ACCESS Curricula Guide for€¦ · competencies and begin the work of designing assessments for each competency. ... This ACCESS Curricula Guide for ... Heavy Construction

Algebra II/Intermediate Algebra,

Precalculus, and (Non-STEM) Statistics

with Embedded Real World Applications

in Assignments and Assessments

Version 1.0

Curricula Guide for ACCESS

M a t h e m a t i c s • A C C E S S C u r r i c u l a G u i d e

1

Version 1.0

Algebra II/Intermediate Algebra,

Precalculus, and (Non-STEM) Statistics

with Embedded Real World Applications

in Assignments and Assessments

Curricula Guide for ACCESS

Aligning Curricula and Career Education

for Student Success (ACCESS)

A C C E S S C u r r i c u l a G u i d e • M a t h e m a t i c s

2

This project was made possible through the generous support of

The William and Flora Hewlett Foundation

The James Irvine Foundation

The Evelyn and Walter Haas Jr. Fund

The Girard Foundation


3T a b l e o f C o n t e n t s

Introduction ......................................................................................................................................................... Page 4

Career Clusters ................................................................................................................................................... Page 9

How to Use This Guide ...................................................................................................................................Page 10

List of ACCESS Math Competencies ......................................................................................................... Page 11

ACCESS Math Competencies ...................................................................................................................... Page 17

Algebra II/Intermediate Algebra Model Assessments for Competencies .......................... Page 17

Precalculus Model Assessments for Competencies ...................................................................Page 87

Statistics (Non-STEM) Model Assessments for Competencies ............................................Page 153

Acknowledgements/Contributors ......................................................................................................... Page 207

Appendix: Mapping ACCESS Competencies to California Content Standards and Common Core Standards .............................................................................................Page 216

For more information, contact Shelly Valdez, EdD, at [email protected].

ACCESS Curricula Guide for


I n t r o d u c t i o n4

I N T R O D U C T I O N

The ACCESS Curricula Guide for Mathematics is the result of many passionate discussions

about Intermediate Algebra (Algebra II), Precalculus, and non-STEM Statistics among

more than 100 K–12 and postsecondary math faculty and Career Technical Education

(CTE) instructors throughout California. These courses were chosen for discussion because

data on math course enrollment at a number

of K–12s, community colleges, and universities

across California indicated that large numbers

of students enroll in them, often to complete

prerequisites for other math courses offered at

those institutions. The goal of these discussions

was to define the exit and entrance competencies

that students should possess in these three

math courses. CTE instructors worked with math

faculty to develop assessments that teachers can

use to measure whether students are meeting

expectations, and make these assessments apply

to real world experiences.

This effort, called ACCESS (Aligning Curricula

and Career Education for Student Success),

launched in September 2008 with the backing

of foundation leaders who believe in the type of

intersegmental work that distinguishes Cal-PASS

from other educational initiatives. The foundations are: The William and Flora Hewlett Foundation,

The James Irvine Foundation, The Evelyn and Walter Haas Jr. Fund, and The Girard Foundation.

Thirteen regions around the state played a role in defining core competencies in

Algebra II, Precalculus, and non-STEM Statistics, and developing assessments to measure

achievement in each course. Based on the experience of the math faculty and their

knowledge of the California State Standards and Community College Student Learning

Outcomes, topics and skills were identified that the faculty felt were important for

students to know upon completing the courses. Once these important topics and skills

This guide is unique in three

distinct ways:

1) It takes an important step

beyond listing the gaps

between high school and

college math skills by

providing sample assessments

that measure mastery of the

core competencies.

2) It is the result of educational

segments working in unison.

3) It answers the often-asked

question from students:

“Where will I ever have to

use this?”


I n t r o d u c t i o n 5

were identified and worded as measurable competencies, they were mapped relative to the

California Content Standards and Common Core Standards for alignment. (See Appendix:

Mapping ACCESS Competencies to California Content Standards and Common Core

Standards, page 216.)

