x 2 3

C 2x 1 4 1 10x 2 3

} x2 2 2x 1 2

D 2x 1 4 2 10x 2 3

} x2 2 2x 1 2

12 s 2 1)

F (4s2 1 2

G (4s2 2 2

H (4s2 2 4s 1 2

J (4s2 1 4s 1 2

13 x4 2 x2 1 9) 2 (2x3 2 4x2 2 x) 1 (2x4 2 3x2 1 6x 2 5)

A 2x4 2 2x3 2 8x2 1 5x 1 4

B 2x4 2 2x3 1 7x 1 4

C 2x4 2 5x3 1 3x2 1 7x 1 4

D 2x4 1 2x3 2 8x2 1 5x 1 4

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SPI 3103.3.2Solve quadratic equations and systems, and determine roots of a higher order polynomial.

You can solve systems of quadratic equations using some of the same methods you use to solve systems of linear equations: graphing, substitution, or elimination.

EXAMPLE 1 Solve the system: 4x2 2 y 2 5 7

9x2 1 y 2 5 45

4x2 2 y 2 5 7

9x2 1 y 2 5 45 Add the two equation.

13x2 5 52

x2 5 4 Simplify.

x 5 2 or x 5 –2 Solve for x.

4(2)2 2 y 2 5 7 Substitute x into either equation.

y 2 5 9 Simplify.

y 5 3 or y 5 23 Solve for y.

The solution is the set of ordered pairs (2, 3), (22, 3), (2, 23), (22, 23).

You can also solve some polynomials by quadratic methods even though they may not appear to be quadratic.

EXAMPLE 2 Solve the polynomial equation x 4 2 4 x 2 2 12 5 0.

x 4 2 4x 2 2 12 5 0 The polynomial has n 5 4 roots.

( x 2 ) 2 2 4(x 2) 2 12 5 0 Notice a pattern.

t 2 2 4t 2 12 5 0 Substitute t for x 2.

(t 1 2)(t 2 6) 5 0 Factor.

t 1 2 5 0 or t 2 6 5 0 Zero Product Rule

t 5 22 or t 5 6 Solve for t.

x 2 5 22 or x 2 5 6 Substitute x 2 for t.

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1 Use the discriminant to determine the nature of the roots of 5 x 2 2 2x 1 1 5 0.

A no real roots

B one real root

C two real roots

D one real and one imaginary root

2 A quadratic function y 5 ax2 1 bx 1 c is graphed below. What is true about the coefficients a, b, and c?

x

y

F b 2 2 4ac $ 0 H b 2 2 4ac 5 0

G b 2 2 4ac , 0 J b 2 2 4ac . 0

3 Donna was solving x2 2 2x 2 8 5 0. She drew the graph of y 5 x2 2 2x 2 8 as shown below. What are the solutions of the equation x2 2 2x 2 8 5 0?

x

y

2

21

A 1, 29 C 2, 24

B 9, 21 D 22, 4

4 What are the solutions of the equation (2x 2 3)2 5 36?

F 3 } 2 , 0

G 9 }

2 , 2

3 } 2

H 26, 6

J 39

} 2 , 2

33 } 2

5 What is/are the solution(s) of the equation 2x2 1 5x 5 ]3?

A ]1

B 2 3 } 2 , ]1

C 1

D 2 3 } 2

6 What are the solutions of the equation x2 2 8x 1 4 5 0?

F 4, 24

G 8, 20.5

H 4 6 2 Ï}

3

J 4 6 2 Ï}

5

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10 Which function is graphed below?

4

8

28

y

x

F y 5 x3 1 11x2 1 34x 1 24

G y 5 2x3 1 11x2 2 34x 1 24

H y 5 2x3 2 11x2 2 34x 2 24

J y 5 x3 2 11x2 1 34x 2 24

11 List all of the zeros of the polynomial function graphed.

4

8

8

4

y

x4

A 23, 23, 21, 1

B 23, 21, 9, 1

C 22, 0

D 23, 23, 21, 1, 1

7 What are the solutions of the equation 3x2 2 4x 1 3 5 0?

A 2 } 3 1

Ï}

5 }

3 i and

2 }

3 2

Ï}

5 }

3 i

B 2 2 } 3 1

Ï}

5 }

3 i and 2

2 }

3 2

Ï}

5 }

3 i

C 2 1 Ï}

5 i and 2 2 Ï}

5 i

D 2 } 3 1

Ï}

13 }

3 and

2 }

3 2

Ï}

13 }

3

8 Which values of x are solutions of the equation x3 2 2x2 2 x 1 2 5 0?

F 22, 21, and 1

G 21, 1, and 2

H 21 and 22

J 1, 2, and 3

9 The graph below is related to the polynomial expression that has factors of:

4

8

4

8

448

y

x8

A (x 2 3), (x 1 2), (x 2 1)

B x, (x 1 2), (x 2 3)

C 2x, (x 1 3), (x 2 1)

D (x 1 3), (x 2 2), (x 1 1)

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15 What is/are the solution(s) to the following system?

y 5 (x 2 2)2

y 5 2x2 1 4

A (0, 4)

B (2, 0)

C (0, 4) and (2, 0)

D (4, 0) and (0, 2)

16 What is the solution to the following system?

(x 2 4)2 1 y 5 21 (x 2 4)2 1 (y 2 1)2 5 4

F (1, 4)

G (0, 21)

H (21, 3)

J (4, 21)

12 What are the rational zeros of y 5 2x4 2 7x3 1 2x2 1 8x 2 3?

F 2 3 }

2 and 1

G 21 and 3 }

2

H 21, 2 1 }

2 , and

3 }

2

J 21, 3 }

2 2

Ï}

5 }

2 ,

3 }

2 1

Ï}

5 }

2 , and

3 }

2

13 The function f (x) 5 x4 2 4x3 1 4x2 1 4x 2 5 has the zero 2 1 i. What are all the zeros of the function?

A 2 1 i and 2 2 i

B 21, 1, and 2 1 i

C 1 and 2 1 i

D 21, 1, 2 1 i and 2 2 i

14 Kelly entered the function y 5 x3 2 x2 1 1 into her graphing calculator and looked at the table shown below. Between which two integers is there a solution of the equation x3 2 x2 1 1 5 0?

x y1

22 211

21 21

0 1

1 1

2 5

F 22 and 21

G 21 and 0

H 0 and 1

J 1 and 2

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2 Which is the simplified form of

x

23 }

2 y

1 }

2 }

x 25

} 2 y

3 }

2 ?

F xy H x } y

G x24

} y J 1 }

x4y2

1 What is 1 x2y21 2 3 }

4x23y2z2

in simplest form?

A x8 }

4z2 C

x3 }

4yz2

B x9 }

4y5z2 D

y }

4x3z2

GO ON

SPI 3103.3.3Add, subtract, multiply, divide and simplify rational expressions including those with rational and negative exponents.

Simplifying Complex Fractions

A complex fraction is a fraction that contains a fraction in its numerator or denominator. A complex fraction can be simplified using either of the methods below:

Method 1: If necessary, simplify the numerator and denominator by writing each as a single fraction. Then divide the numerator by the denominator.

Method 2: Multiply the numerator and the denominator by the least common denominator (LCD) of every fraction in the numerator and denominator. Then simplify.

EXAMPLE

Simplify: 6 } x }

1 }

x22 1

4 } x

Solution

6 } x }

1 }

x22 1

4 } x 5

6 } x }

x2 1 4 } x Negative exponent property

5 6 } x }

x2 1 4 } x

x } x Multiply numerator and denominator by the LCD.

5 6 }

x3 1 4 Simplify.

Answer 6 } x }

1 }

x22 1

4 } x 5

6 }

x3 1 4

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3 What is the simplest form of

2x3 1 5x2 2 3x

}} 2x2 2 x 1 6

?

A x(2x 1 1)

} x 1 2

C 2 x(2x 2 1)

} x 2 2

B x(2x 2 1)

} x 2 2 D 2 x(2x 1 1)

} x 1 2

4 Which is a simplified form of

x2 2 y2

} (x 2 y)22 ?

F 2(x 2 y)3( y 1 x)

G ( y 2 x2)(x 2 y2)

H ( y 2 x)4

J (x 1 y)(x 2 y)3

5 Which is the result when 30x24 }

24y2 is divided

by 18x23 }

15y25 ?

A 24y7

} 25x

B 24xy7

} 25

C 25 } 24xy7

D 25x } 24y7

6 Simplify 1 }

x 1 1 2

1 } x 1 1 } 2 .

F 2x2 2 x 2 2 } 2x(x 1 1)

G 2x2 1 3x 2 2 }} 2x(x 1 1)

H x2 1 x 2 2 } 2x(x 1 1)

J 2x2 1 2x 2 1 } 2x(x 1 1)

7 Which is the sum of 3x 2 4 }

x2 2 9 and

2x 2 1 }

x 1 3

?

A 5 } 3

B x 2 1 } x 1 1

C 5x 2 5 } 3x 1 3

D 2x2 2 4x 2 1 }} (x 1 3)(x 2 3)

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8 Simplify 1 }

x 2 4 2

1 } x 1 5

.

F 9 }} (x 2 4) (x 1 5)

G 2x 1 1 }} (x 2 4) (x 1 5)

H 1 }} (x 2 4) (x 1 5)

J 29 }} (x 2 4) (x 1 5)

9 Simplify 2 }

x2 2 x 1

1 } x2 2 2x 1 1

.

A x 1 2 } x(x 2 1)

B 3x 2 2 } x(x 2 1)2

C 2x 2 1 } (x 2 1)2

D 2x 2 1 } x(x 2 1)2

10 Simplify 1 }

x2 2 x 2 12 1

x } x2 2 16

.

F 2x2 2 2x 1 4 }} (x 1 3) (x 2 4) (x 1 4)

G x2 1 4x }} (x 1 3) (x 2 4) (x 1 4)

H x2 2 2x 2 4 }} (x 1 3) (x 2 4) (x 1 4)

J x2 1 4x 1 4 }} (x 1 3) (x 2 4) (x 1 4)

11 Simplify x 1 3 } x2 2 4

? x 1 2 }

x2 1 3x .

A 1 } x

B 1 } x(x 1 2)

C 1 } x(x 2 2)

D 1 } x 2 2

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12 What is the product of x2 2 x 2 20 and

x2 1 5x 1 4

} x2 2 4x 2 5

?

F (x 1 4)2

G (x 2 5)2

H x 1 4

J 1

13 Simplify x }

x2 2 2x 2 24 4

x2 2 x } x 2 6

.

A 1 }} (x 2 1) (x 1 4)

B 1 }} (x 2 1) (x 2 4)

C 1 }} (x 1 1) (x 1 4)

D x2(x 2 1)

}} (x 2 6)2 (x 1 4)

14

Simplify

x }

x 1 1 1 1 }

x 2 1 }

x 1 1 .

F 2x 1 1 } x2 1 x 1 1

G 21 } x2 1 x 2 1

H 2x 1 1 } x2 1 x 2 1

J x 1 2 } x2 1 x 2 1

15

Simplify

3 }

x 2 2 2 5 }

2 2 4 }

x 2 2 .

A 1

B x 2 4 } x 2 2

C x 2

13 } 5 }

x 2 2

D 25x 1 13 } 2(x 2 4)

16 Which is the simplest form of

1 }

x21 1 y21

}

1 }

x21 2 y21 ?

F 1

G y 1 x

H 1 } y 1 x

J y 2 x

} y 1 x

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1 What is a general rule for the nth term in the arithmetic series below?5 1 9 1 13 1 17 1 · · ·

A 4n C 4n 1 2

B 4n 1 1 D 5n

2 Find the sum of the series5 1 10 1 15 1 · · · 1 70.

F 70 H 1015

G 525 J 1050

SPI 3103.3.4Use the formulas for the general term and summation of fi nite arithmetic and both fi nite and infi nite geometric series.

A series can be written using sigma notation (also known as summation notation). The summation symbol Σ is the capital Greek letter sigma.

fi nite series with n terms: a1 1 a2 1 a3 1 ... 1 an 2 1 1 an 5 o i 5 1

n

ai 5 Sn

infi nite series: a1 1 a2 1 a3 1 a4 1 ... 5 o i 5 1

`

ai

The general formula for a geometric sequence is an 5 a1 ? r n21, where r is the common ratio.

To fi nd the sum of a fi nite geometric series with n terms, use the formula

Sn 5 a1 1 12 rn }

12 r 2

where n is the number of terms, a1 is the fi rst term, and an is the nth term.

EXAMPLE

Write the infi nite series 162 2 54 1 18 2 6 1 … in sigma notation and fi nd the sum to n terms.