T h e P r o b l e m

California high school graduates leave high school believing they are ready for college, but

the data demonstrate that even successful high school students are often ill-prepared for

college. Research over the past 10 years points to specific areas that are needed — both in

class and in the home — to prepare students for the rigors and habits needed to succeed in

college and beyond.

The messages most California high school students receive about standards for

attending a broad-access university or open-access community college are confusing.

Because it is generally perceived that it is easy to enter the community college and

California State University (CSU) systems, there are few intrinsic incentives to work hard

in high school.* Once students enroll in these colleges,

however, they face challenging placement exams, faculty

and university expectations, and graduation requirements

of which they are likely unaware.

Unlike messages students receive from competitive four-

year universities, messages received by students aiming for

what are perceived as less-selective universities provide

little information about the educational level at which they

should achieve. This lack of information is represented by the

number of college freshmen who must remediate in math and/or English and those who drop out.

Remediation rates in college are staggering: 53 percent of students matriculating into the CSU

system and as high as 90 percent at some community colleges.† While a majority of high school

graduates enter college, fewer than half leave with a degree. Many factors influence this attrition,

* Michael Kirst and Andrea Venezia, From High School to College: Improving Opportunities for Success in Post-Secondary Education, San Francisco: Jossey-Bass, 2004.

† Robert Johnstone, “Community College Pre-collegiate Research Across California: Findings, Implications, and the Future,” iJournal, no. 9, 2004.

Remediation rates in

college are staggering:

53 percent of students

matriculating into the

CSU system and as high

as 90 percent at some

community colleges.†


I n t r o d u c t i o n6

but a report by the American Diploma Project states that the preparation students receive in

high school has been found to be the greatest predictor of bachelor’s degree attainment.‡

Although remediation rates at postsecondary institutions have remained steady for 30

years, the population going to college is rising, and the number of students needing remediation

is putting a strain on budgets and making the college experience longer and more arduous.

Students who are unprepared are often unaware of

this fact until they matriculate as freshmen. Already

accepted or registered, they are given placement

tests to determine if they are ready for college-

level work. The CSU accepts students who, by all

indicators, are ready (have taken necessary college-

prep courses), but once at the university, these

students must contend with their lack of preparedness for college-level work. Community

colleges accept all students, regardless of courses taken in high school.

The main source of this disconnect is the lack of communication and collaboration between

high school and higher education.§ High school teachers and college professors rarely talk

to each other about curriculum, learning issues, and expectations. This leads to confusion by

high school students and administrators regarding what it means to be prepared for college.

H e a d i n g To w a r d a S o l u t i o n

With specific instructions from ACCESS project managers, 13 Cal-PASS math Professional

Learning Councils (regional councils made up of teams of discipline-based faculty from

elementary, middle school, high school, community college, and university segments) from

around California deconstructed what they teach in the classroom. Conversation was not

always smooth, as opinions surfaced about what students really need to know in Algebra II,

Precalculus, and (non-STEM) Statistics to prepare them for college and careers.

‡ American Diploma Project, Ready or Not: Creating a High School Diploma that Counts, 2004, www.achieve.org/files/ADPreport_7.pdf.§ David T. Conley, College Knowledge: What It Really Takes for Students to Succeed and What We Can Do to Get

Them Ready, San Francisco: Jossey-Bass, 2005.

The main source of this

disconnect is the lack

of communication and

collaboration between high

school and higher education.§


I n t r o d u c t i o n 7

Research and documents (like those from the math deconstruction project led by Cal-

PASS participants**) ignited passionate discourse among math faculty from high schools,

community colleges, and universities in California. Gathering math faculty from across

educational segments is something Cal-PASS does often,

and is a feat unto itself. The vision of this project was

already realized when 200 math and English teachers

gathered for three days to debate, challenge, argue, and

— finally — come to consensus. Cal-PASS groups worked

tirelessly to articulate what students should know when

they walk into, and then out of, math and English classes

from 11th grade to transfer-level at community college.

Under the ACCESS grant, Cal-PASS Professional

Learning Councils worked throughout the second year of the grant to refine and align the

competencies and begin the work of designing assessments for each competency. They

designed these assessments specifically to be of high interest to students and to apply to real

world work situations with the intention of linking math to careers (contextualized learning).

During the summer of 2010, a convening was held for both teachers and CTE instructors

to embed CTE examples and assessments into the competencies. They used the career

clusters provided by the California Community Colleges Chancellor’s Office to guide them

in this CTE crossover effort.

T h e F i n a l P r o d u c t

This ACCESS Curricula Guide for Mathematics is the summation of math faculty ACCESS

work and represents newly aligned, contextually relevant, and collaboratively developed

curricula. This document is intended to supplement — not supplant — statewide or

institution-based curricula and expectations for multiple audiences, including high school

and postsecondary faculty, curriculum and instruction professionals, and campus-level or

statewide educational leaders.

The vision of this

project was already

realized when 200

English and math teachers

gathered for three days

to debate, challenge,

argue, and — finally —

come to consensus.

** Cal-PASS, Algebra I California Content Standards, Standards Deconstruction Project v. 2.0, 2008; Algebra II California Content Standards, Standards Deconstruction Project v. 2.0, 2008; Geometry California Content Standards, Standards Deconstruction Project v. 1.0, 2008; Pre-Calculus California Content Standards, Standards Deconstruction Project v. 1.0, 2008. www.cal-pass.org/Councils.aspx.

8


I n t r o d u c t i o n

Specifically, this guide

1. documents the necessary competencies of three subsequent levels of math (spanning

from high school to college);

2. provides assessment examples for how faculty can evaluate student competency levels;

3. embeds real world applications into assignments and assessments to contextualize

materials outside of academia; and

4. maps the relationship between ACCESS curricula and California-based and nationally

based standards.

It is our hope that K–12 teachers, as well as math instructors at community colleges

and universities, will find the curricula and assessments in this guide both user-friendly

and useful. Further, those in math teacher preparation programs could use this guide as a

textbook supplement to provide concrete examples of what should be practiced regularly

in the classroom to help students prepare for college and careers.

9


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10


H O W T O U S E T H I S G U I D E

The intended use of the ACCESS Curricula Guide for Mathematics is to be a supplement to

course materials and lesson plans. Math faculty from around the state identified competencies

in Algebra II, Precalculus, and (non-STEM) Statistics that they believe are essential to the goals

of college attainment and success. The competencies are not necessarily organized in the

same way as the California State Standards or Common Core Standards for Algebra II and

Precalculus, nor are they necessarily in the order in which one would teach them in the course.

Each of the competencies includes one or more model assessment items. In no way are

these model assessment items exhaustive, but they are intended to indicate the level of

knowledge and skills expected of the students.

Many competencies include real world example assessments. The provided real world

assessments are just samples; they do not address all possibilities in terms of the CTE

pathways. The purpose of including real world example assessments was to give the

reader examples of how the competencies might apply to various professions. Real world

assessments are not included with every competency; some of the competencies are

prerequisites for the next level of skills or courses and do not lend themselves to real world

examples. Real world assessments are included where appropriate.

One way to use this guide is to first identify the topic in which you are interested

and locate the corresponding competency. Readers also can look at the corresponding

California State or Common Core Standard to identify the relevant competency, and then

use the model assessment items to gauge the level of understanding expected of students.

Teachers should feel free to modify these assessment items as they see fit.

H o w t o U s e T h i s G u i d e


11L i s t o f A C C E S S M a t h C o m p e t e n c i e s

List of ACCESS Math Competencies

Algebra II/Intermediate Algebra ACCESS Competencies

1. Use properties of absolute values to solve multistep equations and inequalities.

2. Solve systems of linear equations in both two and three variables by substitution, elimination/addition, graphs, and matrices.