Solution

The fi rst term is a1 5 162 and the common ratio is r 5 2 1 } 3 . The general term is ai 5 162 p 1 2

1 } 3 2 i 2 1

,

so in sigma notation the infi nite series is

162 2 54 1 18 2 6 1 ... 5 o i 5 1

`

162 p 1 2 1 } 3 2 i 2 1

To fi nd the sum to n terms, substitute a1 = 162 and r = 2 1 } 3 into the formula for Sn.

Sn 5 162 1 1 2 1 2 1 } 3 2

n

} 1 2 1 2

1 } 3 2

2 5 162 1 1 2 1 2 1 } 3 2

n

} 4 }

3 2 5

243 }

2 1 1 2 1 2

1 } 3 2

n 2


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6 Which summation formula correctly represents the arithmetic series shown below? (23) 1 (26) 1 (29) 1 · · ·; n 5 5

F � n51

3

25n H � n51

5

23n

G � n51

5

3n J � n51

5

26n

7 Which infi nite geometric series does not have a sum?

A � n51

¥

5 1 1 __ 4 2 n

B � n51

¥

1 12 3 } 4 2

n

C � n51

¥

2 }

3 1 8 } 7 2

n

D � n51

`

1 }

3 1 9 } 10

2 n

8 Write a formula for the nth term of the sequence below.

22, 1, 2 1 } 2 ,

1 }

4 , 2

1 } 8 , ...

F 22 1 2 1 } 2 2

n

G 22 1 2 1 } 2 2 n 2 1

H (22)n

J (22)n 2 1

3 Find � n51

5

1 1 __ 4 2 n .

A 123 _____ 1024

B 341 _____ 1024

C 1023 _____ 1024

D 1

4 What is a16 for the arithmetic sequence0.8, 2.8, 4.8, ...?

F 28.8

G 30.8

H 32.8

J 34.8

5 What is the sum of the infi nite geometric series 0.3 1 0.03 1 0.003 1 · · ·?

A 3

B 0.5

C 1

D 1 } 3


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1 What is the domain of the function

f (x) 5 3 } x 1 6 ?

A { f 1 x 2 : x Þ 0}

B {x: x Þ 6}

C { f 1 x 2 : x Þ 23}

D {x: x Þ 26}

SPI 3103.3.5Describe the domain and range of functions and articulate restrictions imposed either by the operations or by the contextual situations which the functions represent.

2 What is the range of the function f (x) 5 22(x 2 3)2 1 1?

F f (x) $ 1

G f (x) # 1

H f (x) # 3

J 2` , f (x) , `

A function is a set of numbers where each x value has exactly one corresponding y value. The values of the independent variable constitute the domain of the function and the values of the dependent variable constitute the range of the function.

EXAMPLE 1

Identify the range of the function represented by this graph.

Solution

The range is the set of y values. For this function, 26 # y # 22.

In a real-world situation, the values of the domain or range may need to be restricted.

EXAMPLE 2

A rental car company charges 25 cents per mile plus a flat fee of $30. The customer cannot drive the car more than 100 miles. The charges to a customer who drives x miles is given by C(x) 5 30 1 0.25x. Describe the domain and range of C(x).

Solution

The customer cannot drive a negative number of miles x, so x $ 0. And the customer cannot drive more than 100 miles, so x # 100. Therefore, the domain is 0 # x # 100.

For a customer who has driven the minimum of 0 miles, the cost is C(0), or the flat fee of $30. For a customer who has driven the maximum of 100 miles, the cost is C(100), or $55. Since the cost C(x) is increasing on the interval 0 # x # 100, the cost varies from $30 to $55. Therefore, the range is 30 # y # 55.

x

y

2121

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3 Rosie threw a ball into the air with an initial velocity of 32 feet per second. The height y of the ball after t seconds is given by the function y 5 32t 2 16t 2. The graph of height versus time is shown. What is a reasonable range for this function?

Heig

ht

(ft)

Time (sec)

t

y

4

00 1 2

8

12

16

A All real numbers

B y # 16

C 0 # y # 16

D 216 # y # 16

4 What is the domain of the function represented by this graph?

x

y

21

1

F 22 # x # 2

G 22 # y # 2

H 1 # x # 5

J 1 # y # 5

5 The corners of a square piece of cardboard are cut out and the sides folded up to form an open box as shown in the figure below.

12 in.

x

x

x

xx x

x x

12 in.

The area of the bottom of the box is given by the function A 5 (12 2 2x)2. What is a reasonable domain for this function?

A 0 # x , 144

B 26 , x , 6

C 0 , x , 12

D 0 , x , 6

6 What are the domain and range of the piece-wise function below?

f (x) 5 { x2, 21 # x , 2

3x 2 2, x $ 2

F Domain: {x| x $ 21}

Range: { f (x)| f (x) $ 0}

G Domain: {x| 21 # x , 2}

Range: { f (x)| 0 # f (x) # 4}

H Domain: {x| x $ 2}

Range: { f (x)| f (x) $ 0}

J Domain: {x| x $ –1}

Range: { f (x)| –` , f (x) , `}

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SPI 3103.3.6Combine functions (such as polynomial, rational, radical and absolute value expressions) by addition, subtraction, multiplication, division, or by composition and evaluate at specifi ed values of their variables.

Let f and g be any two functions. A new function h can be defi ned by performing these four basic operations on f and g.

Operation Defi nition

Addition h(x) 5 f (x) 1 g(x)

Subtraction h(x) 5 f (x) 2 g(x)

Multiplication h(x) 5 f (x) p g(x)

Division h(x) 5 f (x)

} g(x)

EXAMPLE 1

Let f (x) 5 5 2 x1/2 and g(x) 5 22x1/2 1 3. Find the following:

a. f (x) 1 g(x) b. f (x) 2 g(x) c. f (x) p g(x) d. f (x)

} g(x)

Solution

a. f (x) 1 g(x) 5 (5 2 x1/2) 1 (22x1/2) 5 5 2 x1/2 22x1/2 5 5 2 3 x1/2

b. f (x) 2 g(x) 5 (5 2 x1/2) 2 (22x1/2) 5 5 2 x1/2 1 2x1/2 5 5 1 x1/2

c. f (x) p g(x) 5 (5 2 x1/2)(22x1/2) 5 5(22x1/2) 2 x1/2(22x1/2) 5 210x1/2 1 2x

d. f (x)

} g(x)

5 5 2 x1/2

} 22x1/2

5 5

} 22x1/2 2

x1/ 2 }

22x1/2 5 2

5 } 2 x21/2 1

1 } 2

Another operation is the composition of functions. The composition of a function g with a function f is h(x) 5 g(f (x)).

EXAMPLE 2

Let f (x) 5 3x21 and g(x) 5 x 1 4. Find the following:

a. f ( g(x)) b. g( f (x))

Solution

a. f ( g (x)) 5 f (x 1 4) 5 3(x 1 4)21 5 3 }

x 1 4

b. g( f (x)) 5 g (3x21) 5 (3x21) 1 4 5 3 } x 1 4


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1 If f (x) 5 7 2 x and g(x) 5 2(5x 1 3)2, which is an equivalent form of f (x) 1 g(x)?

A 25x2 1 16x 1 9

B 25x2 1 15x 1 16

C 50x2 1 59x 1 25

D 50x2 2 760x 1 2888

2 If f (x) 5 1 1 x2/3 and g(x) 5 x 1 8, perform the following operation.f (x) p g(x)

F x5/3 1 8x2/3 1 x 1 8

G x5/3 1 16x2/3 1 x 2 8

H x2/3 1 16x 1 65

J x2/3 2 x 2 7

3 If f (x) 5 2 }

x 2 1 and g(x) 5

x } x 1 1

, what is

g(x) 2 f (x)?

A x2 1 x 1 2 } x2 2 1

B x2 2 3x 1 2 }

x2 2 1

C x2 2 3x 2 2 }

x2 2 1

D 2x2 1 3x 1 2 }} x2 2 1

4 Let f (x) 5 2x4 2 7x3 2 8x2 1 14x 1 8

and g(x) 5 x 2 4. Find f (x)

} g (x)

.

F 2x3 1 x2 1 4x

G x2 2 4x 2 2

H 2x3 2 4x 2 2

J 2x3 1 x2 2 4x 2 2

5 Suppose f and g are functions with domain all real numbers. Which of the following statements is true about the equation?

f (x) 2 g(x) 5 g(x) 2 f (x)

A The equation is always true.

B The equation is true except when x is negative.

C The equation is never true.

D The equation is true when g(x) 5 f (x).

6 If f (x) 5 Ï}

x 2 4 and g(x) 5 x 2 2, what

is the domain of 1 g } f 2 (x) ?

F (4, `)

G (24, `)

H (2`, 4)

J (2`, `)


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7 Use the graphs of the functions f and g fi nd ( f p g) (0).

x

y

O 22428216 212

28

4

8

12

24

f

g

y 5 x

1 2

y 5 2 x 2 4

12

A 28

B 24

C 22

D 0

8 Let f (x) 5 x2 and g(x) 5 (2x 2 5)2. Which of the expressions below results in the following function?

h(x) 5 4x4 2 20x3 1 25x2

F f (x) 1 g(x)

G g(x)

} f (x)

H f (x) p g(x)

J g(x) 2 f (x)

9 Which expression represents f (g(x)) iff (x) 5 x3 1 1 and g(x) 5 5x 1 2?

A (5x 1 2)3 1 1

B 5x3 1 2

C x3 1 5x 1 3

D x3 2 5x 2 1

10 Let f (x) 5 (x2 1 3) and g(x) 5 3x2. To fi nd g( f (x)), Sundie wrote the following solution on the board. Her fi nal answer is incorrect.

g ( f (x)) 5 f (3x2)

5 ((3x2 ) 2 1 3)

5 9x4 1 3

Which statement describes her error?

F She should have added the functions.

G She should have multiplied the functions.

H She found f (g(x)) instead of g( f (x)).

J The entire expression (3x2 1 3) should be squared.


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11 If f (x) 5 2x 2 1 and g(x) 5 |x|, which expression represents (g p f )(x)?

A 2|x| 2 1

B |2x 2 1|

C 2x 1 1

D 2x2 2 |x|

12 Let f (x) 5 5x 2 1 and g(x) 5 (x 1 1)2. What is g( f (4))?

F 19

G 25

H 250

J 400

13 Let f (x) 5 x2 1 3 and g(x) 5 x 1 5. Find g( f (2)) 2 g(3).

A 4

B 44

C 144

D 444

14 If f (g(x)) 5 1 }

Ï}

2x 2 3 , which of the

following could be true?

F f (x) 5 1 } x and g(x) 5

1 }

Ï}

2x 2 3

G f (x) 5 1 } x and g(x) 5 Ï

}

2x 2 3

H f (x) 5 1 }

Ï}

2x and g(x) 5 Ï

}

x 2 3

J f (x) 5 Ï}

2x 2 3 and g(x) 5 1 } x

15 If f (x) 5 x2 and g(x) 5 Ï}

1 2 x2 , what is the domain of f (g(x))?

A [21, 1]

B (21, 1)

C (2`, 21]

D (2`, `)

16 Let f (x) 5 Ï} x . Find the domain of

f ( f 21 (x)).

F all real numbers

G all real numbers $ 0

H all real numbers . 1

J all real numbers . 2


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1 Suppose f and g are functions with domain all real numbers. Which of the following statements is true about g(x) and f (x) given the following?

f (g(x)) 5 g( f (x)) 5 x

A g(x) is equal to f (x).

B g(x) and f (x) are inverse functions.

C g(x) and f (x) are additive inverses.

D g(x) and f (x) are multiplicative inverses.

2 Which of the following pairs of functions are not inverses?

F f (x) 5 x3 1 2 and g(x) 5 3 Ï}

x 2 2

G f (x) 5 1 1 } 2 x 1 2 2 2 and g(x) 5 62 Ï} x 2 4

H f (x) 5 2 Ï} x and g(x) 5 1 } 4 x2

J f (x) 5 x } x 1 1 and g(x) 5 x

} x 2 1

SPI 3103.3.7Identify whether a function has an inverse, whether two functions are inverses of each other, and/or explain why their graphs are reflections over the line y = x.

A relation is a pairing of input values with output values. An inverse relation switches the input and output values of the original relation. So, if (x, y) is an ordered pair of a relation, then ( y, x) is an ordered pair of the inverse relation. The graph of the inverse relation is a reflection of the graph of the original relation. The line of reflection is y 5 x.

The relation g(x) is the inverse of relation f (x) if f (g(x)) 5 g( f (x)) 5 x. When both the original relation f (x) and the inverse relation g(x) are functions, we write g(x) 5 f 21(x). You can check that the inverse of function f (x) is also a function by verifying that no horizontal line intersects the graph of f (x) more than once. (This is called the horizontal line test.) Without a graph, you could instead check that relation f (x) does not assign the same range element to two different domain elements.

EXAMPLE

Verify that f (x) 5 x2 1 2 has the inverse relation

g(x) 5 6 Ï}

x 2 2 , but that g(x) is not a function.

Solution

f (g(x)) 5 (g(x))2 1 25 x 2 2 1 2 5 x

g ( f (x)) 5 6 Ï}}

(x2 1 2) – 2 5 6 Ï}

x2 5 x

Both equal x, so g(x) is the inverse of relation f (x). But on the graph of f (x) 5 x2 1 2, some horizontal lines would cut the graph at two points. For instance, the points (2, 6) and (22, 6) on the graph assign the same range element 6 to two different domain elements, 2 and 22. Therefore, the inverse g(x) is not a function.

y

x4 5 63121

22

1

3

4

5

6

22 O

y 5 x2 1 2

y 5 6 x 2 2

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3 What is the inverse function of f (x) 5 x }

x 1 1 ?

A g(x) 5 x } 1 1 x

C g(x) 5 1 1 x } x

B g(x) 5 x } 1 2 x

D g(x) 5 1 2 x } x

4 What is the inverse function of

f (x) 5 5 2 2x

} 3 ?

7 Which is the graph of the function y 2 2x 5 4 and its inverse?

A

2

4

2

4

224

y

x4

B

4

2

2

4

24

y

x4

C

4

2

4

224

y

x4

D

2

4

424

y

x

F f 21 (x) 5 3 }

5 2 2x

G f 21 (x) 5 5 2 3x

} 2

H f 21 (x) 5 3x 1 5

} 2

J f 21 (x) 5 3x 2 5

} 2

5 What is the inverse function of f (x) 5 1 1 Ï

}

1 1 x ?

A g(x) 5 2 1 Ï} x

B g(x) 5 2 Ï}

1 1 x

C g(x) 5 x2 2 2x

D g(x) 5 x2 Ï}

1 1 x

6 What is f (x) if f 21(x) 5 log2 (x 1 1)?

F f (x) 5 2x 2 1

G f (x) 5 2x 1 1

H f (x) 5 2 (x 2 1)

J f (x) 5 2 (x 1 1)

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SPI 3103.3.8Solve systems of three linear equations in three variables.

Two or more equations are called a system. Here are the steps to solve a linear system in three variables:

Step 1 Rewrite the system as a linear system in two variables by eliminating or substituting for one variable.

Step 2 Solve the system from Step 1 for each variable.

Step 3 Substitute the values from Step 2 into an equation in the original system to find the value of the third variable.

EXAMPLE Solve the system using the substitution method.

x 1 2y 1 z 5 9 Equation 1

22x 2 y 1 4z 5 3 Equation 2

2x 1 y 1 z 5 8 Equation 3

Solution

Step 1 Solve Equation 3 for x.

x 5 y 1 z 2 8

Step 2 Rewrite the system as a linear system in two variables by substituting y 1 z2 8 for x in Equations 1 and 2.

x 1 2y 1 z 5 9 Write Equation 1.

(y 1 z 2 8) 1 2y 1 z 5 9 Substitute y 1 z 2 8 for x.

3y 1 2z 5 17 New Equation 1.

22x 2 y 1 4z 5 3 Write Equation 2.

22(y 1 z 2 8) 2 y 1 4z 5 3 Substitute y 1 z 2 8 for x.

23y 1 2z 5 213 New Equation 2.

Step 3 Solve the system in two variables from Step 2.

3y 1 2z 5 17 Add New Equation 1 to New Equation 2 23y 1 2z 5 213

4z 5 4 Simplify.

z 5 1 Solve for z.

y 5 5 Substitute z into New Equation 1 or 2 to find y.

x 5 22 Substitute y and z into an original Equation to find x.

Answer The solution is (22, 5, 1).

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5 What is the solution to the system of equations shown below?

x 1 3y 2 z 5 24 2x 2 y 1 2z 5 13 3x 2 2y 2 z 5 29

A (0, 1, 7) C (22.5, 23, 7.5)

B (210, 23, 15) D (2, 23, 3)


4x 2 3y 1 z 5 28 22x 1 y 2 3z 5 24 x 2 y 1 2z 5 3

F (1, 22, 4) H (22, 3, 1)

G (21, 4, 22) J (22, 1, 3)

7 Andrea, Bianca, and Cody bake muffins for the Drama Club’s baked-goods sale. Bianca bakes 2 fewer muffins than Andre. Cody bakes 4 more muffins than Bianca. They bake 54 muffins all together. Which system models the problem?

A b 5 a 1 2 c 5 b 2 4 a 1 b 1 c 5 54

B b 5 a 1 2 c 5 b 1 4 a 1 b 1 c 5 54

C b 5 a 2 2 c 5 b 1 4 a 1 b 1 c 5 54

D b 5 a 2 2 c 5 b 2 4 a 1 b 1 c 5 54

GO ON

1 The augmented matrix for a system of equations is given below.

F 4 1 1 1 22

2 23 0

1 * 22 4 3 G

What is the solution of the system?

A (21, 2, 3) C (22, 4, 3)

B (2, 21, 3) D (22, 23, 0)


x 1 2y 1 z 5 9 22x 2 y 1 4z 5 3 2x 1 y 1 z 5 8

F (23, 11, 2) H (21.5, 4, 2.5)

G (212, 1, 25) J (22, 5, 1)


x 1 2z 5 28 x 1 3y 2 z 5 10 2x 2 y 1 z 5 22

A (2, 1, 25) C (22, 21, 5)

B (2, 25, 1) D (22, 25, 3)

4 Find the solution to the system of linear equations.

2x 2 2y 1 z 5 7 x 1 2y 2 3z 5 22 x 1 y 1 2z 5 24

F (21, 1, 3) H (3, 21, 1)

G (1, 23, 21) J (21, 23, 1)

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To graph the solution set of a system of linear or quadratic inequalities in the plane, first ignore the inequality sign and graph each equation in the plane. Remember that the graph of the line y 5 mx 1 b has a y-intercept of b; substitute a value for x to find the y-coordinate of another point on the line. The table below is a review of how to graph quadratic equations.

Properties of Quadratic Graphs

y 5 ax2 y 5 ax2 1 c y 5 ax2 + bx 1 c y 5 a(x 2 b)2 1 c

Vertex (0, 0) (0, c) 1 2 b }

2a , f 1 2

b }

2a 2 2 (b, c)

Axis of Symmetry y-axis y-axis x 5 2

b }

2a x 5 b

Direction of Opening Graph opens up if a > 0 and opens down if a < 0.

y-intercept 0 c c c

To graph the solution set of a system of inequalities, follow these steps:

Step 1 Solve each inequality in terms of y. Graph each inequality in the same plane. Use a solid line or curve for $ and # inequalities; use a dashed line or curve for . and , inequalities.

Step 2 Shade above the line or curve for . and $ inequalities; shade below for , and # inequalities.

Step 3 Identify the region that is common to all the graphs of the inequalities. This region is the graph of the system.

EXAMPLE

Solve the system and graph the solution.

4x 1 y . 9 x 2 2y # 6

In terms of y, the system is: y . 9 2 4x

y $ 23 1 1 } 2 x

The y-intercept of the first line is 9. Another point on the first line is (4, 27). The y-intercept of the second line is 23. Another point on the second line is (6, 0).

Graph the boundary lines. Shade above the dashed line y . 9 2 4x

because of the . sign; shade above the solid line y $ 23 1 1 } 2 xbecause of the $ sign. The solution to the system is the darker area where the graphs overlap.

SPI 3103.3.9Graph the solution set of two or three linear or quadratic inequalities.

y

x82 628 26 24 22

28

26

8

6

4

2

O

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y

x2

2

A x 1 2y $ 10 x 1 y , 3

B x 1 2y $ 10 x 1 y . 3

C x 1 2y # 10 x 1 y , 3

D x 1 2y # 10 x 1 y . 3


y

x

F y , 2x 1 5 y 1 2x . 21

G y . 2x 1 5 y 1 2x . 21

H y , 2x 1 5 y 1 2x , 21

J y . 2x 1 5 y 1 2x , 21

1 Which system of linear inequalities is represented by the graph?

6543

2122

2223242526

23242526

1 2 3 4 5 6

y

x

A 2x 1 y , 5 x 1 y # 1

B y . 22x 1 1 y # x 1 1

C y . 2x 1 1 y $ 0

D y . 2x 1 1 y # x 1 1


1

y

x21

3

F y $ 3x 1 2 H y $ 1 } 3 x 1 2

y # 3x 2 1 y $

1 } 3 x 2 1

G y $ 3x 1 2 J y # 3x 1 2

y # 3x 2 1 y $ 3x 2 1

GO ON

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SPI 3103.3.10Identify and/or graph a variety of functions and their transformations.

1 How does the graph of y 5 1 } x 2 3 relate

to the graph of y 5 1 } x ?

A It shifts y 5 1 } x right 3 units.

B It shifts y 5 1 } x left 3 units.

C It shifts y 5 1 } x up 3 units.

D It shifts y 5 1 } x down 3 units.

Polynomial Functions

A polynomial function in the form f (x) 5 a(x 2 h)n 1 k is a transformation of the parent function f (x) 5 x n. The graph is stretched or compressed by a factor of |a|, translated h units horizontally, and translated k units vertically.

Exponential Functions

An exponential function in the form f (x) 5 abx 1 c is a transformation of the parent function f (x) 5 b x. The graph is stretched by a factor of |a|. If a is negative, the function is refl ected about the x-axis. Notice that if the parent function of f is increasing and a , 0, f is decreasing. The value of c translates the graph c units vertically.

Absolute Value Functions

An absolute value function in the form f (x) 5 a| x2h | 1 k is a transformation of the parent function, f (x) 5 | x| . The graph is translated h units horizontally and k units vertically. The vertex is (h, k).

EXAMPLE

For | a | . 1The graph is vertically stretched, or elongated.

The graph of y 5 a| x | is narrower than the graph of y 5 | x |.

For 0 , | a | , 1The graph is vertically shrunk, or compressed.

The graph of y 5 a| x | is wider than the graph of y 5 | x |.

For a 5 21 The graph of y 5 a| x | is a refl ection in the x-axis of the graph of y 5 | x |.

2 How does the graph of y 5 3 } x relate

to the graph of y 5 1 } x ?

F It shrinks y 5 1 } x vertically by a

factor of 1 }

3 .

G It stretches y 5 1 } x vertically by a

factor of 3.

H It shifts y 5 1 } x right 3 units.

J It shifts y 5 1 } x up 3 units.


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3 Which graph could show the functions y 5 x2 1 2 and y 5 x2 2 4?

A

x

y

B

x

y

C

x

y

D

x

y

4 Which graph could show the functions y 5 2x 1 1 and y 5 2x 2 1?

F

x

y

2 4 6

24

22

6

42

O

8

10

22

G

x

y

2 4 6242628

24

22

6

4

2

O

8

10

22

H

x

y

2 4 6242628

24

22

6

4

2

O

8

10

22

J

x

y

2 4 6242628

24

22

6

4

2

O

8

10

22


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5 Which is the equation of the resulting function when the graph of y 5 x2 1 5 is shifted down seven units?

A y 5 x2 2 2 C y 5 x2 2 12

B y 5 x2 2 7 D y 5 x2 1 12

6 Which is the equation of the resulting function when the graph of y 5 x2 is shifted left four units?

F y 5 x2 2 4 H y 5 (x 1 4)2

G y 5 x2 1 4 J y 5 (x 2 4)2

7 The graph of y 5 2x is shown below.

x

y

2 4 6242628

24

22

6

4

2

O

8

10

22

Which of the following could be the graph of y 5 22x?

x

y

2 4 6

24

22

26

6

42

8

22

D

A

C

B

A graph A C graph C

B graph B D graph D

8 The graph of f (x) 5 2) x ) 1 4 is shown.

x

y

4 824

28

24

O

4

Which of the following is the graph of 2f (x)?

F

x

y

4 824

28

24

O

4

G

x

y

4 824

28

24

O

4

H

x

y

4 824

28

24

O

4

J

x

y

4 824

24

O

4

8


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10 What is the effect on the graph of the function y 5 2x2 1 8 when c is changed to 6?

F It shifts to the right.

G It shifts to the left.

H It shifts up.

J It shifts down.

11 Which of the following sentences is true about the graphs of y 5 2(x 1 7)2 1 1 and y 5 2(x 2 7)2 1 1?

A Their vertices are maximums.

B The graphs have the same shape with different vertices.

C The graphs have different shapes with different vertices.

D One graph has a vertex that is a maximum while the other graph has a vertex that is a minimum.