3. Factor polynomials by grouping.

4. Factor polynomials by using the sum/difference of cubes pattern.

5. Factor polynomials by extending the difference of squares pattern to polynomials of degree higher than 2.

6. Solve quadratic equations in the real and complex number systems by factoring.

7. Solve quadratic equations in the real and complex number systems by completing the square.

8. Solve quadratic equations in the real and complex number systems by using the quadratic formula.

9. Use multiple representations (tables, graphs, and equations) to solve contextualized problems that result in quadratic equations.

10. Add and subtract complex numbers.

11. Multiply complex numbers.

12. Perform division with complex numbers.

13. Simplify rational expressions with higher order polynomials in the denominator by canceling common factors in the numerator and denominator.

14. Add and subtract rational expressions with higher order polynomials in the denominator.

15. Multiply and divide rational expressions with higher order polynomials in the denominator.

16. Divide polynomials by binomials using long division and synthetic division.

17. Simplify a fraction where the numerator, denominator, or both contain a fraction (complex fractions).

18. Use multiple representations (tables, graphs, and equations) to solve contextualized problems that result in equations involving rational expressions.

19. Use the properties of exponents to simplify expressions with rational exponents.

20. Use the properties of exponents to solve equations with rational exponents.

21. Simplify radical expressions by removing repeated factors and performing addition and subtraction operations.


12 L i s t o f A C C E S S M a t h C o m p e t e n c i e s

22. Simplify radical expressions by removing repeated factors and performing multiplication operations.

23. Rationalize the denominator of radical expressions.

24. Use multiple representations (tables, graphs, and equations) to solve problems that result in radical equations.

25. Apply and graph transformations of parent functions (quadratic, cubic, square root, absolute value, exponential, and logarithmic).

26. Identify the parent function (quadratic, cubic, square root, absolute value, exponential, and logarithmic) and transformations for a given transformed function in algebraic or graphical form.

27. Compose two linear, quadratic, cubic, square root, absolute value, exponential, or logarithmic functions.

28. Find the inverse of linear, quadratic, cubic, square root, absolute value, exponential, and logarithmic functions algebraically and graphically.

29. Find the equation of an exponential function given two points on the graph.

30. Use multiple representations (tables, graphs, and equations) to solve contextualized problems that result in one- and two-step exponential and logarithmic equations.

31. Graph simple conics, including:

• acirclegivenitsequation(centeredanywhere)

• anellipsegivenitsequation(centeredattheorigin)

• ahyperbolagivenitsequation(centeredattheorigin)

32. Solve statistical problems:

• Computepermutationsusingfundamentalcountingprinciple.

• Computecombinations.

• Computeprobabilitiesusingcombinations.

• Computeprobabilitiesusingpermutations.

• Expandbinomialexpressionsusingthebinomialtheorem.

33. Compute the general term and sums of arithmetic series, finite geometric series, and infinite geometric series.



Precalculus ACCESS Competencies

1. Perform matrix operations, including addition and multiplication, and calculate the determinants of 2x2 and 3x3 matrices.

2. Describe different types of functions (polynomial, exponential, logarithmic, and trigonometric) using a table, a graph, an equation, and a verbal description.

3. Graph piecewise-defined functions.

4. Find all real zeros (roots) of polynomial functions exactly using the rational root theorem.

5. Find all the zeros (roots) of a polynomial function.

6. Solve polynomial inequalities.

7. Solve rational inequalities.

8. Find the composition of two or more functions, each containing two or more operations (such as linear, quadratic, rational, cubic, square root, cube root, and trigonometric functions).

9. Given a function, determine if an inverse function exists. If so:

• Findtheinverseofthefunctionalgebraicallyandgraphically.

• Identifythedomainandrangeoftheoriginalandinversefunctions.

10. Simplify expressions involving exponents.

11. Expand logarithmic expressions.

12. Condense logarithmic expressions.

13. Solve multistep exponential equations.

14. Solve multistep logarithmic equations.

15. Sketch the graph of a conic, identifying center, vertices, foci, and asymptotes (if present).

16. Graph trigonometric functions of the form ( ) .y A f Bx C k= + +

• Determinetheamplitudeofasineorcosinefunctionfromitsequation.