9 Which is the graph of y 5 22x ?

A y

x

2

1

B y

x

3

1

C y

x1

1

D y

x1

1

GO ON


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SPI 3103.3.11Graph conic sections (circles, parabolas, ellipses and hyperbolas) and understand the relationship between the standard form and the key characteristics of the graph.

Circle The equation of a circle with center (h, k) and radius r units is (x 2 h)2 1 (y 2 k)2 5 r 2

Parabola Standard Form y 5 a (x 2 h)2 1 k x 5 a (y 2 k)2 1 h

Vertex (h, k) (h, k)

Axis of Symmetry x 5 h y 5 k

Focus 1 h, k 1 1 } 4a 2 1 h 1

1 } 4a , k 2

Directrix y 5 k 2 1 } 4a x 5 h 2

1 } 4a

Ellipse

c2 5 a2 2 b2

Standard Form (x 2 h)2

} a2

1 (y 2 k)2

} b2 5 1

(y 2 k)2

} a2

1 (x 2 h)2

} b2 5 1

Direction of Major Axis horizontal vertical

Foci (h 6 c, k) (h, k 6 c)

Length of Major Axis 2a units 2a units

Length of Minor Axis 2b units 2b units

Hyperbola

c2 5 a2 1 b2

Standard Form (x 2 h)2

} a2

2 (y 2 k)2

} b2 5 1

(y 2 k)2

} a2 2

(x 2 h)2

} b2 5 1

Direction of Transverse Axis

horizontal vertical

Foci (h 6 c, k) (h, k 6 c)

Vertices (h 6 a, k) (h, k 6 a)

Length of Transverse Axis 2a units 2a units

Length of Conjugate Axis 2b units 2b units

Asymptotes y 5 6 b } a (x 2 h) 1 k y 5 6

a }

b (x 2 h) 1 k

1 Find the equations of the asymptotes of

the hyperbola y2

} 5 2 x2

} 4 5 1.

A y 5 Ï

}

2 } 5 x; y 5 2

Ï}

2 } 5 x

B y 5 Ï

} 5 }

2 x; y 5 2

Ï}

5 } 2 x

C y 5 2 }

Ï}

5 x; y 5 2

2 }

Ï}

5 x

D y 5 5 }

Ï}

2 x; y 5 2

5 }

Ï}

2 x

2 The graph of x2

} 19

1 y2

} 100

5 1 is an ellipse.

Which set of points represents the foci of the ellipse?

F (0, 69) H (6 Ï}

6 , 0)

G (69, 0) J (0, 6 Ï}

6 )


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3 Which equation represents the graph of a circle with center (21, 22) and radius 8?

A x2 2 2x 1 y2 2 4y 2 69 5 0

B x2 1 2x 1 y2 1 4y 2 69 5 0

C x2 1 2x 1 y2 1 4y 2 59 5 0

D x2 2 2x 1 y2 2 4y 2 59 5 0

4 Find the vertex and axis of symmetry for

the function y 5 1 }

3 (x 1 5) 2 2 2.

F vertex (22, 5);axis of symmetry x 5 22

G vertex (2, 25);axis of symmetry x 5 2

H vertex (25, 22);axis of symmetry x 5 25

J vertex (25, 2);axis of symmetry x 5 25

GO ON

5 Which of the following functions is graphed below?

y

x

8

(�5, 0)(0, 4)

(5, 0)2

(0, �4)

(0,��41)

(0, ��41)

A x2 }

16 2

y2

} 25

5 1

B y2

} 16

2 x2

} 25

5 1

C x2 }

4 2

y2

} 5 5 1

D y2

} 4 2

x2 } 5 5 1

6 Which of the following functions is graphed below?

21

(11Ïw2, 23)

1

y

x

(12Ïw2, 23)

(1, 21)

(1, 23)

(1, 25)

F (x 1 1)2

} 4 1

(y 2 3)2

} 2 5 1

G (x 2 1)2

} 4 1

(y 1 3)2

} 2 5 1

H (x 2 1)2

} 2 1

(y 1 3)2

} 4 5 1

J (x 1 1)2

} 2 1

(y 2 3)2

} 4 5 1


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SPI 3103.3.12Interpret graphs that depict real-world phenomena.

To analyze graphs and draw conclusions, you must first identify the quantities involved in the problem and how they are related to each other. It is important to determine what the intercepts of the graph represent in a given situation, and it is also important to determine what the minimum or maximum value of the graph represents. Pay close attention to the units and scales used on the graph’s axes.

EXAMPLE

This graph shows the height of a baseball from the time it is thrown until it hits the ground. After how many seconds does the ball first reach a height of 20 feet?

Heig

ht

(ft)

Time (s)

x

y

Solution

Look at the graph. Find 20 on the y-axis since the height in feet is shown along this axis.

Notice that the graph has 2 points where the y-coordinate is 20, one on the way up and one on the way down. We are looking for the first time the ball reaches 20 feet, so find the value of the x-coordinate at the first point where the y-coordinate is 20. This x-value is about 0.75. Therefore, the ball first reaches a height of 20 feet after about 0.75 seconds.

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4 A model of the cost per guest charged by a caterer for a birthday dinner is shown in the graph below.

20

40

60

80

20 40 60 80Number of guests

Co

st

per

gu

est

0

0x

y

If the party host wants to keep the average cost below $20 per guest, what is the fewest number of guests he can have at the party?

F about 20 H about 60

G about 40 J about 80

3 A company’s monthly profits are modeled in the graph below.

200

400

600

800

20 40 60 80Advertising ($1000’s)

Pro

fit

($1000’s

)

0

0x

y

What amount should the company spend on advertising to maximize its profit?

A $35,000 C $200,000

B $75,000 D $815,000

GO ON

1 Steven is the quarterback on the football team. He throws the ball to the receiver. The ball is thrown and caught 5 feet above the ground. The path of the ball is part of a parabola as shown.

Heig

ht

(ft)

010 20 30 40 4550 15 25 35

20

15

10

5

Distance (ft)

x

y

What is the maximum height the ball reaches?

A 15 1 }

2 ft C 20

1 }

2 ft

B 17 ft D 25 ft

2 Samantha threw a paper airplane into the air. The flight path was part of a parabola, as shown below.

Heig

ht

(ft)

10 2 3 4 95 6 7 8

Distance (ft)

x

y

8

10

12

14

16

18

4

6

2

0

What was the maximum height the plane reached?

F 4 ft H 20 ft

G 9 ft J 24 ft

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SPI 3103.3.13Solve contextual problems using quadratic, rational, radical and exponen-tial equations, fi nite geometric series or systems of equations.

1 An electric cable is strung from a power pole to a building. The height y, in feet, of the cable x feet from the pole is approximated by the function y 5 x2 2 8x 1 30. What is the closest that the cable comes to the ground?

A 30 ft

B 23 ft

C 14 ft

D 8 ft

2 Ramona launches a model rocket from the ground. She calculates the height of the rocket t seconds after launch using the function y 5 216t2 1 128t, where y is measured in feet above the ground. If she were to move the rocket and the launcher to the top of a 400-foot building, what function would she then use to calculate the height of the rocket above the ground?

F y 5 216(t 2 400)2 1 128(t 2 400)

G y 5 216(t 1 400)2 1 128(t 1 400)

H y 5 216t2 1 128t 1 400

J y 5 216t2 1 128t 2 400

Exponential functions can be used to model many real-world problems involving exponential growth or decay, such as compound interest and depreciation. In general, exponential growth or decay can be modeled with the equation y 5 a(1 1 r)t where a is the initial amount, r is the percent increase or decrease, and t is the time.

The formula for compound interest is A 5 P 1 1 1 r } n 2 nt

, where A is the account balance, P is the

principal deposited, r is the annual interest rate, n is the number of times compounded annually, and t is the time in years.

EXAMPLE

$500 is invested in an account that pays 4% annual interest, compounded monthly. What is the balance in the account after 8 years?

Solution

A 5 P 1 1 1 r } n 2 nt

5 500 1 1 1 0.04

} 12

2 12 • 8

< 688.20

The balance in the account will be $688.20.


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5 A store is having a drawing at its open house. The prizes are 16 gift cards worth a total of $625. Some gift cards are worth $25 and some are worth $50. Which system could represent this situation?

A 25x 1 50y 5 16 25x 1 50y 5 625

B x 1 y 5 16x 1 y 5 625

C x 1 y 5 625 25x 1 50y 5 16

D x 1 y 5 1625x 1 50y 5 625

6 Mr. Haynes deposits $5000 in an account that pays 3.75% annual interest. Which equation gives the amount in the account after t years if interest is compounded annually?

F A 5 5000(0.0375)t

G A 5 5000(1 1 0.0375)t

H A 5 5000 1 1 1 0.0375 } 2 2 2t

J A 5 5000 1 1 1 0.0375 } 12

2 12t

7 Which equation models the value after t years of a boat that was purchased new for $23,000 and depreciates 12% each year?

A y 5 212(23,000)t

B y 5 23,000(0.88)t

C y 5 23,000(0.12)t

D y 5 23,000(1.12)t

3 Grace drops a tennis ball from the top of a building. The height of the ball as it falls can be found using the equation h(t) 5 216t2 1 200, where t is the time measured in seconds and h(t) is the height measured in feet. Which best describes the change in Grace’s situation if the equation is changed to h(t) 5 216t2 1 161?

A Grace drops the ball 39 seconds sooner.

B Grace drops the ball with less velocity.

C When Grace drops the ball, it falls 39 feet.

D Grace drops the ball from a building that is 161 feet tall.

4 Steve invested $15,000 in two different interest-earning accounts. The fi rst earned 8% per year while the other only earned 5.5%. After a year, he earned a total of $1055 in interest from the two accounts combined. Which system of equations could you use to fi nd how much Steve invested in each account?

F { x 1 y 5 1055

8x 1 5.5y 5 15,000

G

{ x 1 y 5 15,0008x 1 5.5y 5 1055

H

{ x 1 y 5 15,0000.08x 1 0.055y 5 1055

J { x 1 y 5 15,000

1.08x 1 1.055y 5 1055

GO ON


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SPI 3103.3.14Solve problems involving the binomial theorem and its connection to Pascal’s Triangle, combinatorics, and probability.

Binomial Theorem

If nCr 5 n! }

(n 2 r)!r! , then (x 1 y)n 5 nC0x ny 0 1 nC1x n 2 1y1 1 ... 1 nCn 2 1 x1y n 2 1 1 nCnx0y n

EXAMPLE 1

In the expansion of (a 1 b)5, what is the coefficient of the a3b2 term?

Solution

Apply the Binomial Theorem with n 5 5. Since in a3b2 the power on b is 2, use r 5 2 to find the coefficient.

5C2 5 5! }

3! 2! 5

5 4 3! }

3! 2 1 5 10

In a binomial experiment, a single event with two possible outcomes is repeated several times. The outcome of the experiment is the number of times each type of outcome occurs. For example, in spinning a spinner with dark and light sections 5 times, one possible outcome for the experiment is landing on 3 dark sections and 2 light sections. If the probability of landing on a dark section is p, the probability of each possible outcome of n trials of the binomial experiment is given by the terms of the expanded polynomial expression

( p 1 (1 2 p))n 5 nC0 pn(1 2 p)0 1 nC1pn 2 1(1 2 p)1 1 ... 1 nCn 2 1p1(1 2 p)n 2 1 1 nCn p0(1 2 p)n

where the first term is the probability of landing on exactly n dark sections and 0 light sections, the second term is the probability of landing on exactly n 2 1 dark sections and 1 light section, and so on.

EXAMPLE 2The spinner shown is spun 5 times. What is the probability that exactly 3 of those spins will land on the dark section?

Solution

If 5 spins have exactly 3 that land in the dark section, then 2 land in a light section. One factor of the probability is p3( p 2 1)2. Because the exponent on ( p 2 1) is 2, this term also has a factor of 5C2. Using the result of the preceding example for 5C2,

5C2 1 1 } 3 2

3 1 1 2

1 }

3 2 2 5 10 1 1 }

3 2 3 1 2 }

3 2 2 5 10

22 }

35 5 40

} 243

ø 0.165

The probability that of 5 spins exactly 3 will land in the dark section is about 16.5%.

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1 You must correctly choose 6 numbers, each an integer from 1 to 49, to win a raffle. What is the probability of winning the raffle? Assume the numbers are chosen randomly. The order of the winning numbers is not important.

A 1 } 3,983,816

C 1 }}

1,068,347,520

B 1 } 13,983,816

D 1 }}

10,068,347,520

2 Carla randomly chooses 5 wooden alphabet tiles from a bag. In the bag, there are 26 tiles, one for each letter of the alphabet. What is the probability that Carla chooses A, B, C, D, and E in any order?

F 1 } 780

G 1 }

32,540

H 1 } 65,780

J 1 }

7,893,600

3 There are 15 people in a room, each having a unique birth date. Teams of 3 are chosen randomly. What is the probability that the first 3 people are chosen in order of their birth dates?

A 1 } 455

C 1 }

2730

B 1 } 1555

D 1 }

7300

4 Cade chooses lettered tiles from a bag containing 26 lettered tiles, one for each letter of the alphabet. What is the probability that he chooses the letters in the order CADE?

F about 0.003 H about 0.00003

G about 0.0003 J about 0.000003

5 How many terms does the binomial expansion of (3x2 1 y3)18 contain?

A 18

B 19

C 36

D 54

6 What are the first 4 terms in the expansion of (x 1 3y)6?

F x6 1 18x5y 1 135x4y2 1 540x3y3

G x6 1 18x5y 1 180x4y2 1 360x3y3

H x6 1 18x5y 1 270x4y2 1 360x3y3

J x6 1 18x5y 1 360x4y2 1 540x3y3

7 Which is the coefficient of y4 in the expansion of ( y 1 4)9?

A 126

B 1024

C 32,256

D 129,024

8 Which is the coefficient of x3 in the expansion of (2x3 1 3)8?

F 3078

G 16,128

H 34,992

J 108,864

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A trigonometric ratio for an acute angle u is a ratio of the lengths of two sides of a right triangle. When the triangle is drawn inside a unit circle, the trigonometric ratios can be defined in terms of the coordinates of a point on the circle.

sin u 5 opposite

} hypotenuse

5 y }

1 5 y

cos u 5 adjacent

} hypotenuse

5 x }

1 5 x

tan u 5 opposite

} adjacent

5 y } x

The resulting equations define trigonometric functions that also apply to angles greater than 90 degrees.

EXAMPLE

Find the sine, cosine, and tangent of each of the following angles: 60 , 135 , 210 , and 330 .

Solution

u 5 60 , the corresponding point on the unit circle is 1 1 } 2 ,

Ï}

3 }

2 2 .

Therefore, sin 60 5 Ï

}

3 }

2 , cos 60 5

1 }

2 , and tan 60 5

Ï}

3 /2 }

1/2 5 Ï

}

3 .

u 5 135 corresponds to 1 2 Ï

}

2 } 2 ,

Ï}

2 }

2 2 .

5 Ï

}

2 }

2 , cos 135 5 2

Ï}

2 } 2 ,

and tan 135 5 2 Ï

}

2 /2 }

Ï}

2 /2 5 21.

5 2 1 } 2 , cos 210 5 2

Ï}

3 } 2 , and

tan 210 5 21/2

} 2 Ï

}

3 /2 5

1 }

Ï}

3 .

5 2 1 } 2 , cos 330 5

Ï}

3 }

2 , and

tan 330 5 21/2

} 2 Ï

}

3 /2 5

1 }

Ï}

3 .

SPI 3103.4.1Exhibit knowledge of unit circle trigonometry.

y

x

608

308

3308

308

458

1358

2108

608 3 2 2

2 2 2

12,

12, 2(2 )

,

3 2

12, 2(2 )

(2 ) ( )

3 2

(x, y)

y

x

1

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x

y

r 5 1

(x, y)

1208


A x 5 2 1 } 2 C x 5 2cos 60°

B y 5 Ï

}

3 }

2 D y 5 2sin 60°

2 The angle shown below is in standard position.

x

y

r 5 1 (x, y)

308

What is the value of x?

F Ï}

2 } 2 H

Ï}

3 }

3

G Ï

}

3 }

2 J

1 }

2


x

y

r 5 1

(x, y)

21358

What is the value of x?

A 2 Ï

}

2 }

2 C 2

1 } 2

B Ï

}

2 } 2 D

1 }

2

4 The angle shown below is in standard position.

x

y

r 5 1(x, y)

3308


F x 5 2 Ï

}

3 } 2 H x 5 cos 30°

G y 5 2 1 } 2 J y 5 2sin 30°

GO ON

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SPI 3103.4.2Match graphs of basic trigonometric functions with their equations.

For all nonzero real numbers a and b, the basic trigonometric functions are shown in the table.

Function Equation Domain Range Period Graph

Sine y 5 a sin bx all real numbers y $ 2a or y # a 2p

} ) b )

x

y

O

1

1

2 2

2period

y 5 sin x

0.5

0.5

Cosine y 5 a cos bx all real numbers y $ 2a or y # a 2p

} ) b )

2period

x

y

O

1

1

2 2

0.5

0.5

y 5 cos x

Tangent y 5 a tan bx all real numbers except multiples

of p

} ) b )

all real numbers p

} ) b )

period

5

22 22 O

y 5 tan x

x

y10

210

25

Cosecant y 5 a csc bx all real numbers except multiples

of p

} ) b )

y # 2a or y $ a 2p

} ) b )

x

y

3

1

32

2

period

y 5 csc x

1O

2

2

2

Secant y 5 a sec bx all real numbers except odd multiples of

p }

2 ) b )

y # 2a or y $ a 2p

} ) b )

x

2 O

2

y 5 sec x

2

y

period

Cotangent y 5 a cot bx all real numbers except multiples

of p

} ) b )

all real numbers p

} ) b )

period

5

O

y10

210

25

222 2

y 5 cot x

x

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x

y

O

21.5

1.5

22 2 2

A sine

B cosine

C tangent

D cosecant


x

y

O

2

22

24

4

2222 23

22

F 1 } 2 tan (x)

G 2 tan (x)

H tan 1 1 } 2 x 2

J tan (2x)


x

y

22 22 O

A cosecant

B secant

C cotangent

D tangent


x

y

O

21

22

1

2

2 3 4

F 1 }

2 sin 1 1 }

2 x 2

G 2 sin (2x)

H 1 }

2 sin 2x

J 2 sin 1 1 } 2 x 2


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SPI 3103.4.3Describe and articulate the characteristics and parameters of parent trigonometric functions to solve contextual problems.

For functions of the form y 5 A sin Bx 1 D or y 5 A cos Bx 1 D, the amplitude is ) A ) , the

period is 2p

} B

, and the vertical shift is D.

EXAMPLE

The average time of sunrise over a 12-month interval can be modeled by the equation A cos Bx 1 D. Use the graph at the right to determine the following:

a. the vertical shift D

b. the amplitude A

c. the value of B

Solution

a. The vertical shift D is the average of the minimum and maximum values: D 5 5.0 1 8.0

} 2

5 6.5.

b. The amplitude A is half the difference between the maximum and minimum values:

A 5 8.0 2 5.0

} 2 5 1.5.

c. The graph repeats at x 5 12 months. Thus, 2p

} B 5 12, so B 5 2p

} 12

5 p

} 6 .

Tim

e o

f su

nri

se

Time of Average Monthly Sunrise

1

2

3

4

5

6

7

8

9

1 2 3 4 5 6 7 8 900 10 11 12

Month of year

y

x


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3 The height of one tip of a plane’s propeller above the ground can be modeled by a sine curve.

2

4

6

8

0.05 0.1 0.15 0.2 0.25Time (s)

Heig

ht

(ft)

00 x

y

What are the period, T, and amplitude, A, of the function?

A T 5 0.025 s; A 5 3 ft

B T 5 0.025 s; A 5 6 ft

C T 5 0.05 s; A 5 3 ft

D T 5 0.05 s; A 5 6 ft

4 The temperature of an air-conditioned room over a 1-hour time period can be modeled by a sine curve.

72

76

80

84

10 20 30 40 50Time (min)

Tem

pera

ture

(�F

)

00 x

y


F T 5 10 s; A 5 3° F

G T 5 12 s; A 5 3° F

H T 5 10 s; A 5 6° F

J T 5 12 s; A 5 6° F


1 The hours of daylight in a town in Alaska can be modeled by a sine curve.

4

8

12

16

20

200 400 600Day

Tim

e (

hr)

00 x

y


A T < 360 days; A < 8 hours

B T < 360 days; A < 16 hours

C T < 720 days; A < 8 hours

D T < 720 days; A < 16 hours

2 The distance from the top of a carousel horse to the ground when the ride is in motion can be modeled by a sine curve.

2

4

6

8

10 20 30 40 50Time (s)

Heig

ht

(ft)

00 x

y


F T 5 10 s; A 5 2 ft

G T 5 10 s; A 5 1 ft

H T 5 8 s; A 5 2 ft

J T 5 8 s; A 5 1 ft


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SPI 3103.3.5.1Compute, compare and explain summary statistics for distributions of data including measures of center and spread.

A number used to represent the center or middle of a set of data values is a measure of central tendency. Mean, median, and mode are measures of central tendency.

The mean is the average of the data values. Find the mean by adding all the values and dividing by the number of values.

The median is the middle value. Find the median by ordering the numbers from smallest to largest. If the number of values is odd, the median is the middle number. If the number of values is even, the median is the average of the middle two numbers.

The mode is the most frequently occurring value. If no value appears more than once, the data has no mode. It is possible for a data set to have multiple modes.

A number used to represent the spread of a set of data values is a measure of dispersion. Range, interquartile range, variance, and standard deviation are measures of dispersion.

The quartiles divide the data into four parts, each with an equal number of data elements. Quartile 1 is the median of the lower half the data. Quartile 2 is the same as the median of the data set. Quartile 3 is the median of the upper half of the data.

The interquartile range is calculated as Quartile 3 2 Quartile 1.

The variance, denoted by s2, is the average of the squared differences of a numerical data set x1, x2, . . ., xn from the mean, } x :

s2 5 (x1 2 } x ) 2 1 (x1 2 } x ) 2 1 … 1 (xn 2 } x ) 2

}}} n

The standard deviation, denoted by s, is the square root of the variance of a numerical data set x1, x2, . . ., xn:

s 5 Ï}}}

(x1 2 } x ) 2 1 (x1 2 } x ) 2 1 … 1 (xn 2 } x ) 2

}}} n

EXAMPLE

Find the variance and standard deviation of the data set. 30, 8, 12, 35, 62, 48, 50

Find the mean of the data set. } x 5 30 1 8 1 12 1 35 1 62 1 48 1 50

}}} 7 5 245

} 7 5 35

Find the variance.

s2 5 (30 2 35)2 1 (8 2 35)2 1 (12 2 35)2 1 (35 2 35)2 1 (62 2 35)2 1 (48 2 35)2 1 (50 2 35)2

}}}}}}} 7 < 343.71

Find the standard deviation. s 5 Ï}

343.71 < 18.54

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SPI 3103.3.5.1 (continued)

5 A data set has a mean of 8, a median of 6 and a standard deviation of 3.2. Suppose every value in the data set is multiplied by 4. What is the mean, median and standard deviation of the new data set?

A 12, 10, 3.2 C 32, 24, 3.2

B 12, 10, 7.2 D 32, 24, 12.8

6 Emily works at a car dealership. The list below shows the number of hours she worked each day for a 6-day period.

7, 8, 7, 6, 8, 6

If the mean of these data is 7, what is the standard deviation for these data? Round your answer to the nearest hundredth.

F 0.67 H 0.91

G 0.82 J 0.99

7 Find the mean and the variance of the data set below.

35, 45, 30, 35, 40, 25

A 35 and 6.5 C 35 and 41.7

B 35 and 13 D 35 and 169

8 The data set below represents the number of pages for five different food magazines issued in June.

118, 146, 98, 102, 152

Find the variance and standard deviation for the data. Round to the nearest tenth.

F 22.2 and 492.2

G 22.2 and 123.2

H 492.2 and 123.2

J 492.2 and 22.2

1 Kohei found the mean and standard deviation of the numbers below. If he multiplies each number by 5, which of the following will result?

9, 5, 2, 1, 4, 5, 10

A The standard deviation will be multi-plied by 5.

B The standard deviation will not change.

C The mean will increase by 5.

D The mean will not change.

2 Find the mean and standard deviation for the data set below.

5, 6, 7, 11, 15, 16, 17

F 11 and 4.7 H 11 and 9.4

G 11 and 5.1 J 11 and 22

3 The median of a data set is 15. Each value in the data set is increased by 4. What is the new median?

A 15 C 30

B 19 D 60

4 The data set below shows the number of miles that 8 runners ran during 1 week.

21, 15, 12, 18, 8, 14, 12, 16

Find the mean, median, and mode.

F 12, 14, and 14.5

G 14.5, 12, and 14.5

H 12, 15, and 14.5

J 14.5, 14.5, and 12GO ON

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SPI 3103.5.2Compare data sets using graphs and summary statistics.

You can use summary statistics to compare data sets.

EXAMPLE

The table shows the birth weights of a sample of babies born at a hospital on two different dates.