• Determinetheperiodofasine,cosine,ortangentfunctionfromitsequation.

• Determinetheverticalshiftofasine,cosine,ortangentfunctionfromitsequation.

• Determinethephaseshiftofasine,cosine,ortangentfunctionfromitsequation.

• Sketchbyhandthegraphofasine,cosine,ortangentfunction.

17. Given the graph of a trigonometric function, write the equation (sine, cosine, tangent).

18. Graph inverse trigonometric functions.

19. Prove trigonometric identities.

20. Solve trigonometry equations.



21. Solve for all unknown parts of a given triangle:

• byusingrighttriangletrigonometry

• byusingtheLawofSines

• byusingtheLawofCosines

22. Convert between polar and rectangular coordinates.

23. Graph equations written in polar form.

24. Graph equations written in parametric form.

25. Given the components of a two-dimensional vector, determine the magnitude and direction of the vector.

26. Perform the vector operations of addition, subtraction, and scalar multiplication.

27. Use factorial notation.

28. Expand binomial expressions.



Statistics (Non-STEM) ACCESS Competencies

1. Determine whether a variable is categorical or quantitative, given a situation.

2. Identify the sampling method used (random, systematic, convenience, stratified, cluster, or voluntary response), given a scenario in which data were collected from a population.

3. Identify the basic terminology of experimental design used in a given scenario.

4. Identify biases in experimental designs or in the creation of a sample.

5. Determine an appropriate experimental design for a given scenario.

6. Construct and describe an appropriate visual display given a data set (histogram, bar graph, stem plot, scatter plot, box and whisker plot, and pie chart).

7. Identify which measure of the center (mean, median, or mode) is most appropriate for a given situation and what the relationship between the mean, median, and mode indicates about the distribution of the data.

8. Determine standard deviation from multiple representations of data (table, graph, and word problems) and explain standard deviation in the context of a given situation.

9. Use the normal distribution to solve for the probability of an event or solve for the boundaries for a particular event.

10. Determine the size of a sample space using counting principles, permutations, and combinations.

11. Compute basic probability of independent events and solve problems involving the binomial distribution:

• DeterminesituationsthatareBernoullitrials.

• Calculatetheprobabilityofanevent(success)and itscomplementaryprobability(failure).

• Discern thedifferencebetweengeometricprobabilityandbinomialprobabilityofBernoulli trials.

• Applytheformulaforbinomialprobabilityforx successes in n trials

( ) .x n xn rP x C p q −=

• Use the normal model ( , )N µ σ , where npµ = and npqσ = to approximate a binomial probability when the number of trials and desired successes is inordinately large.

12. Apply the Central Limit Theorem to sampling distributions.

13. Estimate population parameters using confidence intervals for means and proportions.



14. Perform appropriate hypothesis test (state the hypotheses, determine the significance level, check conditions/criteria, calculate the test statistic, determine the p-value, make a decision and interpret it in the context of the problem) for a given situation.

15. Explain the relationship between parameters and statistics.

16. Given a p-value, interpret its meaning in the context of the variables of a problem:

• Definethep-value.

• Identifytheappropriatenullandalternativehypotheses,andusethepropernotationof H0 and Ha.

• Interpretagivenp-value in the context of a hypothesis testing situation as it relates to the rejection of or failure to reject H0.

17. Find and apply the equation of the regression line when linear regression is appropriate:

• Determinewhenlinearregressionisappropriateforasetofbivariatedata.

• Findthelinearcorrelationcoefficientr and the coefficient of determination r 2.

• Usetheregressionlinetomakeappropriatepredictions.

• Interpretr, r 2, and slope in the context of a given situation.


A l g e b r a I I / I n t e r m e d i a t e A l g e b r a C o m p e t e n c i e s 17

Algebra II/Intermediate

AlgebraCompetencies


A l g e b r a I I / I n t e r m e d i a t e A l g e b r a C o m p e t e n c i e s18

C O M P E T E N C Y 1

Use properties of absolute values to solve multistep equations and inequalities.