Birth Weights (ounces)

March 1 September 177, 83, 90, 107,

109, 110, 112, 114,

115, 120, 120, 125,

126, 130, 131, 131,

147, 147, 147, 160

88, 90, 90, 98, 108,

110, 114, 115, 119,

119, 120, 120, 120,

126, 128, 130, 130,

140, 152, 157

Compare the means and the standard deviations of the two data sets.

Solution

Enter the data sets into a calculator and use the CALC: 1-Var Statistics feature.

March 1 data set: Mean 5 120.05 oz Standard Deviation 5 21.00 oz

September 1 data set: Mean 5 118.7 oz Standard Deviation 5 18.27 oz

The second data set has a smaller mean and a smaller standard deviation. Overall, the babies born on September 1 were smaller and their weights were more closely clustered near the mean.

1 The graph shows the number of students who tried out for the basketball teams at a local high school.

02006 2007 2008 2009

Year

Nu

mb

er

of

Stu

de

nts

Girl’s

Boy’s

10

20

30

40

50

Which basketball team had the greater mean number of students who tried out from 2006 to 2009?

A the boy’s team

B the girl’s team

C The means are the same.

D The answer cannot be determined.


Set A {6, 10, 7, 7, 9, 2, 8}

Set B {8, 12, 9, 9, 11, 4, 10}

F The standard deviations are equal.

G The means are equal.

H The medians are equal.

J The modes are equal.

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5 Mark earned the following test scores in his Biology class.

84, 79, 88, 75, 74

Mark earned a 98 on his last Biology test. By how many points did Mark’s average test score increase after the sixth test?

A 1

B 2

C 3

D 4

6 Which data set is most tightly clustered about its mode?

F [4, 4, 4, 6, 6, 6, 6]

G [6, 6, 6, 6, 7, 8, 8]

H [2, 2, 6, 6, 6, 10, 10]

J [4, 4, 6, 6, 6, 8, 8]

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3 Jeff wanted to compare the prices of MP3 players at two stores. He checked the price of seven MP3 players at Store X, and then checked the prices of the same MP3 players at Store Y.

60

60 75 80 100 125

80 100 120 140 160

Store X

70 85 100 110 150Store Y

Price of MP3 Player ($)

Which statement is true?

A The median price of MP3 players at Store X is lower than the median price at Store Y.

B The median price of MP3 players at Store X is higher than the median price at Store Y.

C The median price of MP3 players is the same at both stores.

D The median prices of MP3 players cannot be determined from the box plots.

4 Which statistical measure is not changed when the 7 in the following data set is changed to 15?

16, 10, 12, 9, 20, 7, 11, 15, 15

F mean

G median

H mode

J weighted average

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You can often describe the relationship of data displayed in a scatter plot. Look for a pattern in the plot that might indicate a linear relationship, or a non-linear relationship, such as quadratic, exponential, or logarithmic.

EXAMPLE 1

Describe the relationship shown in the data in the scatter plot.

20 4 6 8 10 x

0

1000800600400200

y

Solution

The trend of the data is to first increase slowly, and then to increase more rapidly. This pattern fits an exponential relationship.

EXAMPLE 2

Describe the relationship shown in the data in the scatter plot.

y

0

252015105

50 10 15 20 25 x

Solution

The trend of the data is to first increase rapidly, and then to increase more slowly. This pattern fits a logarithmic relationship.

SPI 3103.5.3 Analyze patterns in a scatter-plot and describe relationships in both linear and non-linear data.

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1 The data in the graph show the height above ground of an object in free-fall. What type of relationship exists between the data?

100

200

300

400

500

600

700

800

900

1 2 3 4 5 6 7 8 9

Heig

ht

(ft)

00

Time (s)

A linear C logarithmic

B quadratic D exponential

2 Laurie recorded the high temperature in Knoxville each Saturday for two months and recorded the data in the graph below.

Week

Tem

pera

ture

(8F

)

0 1 2 3 4 5 6 7 8

70

60

50

0 x

y

What is the relationship between the week number and the high temperature on that Saturday?

F The high temperature is dependent on the week number.

G The week number is dependent on the high temperature.

H The week number and the high temperature are each dependent on the other.

J There is no relationship.


3 The graph shows the number of bean plants that have sprouted after a number of days.

Pla

nts

havin

g s

pro

ute

d

02 4 6 8 100

2

4

6

8

10

Days

Which type of relationship appears to be the least likely to exist among the data?

A linear C quadratic

B exponential D logarithmic

4 The graph shows the number of mockingbirds counted in a state park in the first 7 months of last year.

y

x2 3 4 5 6 7 810

2

1

0

3

4

5

6

7

8

910

Nu

mb

er

of

mo

ckin

gb

ird

s

Month

Which type of relationship appears to be the most likely to exist among the data?

F linear H cubic

G quadratic J quartic

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SPI 3103.5.4Apply the characteristics of the normal distribution.

A normal probability distribution, often called a bell curve, is symmetric about a central peak and extends infinitely far to the left and right. The location of the peak on the horizontal axis is the mean of the distribution. The spread of the peak is described by the standard deviation, which represents a distance from the mean. The greater the standard deviation, the greater the spread. With a normal distribution, about 68% of the area under the curve lies within 1 standard deviation from the mean,

1 std.dev.

1 std.dev.

mean

68%

About 95% of the area under the curve lies within 2 standard deviations from the mean, and about 99.7% lies within 3 standard deviations. For a given interval on the horizontal axis, the area under the curve over that interval represents the probability that an outcome will fall within the interval.

EXAMPLEA variable has a normal probability distribution with a mean of 8 and a standard deviation of 2.5. Sketch the curve and find the probability that on a single trial, the variable takes on a value between 8 and 13.

Solution

3 8 13

The interval from 8 to 13 represents the values that lie within 2 standard deviations of the mean and are greater than the mean. The probability that the variable falls in that interval is about half of 95%, or 47.5%.

Some variables, such as height and weight in a random sample of people, tend to have normal distribution. Other variables tend not to be normally distributed. People’s salaries, housing prices, and city populations, for example, all have asymmetric distributions, with more extreme values above the mean than below it.

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1 A standard normal distribution:

A has a mean of 1 and a standard deviation of 0

B has a mean of 1 and a standard deviation of 1

C has a mean of 0 and a standard deviation of 1

D has a mean of 0 and a standard deviation of 0

2 Approximately how much of the area under a standard normal curve is within one standard deviation of the mean?

F 99% H 80%

G 95% J 68%

3 Which of the following is least likely to be normally distributed?

A daily rainfall, in inches, in a small town in Arizona

B the number of traffic tickets issued daily in New York City

C the number of seconds it takes some-one to thread a needle

D the number of fleas on a wolf

4 The SAT is designed so that the scores are normally distributed with a mean of 500 and a standard deviation of 100. Approximately what percent of the scores are below 600?

F 98% H 16%

G 84% J 50%

5 The scores on a math test are normally distributed with a mean of 65 and a standard deviation of 13. Find the score that is two standard deviations below the mean.

A 26 C 52

B 78 D 39

6 A set of data is normally distributed with a standard deviation of 9. If the value 101 is two standard deviations above the mean, what is the mean?

F 83 H 119

G 95 J 92

7 A set of data is normally distributed. The value one standard deviation below the mean is 48, and the value one standard deviation above the mean is 64. What are the mean and standard deviation?

A mean 5 52; standard deviation 5 4

B mean 5 56; standard deviation 5 2 Î}

2

C mean 5 56; standard deviation 5 8

D mean 5 56; standard deviation 5 16

8 A set of data are normally distributed with a mean of 58 and a standard deviation of 7. Which value is two standard deviations above the mean?

F 65 H 44

G 51 J 72

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SPI 3103.5.5Determine differences between randomized experiments and observational studies.

In a survey, a specific group of people is asked a series of questions about the topic under consideration. When designing a survey, it is important to keep several factors in mind:

Surveys can be given in person, online, over the phone, or by mail. Survey results can be affected by the wording of the question. Survey results may be biased if the population is selected so that respondents are more or less

likely than the general public to answer questions in a certain way. Survey results may be affected by respondents not answering truthfully or completely.

In an observation, the group being studied is observed without any intervention on the part of the researcher.

In an experimental study, the researcher will collect data, intervene in some way in the population being studied, and then collect data again to study the effects of the intervention.

An experimental study is a controlled experiment if the intervention is not performed on some of the population (the control group) so that the results found in the group that had the intervention can be compared to the control group.

There are several different ways to sample a population for a survey or study.

Random samples and biased samples each have their place in data collection. In a random sample, no part of the population is favored over another, and so a random sample is more likely to lead to sound conclusions. In a biased sample, one or more parts of the population have a greater likelihood of being included. However, biased sampling is generally easier and cheaper than random sampling and can be necessary when random sampling is not feasible.

Random Samples Biased Samples

Type Type

Simple RandomSample

ConvenienceSample

Those members of the popu-lation that are easily accessedare chosen for the sample.

Members of the populationwho want to participate makeup the sample.

Voluntary ResponseSampleStratified Random

Sample

Systematic RandomSample

The population is divided intosimilar groups. Then a simplerandom sample is chosenfrom each group.

A member of the populationis chosen for the sample at aregular interval.

Every member of thepopulation has an equalchance of being chosen forthe sample.

Definition Definition

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4 A telemarketer calls every hundredth name in the phone book. Which type of sampling method did he use?

F random sampling

G convenience sampling

H systematic sampling

J self-selected sampling

5 A company wants to know if employees are satisfied with the company’s vacation policy. Which sampling method will give the most representative sample?

A place survey cards and a survey box outside the human resources department

B survey all employees with children

C survey every tenth employee

D survey every tenth employee who arrives to work on Friday

6 Members of the student council want to conduct a survey to see if students would like to change the school mascot. Which method of sampling is least likely to produce a biased sample?

F survey students in the school cafeteria

G survey every fifth student who enters the building before school

H survey friends of the student council members

J survey every student from the school who attends a varsity basketball game

1 A principal wants to know whether students would like pizza as a lunch option every day. She surveyed every other student who entered the cafeteria during lunch. What is the sample?

A all football players

B every other student who entered the cafeteria during lunch

C all students who buy lunch in the cafeteria

D all students in the school

2 A research agency wants to determine how many hours per day people in the U.S. spend watching television. Which sampling method will give the most representative sample?

F survey 1000 households in Alabama

G survey 1000 randomly selected households in the U.S.

H survey 1000 randomly selected households in the Northeast

J survey 1000 households in five states

3 A pharmaceutical company wants to test the effects of a new cold medicine. The company advertises for test subject volunteers in the local newspaper. Which type of sampling method is used?

A random sampling

B convenience sampling

C systematic sampling

D self-selected sampling


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SPI 3103.5.6Find the regression curve that best fi ts both linear and non-linear data (using technology such as a graphing calculator) and use it to make predictions.

To determine a regression curve that best fits a set of data, first plot the data on a scatter plot. Next, consider the general shape of the scatter plot to select an appropriate curve to find using a graphing calculator. If the shape of the scatter plot is not clear, you may need to try two or more curves to see which best fits the data.

Once you have found a regression curve, you can use it to make predictions.

EXAMPLE

The table shows the distance Amanda bicycled during a 30-minute training ride each week for 10 weeks. Find the regression curve that best fits the data, and predict the distance Amanda will ride in week 15.

Week 1 2 3 4 5 6 7 8 9 10

Distance (mi) 5.8 7.3 7.7 8.8 9.3 9.5 10.1 9.9 10.3 10.4

Solution

Make a scatter plot of the data.

The trend of the data is to increase more rapidly at first, and then to increase more slowly. This describes a logarithmic relationship. A logarithmic regression curve would most likely be the best fit for the data.

Enter the data into a graphing calculator by assigning to one list the weeks (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) and to another list the corresponding times (5.8, 7.3, 7.7, 8.8, 9.3, 9.5, 10.1, 9.9, 10.3, 10.4).

Use the logarithmic regression selection to find the equation for the regression line:y 5 5.8 1 2.1 ln x. Evaluate the equation at x 5 15 to predict Amanda’s distance in week 15. The value is about 11.5 miles.

200

2

4

6

8

10

4 6 8 10x

y

Dis

tan

ce (

mi)

Week


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1 The scatter plot shows the number of members of a social networking Web site t days after it is launched.

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 80

0

Based on the line of best fi t, y 5 4.75x 1 5.29, about how many members did the site have nine days after it was launched?

A 43 B 50 C 48 D 45

2 The scatter plot represents the ages of ten people and their yearly incomes.

0Age of Person

Yearl

y In

co

me

(in

th

ou

san

ds o

f d

ollars

)

10 20 30 40 50 60 70 800

20

40

60

80

100

120

140

Age of Person vs. Yearly Income

If another person earns $102,000 per year, which is the best prediction for the person’s age, based on the line of best fi t?

F 50 H 66

G 60 J 80

3 The scatter plot shows Tiffany’s time to the nearest 0.2 second for a one-mile run each week for eight weeks. Find the linear regression equation, and use it to predict Tiffany’s time in week 10.

x

y

2

9

10

11

12

1210864Week

Time for One-Mile Run

Min

ute

s

A y 5 20.18x 1 11.5, 9.7 min

B y 5 20.13x 1 10.8, 9.5 min

C y 5 20.20x 1 11.0, 9.0 min

D y 5 20.70x 1 15.5, 8.5 min

4 A bookstore owner tracked the number of sales the store made each hour after opening.

Hour 1 2 3 4 5 6 7 8

Number of Sales 8 15 21 26 28 34 39 43

Choose the best estimate for the equation of the regression line and use it to predict the number of sales in the tenth hour.

F y 5 4.8x 1 5; 53

G 4x 1 4; 44

H 5.5x 1 2.5; 58

J 5.6x 1 3; 59


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SPI 3103.5.7Determine/recognize when the correlation coeffi cient measures goodness of fi t.

The correlation coeffi cient r of a regression equation indicates the nature and strength of the asso-ciation between the two variables. The correlation coeffi cient ranges from 0 to 1 when the regression curve is increasing, and from 0 to 21 when the regression curve is decreasing. Values near 21 or 1 indicate a strong relationship, while a value near 0 indicates little or no relationship. When a calculator performs least-squares regression, it stores the correlation coeffi cient as the variable r.

The following graphs illustrate different values for r obtained when fi nding a linear regression curve for the data.

x

r near 1

y

r near 0

x

y

r near 2 1

x

y

1 A biologist writes four different models for a scatter plot. The correlation coeffi cient for each model is listed below. Which correlation coeffi cient has the best fi t?

A r 5 20.92

B r 5 20.06

C r 5 0.13

D r 5 0.79

2 Celeste calculates a regression line with a correlation coeffi cient of 20.09 using the data shown in the scatter plot below.

x

y

What can you conclude about Celeste’s regression line?

F It is an exact fi t to the data.

G It is a good fi t to the data.

H It is a fair fi t to the data.

J It is a poor fi t to the data.


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4 A student found the regression line for a set of data shown in the graph below. If the correlation coeffi cient of the line is 20.92, which set of data did the student most likely use?

Te

mp

era

ture

(8F

)

02 4 6 8 100

10

20

30

40

50

Time (h)

Temperatures in Jackson

y

x

F x y2 44

5 41

9 33

G x y2 35

5 40

9 42

H x y1 50

4 30

7 10

J x y1 42

4 36

7 28

3 Select the graph most likely to represent a regression line of the data with a correlation coeffi cient of 20.95.

A

B

C

D


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1 Philip is randomly drawing a card from a standard deck of 52 playing cards. What is the probability that he will draw either a 7 or a club?

A 1 } 52

C 17

} 52

B 4 } 13

D 9 }

26

2 From a standard deck of 52 playing cards, Kelly randomly picks a card, replaces it, and then selects another card. What is the probability that the two cards she chooses are both hearts?

F 1 } 2 H

1 }

8

G 1 } 4 J

1 }

16

SPI 3103.5.8Apply probability concepts such as conditional probability and independent events to calculate simple probability.

Theoretical probability is computed by dividing the number of ways a particular event can happen by the number of events in a sample space. The probability that an event E will occur is written as P(E ). Probability can be expressed as a fraction, decimal, or percentage. A value of 0 or 0% means that the event cannot happen. A value of 1 or 100% means the event is certain to happen.

The conditional probability of B given A, P(B|A), is the probability that B occurs, given

that A has occurred. For any two events A and B, P(B|A) 5 P(A and B)

} P(A)

.

EXAMPLE 1

A number cube is rolled. What is the probability that the roll is a 4, given that it is an even number?

Intuitively, a number cube has three even numbers, 2, 4 and 6, all equally likely.

So the probability of a 4, given that the roll is even, is 1 }

3 . Using the conditional probability formula,

P(4 | even) 5 P(even and 4)

}} P(even)

5 1 }

6 }

1 }

2 5

1 }

6 p 2 }

1 5

2 }

6 5

1 }

3

Two events, A and B, are independent if the occurrence of one does not affect the probability of theoccurrence of the other. For independent events, P(B|A) 5 P(B) and P(A and B) 5 P(A) p P(B).

EXAMPLE 2

For a machine that stamps out machine parts, 2% of the parts come out defective. If two parts are picked at random from a week’s worth of production from that machine, what is the probability that both parts are defective? Treat each part as an independent event.

P(defective first part) 5 P(defective second part) 5 0.02. By the rule for independent events, P(both parts defective) 5 0.02 p 0.02 5 0.0004. The probability that both parts are defective is 0.04%.


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3 Lane has a marble collection that he keeps in a cloth bag. The number of each color of marble is shown in the table below. Every color he has is listed. If he randomly selects two marbles from the bag, one at a time without replacement, what is the probability that both marbles are green?

Color NumberRed 13

Blue 14

Green 24

Yellow 23

A 144 } 1369

B 276 } 2701

C 12 } 37

D 1727

} 2701

4 A bag contains 15 golf balls, 4 of which are defective. If two balls are selected randomly from the bag, what is the probability that both are defective?

F about 0.0075

G about 0.057

H about 0.57

J about 0.75

5 In a high school, the probability that a student is taking Spanish and French is 0.04. The probability that a student takes Spanish is 0.36. Find P(French|Spanish).

A 0.320

B 0.125

C 0.11 } 1

D 0.0144

6 A bag contains red and blue marbles. Two marbles are selected without replacement. The probability of selecting a red marble and a blue marble is 0.20. The probability of selecting a red marble fi rst is 0.75. What is P(blue|red)?

F 0.550

G 0.36 }

36

H 0.26 } 6

J 0.150

7 Events A and B are dependent. What is P(A and B) given that P(A) 5 0.3 and P(B) A) 5 0.5?

A 0.15

B 0.2

C 0.8

D 1



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Post Test 97Algebra II Tennessee

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Post Test

1 The graph of a quadratic function is shown. Which of these best describes the zero(s) of the function?

21

1

2

3

4

21222324 0 1 2 3 4

22

23

24

y

x

A 1 real zero

B 1 imaginary zero

C 2 real zeros

D 2 imaginary zeros

2 Which function is shown below?

x

y

21

0.5

1

p2

3p2

p 2pO

F sin 2x

G 2 sin x

H 1 } 2 sin 2x

J 2 sin 2x

3 The angle shown below is in standard position. What is the value of x?

x

y

r 5 1

(x, y)

1508

A 2 1 } 2 C 2

Ï}

3 } 2

B 1 }

2 D Ï

}

3 } 2


x

y

r 5 1

(x, y)

458


F x 5 1 }

2

G y 5 Ï

}

2 }

2

H sin 45° 5 cos 45°

J y 5 sin 45°

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98 Post Test Algebra II Tennessee

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Post Test (continued)

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2x 2 y 1 2z 5 27 2x 1 2y 2 4z 5 5 x 1 4y 2 6z 5 21

A (23, 21, 1)

B (23, 21, 2)

C (22, 21, 21)

D (23, 21, 21)

6 The graph shows the daily production cost for a company that manufactures microwave ovens, where x is the number of ovens produced. Which statement is true about this situation?

Co

st

(do

llars

)

010 20 300

700

800

600

Ovens

x

y

F The minimum cost is $700.

G The minimum cost is $20.

H To minimize the cost, the company should produce 700 ovens.

J To maximize profit, the company should produce 800 ovens.

7 If f (g(x)) 5 (5x2 1 2)2, which of the following could be true?

A f (x) 5 5x 1 2 and g(x) 5 x2

B f (x) 5 (5x)2 and g(x) 5 2x2

C f (x) 5 5x2 1 2 and g(x) 5 x2

D f (x) 5 x2 and g(x) 5 5x2 1 2

8 The table shows the birth weights of a sample of babies born at a hospital on two different dates.

Birth Weights (ounces)

March 1 September 1

77, 83, 90, 107,

109, 110, 112, 114,

115, 120, 120, 125,

126, 130, 131, 131,

147, 147, 147, 160

88, 90, 90, 98, 108,

110, 114, 115, 119,

119, 120, 120, 120,

126, 128, 130, 130,

140, 152, 157

Which statement is true?

F The mean birth weight of babies born on March 1 is greater than the mean birth weight of babies born on September 1.

G The mean birth weight of babies born on March1 is less than the mean birth weight of babies born on September 1.

H The mean birth weights are equal.

J The mean cannot be determined from the given data.

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12 The table below shows the observed height y (in feet) of an ocean tide x hours after the initial observation.

x y

0 3.8

2 2.5

4 1.2

6 0.7

8 2.1

10 3.5

12 4.2

14 3.3

16 1.8

18 1.0

20 1.7

22 3.1

24 3.8

Use the regression capabilities of a graphing calculator to find a sine function that models the data.

F y 5 1.6 sin (0.5x 12.1) 1 2.5

G y 5 1.6 sin (0.5x) 1 4.6

H y 5 1.6 1 [sin (0.5x 12.1) 1 2.5]

J y 5 1.6 sin (2.6x) 1 2.5

9 Divide by using long division:

(2x2 1 4x3 2 4 2 10x) 4 (2x 1 3)

A 2x2 1 2x 1 2 1 2 }

2x 1 3

B 2x2 2 2x 2 1 2 1 }

2x 1 3

C 2x2 1 2x 1 1 2 1 }

2x 1 3

D 2x2 2 2x 2 2 1 2 }

2x 1 3

10 A set of data is normally distributed with a mean of 105 and standard deviation of 16. Which value is one standard deviation above the mean?

F 97

G 121

H 89

J 101

11 Which of the following can be used to find the coefficient of the 9th term of (a 1 b)15?

A 15C8

B 15C9

C 8C15

D 9C15

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Algebra II Tennessee100 Post Test

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14 Students from a high school were asked which pet is their favorite. Find P(cat|male).

Male Students

Female Students

Cat 18 46

Dog 58 34

Reptile 32 8

Bird 6 15

Fish 10 15

Other 5 2

F about 0.14

G about 0.07

H about 0.28

J about 0.04

13 Which graph best represents the solution to this system of inequalities?

y , 2x 21 y $ 23x 1 2

A y

xO 2

2

B y

x2

2

O

C y

xO 2

2

D y

x2

2

O

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16 The stem-and-leaf plots show the quiz scores for Mrs. West’s first period and third period algebra classes.

First Period Third Period

6

7

8

9

5 8

2 4 5 5 8 8

0 2 5 6

0 4 8

5

6

7

8

9

6

4 6

5 6 6 8 9

4 5 5 6

4 8 8

Which class period had the greater median quiz score?

F first period

G third period

H The median quiz scores are equal.

J The median quiz scores cannot be determined.

15 Richard rolls two six-sided dice. One die is black and one is white. What is the probability that the white die shows a 3 and the sum is greater than 7?

A 1 } 18

B 1 } 12

C 1 } 9

D 1 } 4

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19 Ilana is training for a marathon. The data sets below show the number of miles she runs each day over a two-week period.

Week 1: 8, 6, 12, 11, 15, 9

Week 2: 10, 8, 12, 8, 6, 17

Which of the following statements is not true about the data sets?

A The means are equal.

B The median of the data set for week 1 is greater than the median of the data set for week 2.

C The standard deviations are equal.

D The range of the data for week 1 is smaller than the range of the data for week 2.

20 A cellular phone service company charges 20 cents for each minute or part of a minute of airtime. If the total charge y is a function of the number of minutes x that Cora uses her phone, what is the range of this function?

F {0, 1, 2, 3, 4, . . .}

G 0 # y # 20

H {0, 20, 40, 60, . . .}

J y $ 0

17 Describe how the graph of y 5 f (x 2 3) relates to the graph of y 5 f (x).

A The graph of y 5 f (x 2 3) is the graph of y 5 f (x) shifted left 3 units.

B The graph of y 5 f (x 2 3) is the graph of y 5 f (x) shifted right 3 units.

C The graph of y 5 f (x 2 3) is the graph of y 5 f (x) shifted up 3 units.

D The graph of y 5 f (x 2 3) is the graph of y 5 f (x) shifted down 3 units.

18 If f (x) 5 2 1 x1/5 and g(x) 5 x 2 2, perform the following operation.

f (x) ? g(x)

F x6/5 1 2x 2 2x1/5 2 4

G 3x 1 4

H x12/5 1 4

J 3x 2 x1/5 1 4

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23 A person’s blood pressure can be modeled by a sine curve.

Time (s)

Pre

ssu

re (

mm

Hg

)

1 2 3 40

20

40

60

80

100

120

0 x

y

What are the period and amplitude of the function?

A period 5 1 s; amplitude 5 20 mmHg

B period 5 1 s; amplitude 5 40 mmHg

C period 5 5 s; amplitude 5 20 mmHg

D period 5 5 s; amplitude 5 40 mmHg

24 Use the discriminant to determine the nature of the roots of 3x2 2 7x 1 5 5 0.

F no real roots

G 1 real root

H 2 real roots

J 1 real and 1 imaginary root

21 In the set of numbers

5 24 2 3i, 2 Ï}

2 , 0, 1 }

2 i, , 5.4 6 , which are

complex numbers?

A 5 24 2 3i, 1 }

2 i 6

B 5 23i, 1 }

2 i 6

C 5 24 2 3i, 2 Ï}

2 , 0, 1 }

2 i, , 5.4 6

D 5 24 2 3i, 2 Ï}

2 , 1 }

2 i, 6

22 The graph of which function has an asymptote?

F y 5 Ï}

x 2 1

G y 5 1 }

x2 2 1

H y 5 x2 2 1

J y 5 x2 2 1

} 4

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27 Anna is conducting a survey to determine the percentage of teenage girls in her town who participate in sports. Which group of people would be the most representative sample for her survey?

A every fifth person leaving the local sporting goods store

B every other female student entering the local high school

C all of the female students on the soccer team

D every other student entering the local high school

28 The data in the table show the amount of snowfall (in inches) for the years from 2005–2009.

YearSnowfall Amount

2005 32

2006 38

2007 43

2008 44

2009 48

Which linear equation is the best estimate for the regression line for the data, where x 5 5 represents 2005?

F y 5 23.8x 1 14

G y 5 3.8x 1 14

H y 5 6x 1 3

J y 5 5x 1 30

25 Why might this graph be misleading?

Nu

mb

er

so

ld

(th

ou

sn

ds)

0

1200

800

Year

Shoe Sales

1990

1994

1998

2002

2006

A The years on the horizontal axis do not cover enough of a time span.

B The shoe sales are increasing recently, so the graph will continue to rise.

C The large increments on the vertical axis compress the graph and make it appear that there was little change.

D Changes in data over time should not be displayed in a line graph.

26 Find all the solutions of the equation (2x 2 1)2 1 11 5 4.

F 23

G 6 Ï

}

6 } 2 i

H 1 }

2 6

Ï}

7 }

2 i

J 2 1 }

2 6

Ï}

7 }

2 i

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31 The times for the 8 girls in the final of the 100-meter hurdle race at a track meet are listed below according to the lane the girls ran in. Is the mean a good representation of the times? Why or why not?

Lane #1 – 16.33 sec

Lane #2 – 16.38 sec

Lane #3 – 15.88 sec

Lane #4 – 15.59 sec

Lane #5 – 21.45 sec

Lane #6 – 16.47 sec

Lane #7 – 16.72 sec

Lane #8 – 16.74 sec

A Yes, it represents the data well.

B No, it is too high to represent the data.

C No, it is too low to represent the data.

D No, it is the same as one data point.

32 What is |22 1 i|?

F Ï}

5

G 3

H 2 1 i

J 4 1 i

29 What is the product of 4 2 5i and 21 1 2i?

A 214 1 13i

B 24 1 3i

C 6 1 13i

D 6

30 Write a formula for the nth term of the geometric sequence below.

1, 3, 9, 27, 81, . . .

F an 5 1 1 } 3 2 n21

G an 5 2n

H an 5 3n21

J an 5 3n

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35 Which statement best describes the relationship between the distance people live from work and the time it takes for them to get there?

50

40

30

20

10

45

35

25

15

5

5Distance (mi)

Tim

e (

min

)

Distance From Work

10 15 20 25

A There is no correlation.

B There is a strong positive correlation.

C There is a strong negative correlation.

D There is a weak negative correlation.

36 If f (x) 5 1 }

x 1 1 and g(x) 5

|x 2 1| } x , what is

( f 2 g)(2)?

F 2 1 }

6

G 2 1 }

2

H 5 } 6

J 0

33 Lisa checked the outside temperature every hour over a ten-hour period. Below is a graph of her data. She wants to find an equation to fit the data. Which type of model is most reasonable?

10

20

30

40

50

60

70

80

90

1 2 3 4 5 6 7 8 9

Tem

pera

ture

(8F

)

00

Hour

A linear C exponential

B quadratic D logarithmic

34 What is the domain of the function

f (x) 5 x2 2 4 }

2x2 2 5x 1 2 ?

F 5 x ) x Þ 1 }

2 , 2 6

G {x | x Þ 22, 2}

H 5 x ) x Þ 1 }

2 6

J {x | x is a real number}

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38 For which value of a is the statement 2a # a false?

F 5

G 2 } 3

H 0

J 2 1 }

2

39 What is the sum of 7x3 1 6x2 25x 2 5 and 26x3 2 8x2 1 10x 2 7?

A x3 1 2x2 1 5x 1 12

B x3 2 2x2 1 5x 2 12

C 2x3 2 2x2 1 5x 2 12

D x3 2 2x2 2 5x 1 12

37 The graph of y 5 2x2 1 3 is shown. Which parabola in this figure could be the graph of y 5 2x2 1 6?

x

y

y 5 2x 2 1 3

a

b

c

d

A a

B b

C c

D d

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41 A researcher writes four different models for a scatter plot. The correlation coefficient for each model is listed below. Which correlation coefficient indicates the best fit?

A r 5 20.75

B r 5 20.07

C r 5 0.15

D r 5 0.83

40 Mr. Jansen is coaching baseball. He hits a pop fly to the second baseman. The ball is hit and caught four feet above the ground. The path of the ball is part of a parabola as shown.

Heig

ht

(ft)

010 20 30 4550 15 25 35 40

40

35

30

25

20

15

10

5

Distance (ft)

x

y

What is the maximum height the ball reaches?

F 25 ft

G 40 1 }

2 ft

H 43 ft

J 50 ft

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43 Which linear equation is the best estimate of the line of best fit for the data in the table?

x 0 1 2 3 4 5

y 16 21 28 36 43 49

A y 5 5x 1 16

B y 5 6.8x 1 15

C y 5 26x 1 15

D y 5 7x 1 16

44 The base of a triangle is 4 centimeters more than 3 times the height. The area is 66 square centimeters. Austin wrote the

equation 3 }

2 h 2 1 2h 2 66 5 0, where h

represents the height of the triangle. He

then entered y 5 3 }

2 x 2 1 2x 2 66 into his

graphing calculator and looked at the table shown. What is the height of the triangle?

x y

5 218.5

6 0

7 21.5

8 46

F 5 cm H 7 cm

G 6 cm J 8 cm

42 Which equation represents the ellipse in the graph below?

y

x5 0

2.5 (0, 2)

(0, 2)2.5

5

F y2

} 4 2

x2 }

8 5 1

G x2

} 4 1

y2

} 8 5 1

H y2

} 4 1

x2 }

8 5 1

J x2

} 4 2

y2

} 8 5 1

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47 Mr. Anderson, Ms. Barth, and Ms. Cole each teach an Algebra class. Mr. Anderson’s class has six more students than Ms. Cole’s class. Ms. Barth’s class has two fewer students than Mr. Anderson’s class. There are a total of 64 students in the three classes combined. Which system models the problem?

A 5 a 5 c 1 6 b 5 a 1 2

a 1 b 1 c 5 64

B 5 a 5 c 2 6 b 5 a 2 2

a 1 b 1 c 5 64

C 5 a 5 c 1 6 b 5 2 2 a

a 1 b 1 c 5 64

D 5 a 5 c 1 6 b 5 a 2 2

a 1 b 1 c 5 64

45 What are the solutions of the equation 8x2 2 2x 2 3 5 0?

A 3 } 4 , 2 1 }

2

B 1 } 2 , 2

3 }

4

C 3 }

8 , 21

D 3, 21

46 A candidate poll is to be conducted during an election for state governor. Which sampling method would result in the least biased poll?

F calling 1000 randomly selected registered Democrats statewide

G calling 1000 randomly selected registered voters in the largest city in the state

H calling 1000 randomly selected registered Republicans statewide

J calling 1000 randomly selected registered voters statewide

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50 Solve Ï}

5x 2 8 5 Î}

x2 2 x .

F {2, 4}

G {22, 4}

H {2}

J {4}

48 Corina earns different amounts as a hostess and a server. Last week she earned $189.50 for 10 hours as a hostess and 18 hours as a server. This week she earned $167.25 for 15 hours as a hostess and 9 hours as a server. How much per hour does Corina earn as a server?

F $5.58

G $6.50

H $11.15

J $13.00

49 Given the equation Ï}

a2 b4 5 3 Ï}

a3 b6 is true for all real values of a greater than or equal to 0, which statement is valid for real values of b?

A b $ 0

B b 5 0

C b # 0

D b is any real number.

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52 Which graph represents the equation

x2

} 4 2

y2

} 9 5 1?

F

2

2

2468

468

4 6 8468

y

x

G

2

468

468

4 6 8468

y

x

H

2

42

68

468

42 6 8268

y

x

J

2

42

68

468

2 6 868

y

x

51 The height of a rectangular prism is (x 1 5) cm and the area of the base is (3x2 2 2x 1 1) cm2. Which expression represents the volume of the rectangular prism?

A 3x3 1 13x2 2 10x 1 5

B 3x3 1 13x2 2 10x 1 6

C 3x3 1 6x2 2 6x 1 6

D 3x3 1 13x2 2 9x 1 5

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55 Approximately what percent of the area under a standard normal curve is within two standard deviations of the mean?

A 68%

B 95%

C 99%

D 86%


x

y

22

21

1

2

22p 2p p 2pO

F cosecant

G cosine

H cotangent

J tangent

53 What is the inverse function of f (x) 5 (5 2 x)3?

A g(x) 5 5 2 3 Ï}

x

B g(x) 5 3 Ï}

5 2 x

C g(x) 5 3 Ï}

5 2 x

D g(x) 5 5 1 3 Ï}

x

54 Find o n51

15

(3n 2 8).

F 32

G 37

H 240

J 277

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59 Multiply x(x2 1 x)

} x2 2 2x 2 3

? x2 2 x 2 6

} x2 1 2x

.

A x C x }

x 2 3

B x3 D x3 }

(x 2 3)2

60 Simplify: (6 1 3i) 2 (4 2 2i).

F 10 1 i

G 10 1 5i

H 2 1 5i

J 2 1 i

57 Which is a simplified form of 1 2 x2

} (x 2 1)23 ?

A 2(x 2 1)4(1 1 x)

B (1 2 x2)(x 2 1)3

C 2(1 2 x)4

D 2(1 1 x2 )(x 2 1)3

58 Which of the following pairs of functions are not inverses? Assume x $ 0.

F f (x) 5 x3 1 2 and g(x) 5 3 Ï}

x 2 1

G f (x) 5 1 1 } 2 x 1 2 2

2 and g(x) 5 2 Ï}

x 2 4

H f (x) 5 2 Ï} x and g(x) 5

1 }

4 x2

J f (x) 5 (x 1 1)2 and g(x) 5 Ï} x 2 1

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62 Bacteria in a culture are reproducing exponentially with time, as shown in the table below.

Bacteria Growth

Hour Bacteria

0 50

1 100

2 200

Which of the following equations expresses the number of bacteria, y, present at any time, t?

F y 5 50 1 2t

G y 5 50 ? 2t

H y 5 2t

J y 5 100 ? 2t

61 The box plots show the commuting times (in minutes) for employees from two downtown companies. Twenty-five employees from each office were surveyed.

5 23 38 45 55

Company A

Company B

17 30 42 50 60

B

0 10 20 30 40 50 60

Which statement is not true?

A More employees from Company B than from Company A commute over 30 minutes.

B The median commuting time for Company A is less than the median commuting time for Company B.

C The range of commuting times is greater for Company B than for Company A.

D There are at least 12 employees from Company A whose commuting time is less than or equal to 38 minutes.

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65 Which graph best represents the solution of the system of inequalities?

y . 2 1 }

2 x 1 2

y # 2x 2 6

A y

x2

2

B y

x2

2

C y

x2

2

D y

x2

2

63 A model rocket is launched so that the height above the ground, in feet, after t seconds is given by the function y 5 26.4t2 1 96t. What is the maximum height reached by the rocket?

A 360 ft

B 358.4 ft

C 350 ft

D 96 ft

64 Which is the difference between

5x 1 2

} x2 1 4x 1 3

and 4x 2 1

} x 1 3

?

F 5 } 4

G (x 1 4)2

} (x 2 3)2

H 24x2 1 8x 1 1 }} (x 1 3)(x 1 1)

J 24x2 1 2x 1 3 }} (x 1 3)(x 1 1)

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Tenn Algebra 2 EOC Practice Workbook

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