Prior Knowledge:

• Solvealgebraicequations.• Solvealgebraicinequalities.• Evaluateabsolutevalueexpressionsforagivenvalueofthevariable.• Graphsolutionsofaninequalityontherealnumberline.• Interprettheabsolutevalueofanumberasadistancefromzeroonanumberline.• Representsolutionsetsusinginequalities.• Convertsolutionsetswritteninintervalnotationtoinequalitynotationandviceversa.• Solvecompoundinequalities.• Translateabsolutevalueinequalitiesintocompoundinequalities.• Identifyabsolutevalueequationsthathavenosolution(e.g., − = −3 9x ).• Identifyabsolutevalueinequalitiesthathavenosolution(e.g.,

+ <2 0x ).

Model Assessments

Assessment Example 1:

Aperson’sbodytemperature,t,isconsiderednormalifitfallswithin1.5°Fof98.6°F.Writeanabsolutevalueinequalitytoexpressthisfact.

Model Assessment Answer(s):

− ≤98.6 1.5.t


Expressthefactthatxdiffersfrom-3bymorethan5asaninequalityinvolvinganabsolutevalue.Solvetheinequalityforx.


+ >3 5.x

Solution: < − >8or 2.x x

Real World Application Reference:

• BuildingTradesandConstruction• EngineeringandDesign




Solveforx: − − + < −2 3 5 1 7.x


> <1

3or .3

x x


a) Aconstructioncompanyisbuildinghouses.Theframingcrewneedstoplacestuds3feetapart with a tolerance of a quarter inch to pass inspection.Write an absolute valueinequalityforthissolutionsetandsolve.Explainyourreasoningtotheforemanoftheproject.

b) ABCManufacturingmakesdoorsandwindows.Eachdoormustbe42inchesinwidthwithatoleranceofhalfanincheitherwayinordertofitintheframe.Writeaninequalitytosolvethisproblem.Explainyourreasoning.Ifdoorsare84inchestall,whatisthedifferenceinmaterialsthatwouldbenecessarytoconstruct1,000doorsatthetwoextremes?


a) Let xbethedistancebetweenthestuds.Thentheinequalitythatmustbesolvedis

136

4,x − ≤

andthesolutionsetis

3 136 ,

4 435 x≤ ≤

sothedistancebetweenthestudsmustbeatleast35inchesbutnomorethan36inches.



b) Let xbetheactualwidthofthedoor.Thentheinequalitythatmustbesolvedis

142

2,x − ≤

andthesolutionsetis

≤ ≤83 85

,2 2

x

sothedoor’swidthmustmeasureatleast41½inchesbutnomorethan42½inches.Ifwelet ybetheheightofthedoors,thentheinequalityis

184

2,y − ≤

andthesolutionsetis

≤ ≤167 169

,2 2

y

sothedoor’sheightmustmeasureatleast83½inchesbutnomorethan84½inches.Theamountofmaterialinthesmallestallowabledoorwouldbe

41½in.x83½in.=3,465.25sq.in.(about24.064sq.ft.).

Thematerialinthelargestallowabledoorwouldbe

42½in.x84½in.=3,591.25sq.in.(about24.939sq.ft.).

The difference is 0.875 square feet, so multiplying that by 1,000 doors would be875squarefeetofadditionalmaterialifallofthedoorsareaslargeaspossibleratherthanassmallaspossible.




Solve systems of linear equations in both two and three variables by substitution, elimination/addition, graphs, and matrices.

Prior Knowledge:

• Solveandgraphalinearequationintwovariables.• Solveandgraphalinearinequalityintwovariables.• Substitutearationalnumberorexpressionforavariable.• Identifythecoefficientsfromanequationinstandardform.

Model Assessments


Apopularnewbandwillbeperformingatalocalconcertvenue.Thereare800ticketsavailable.Tickets for reservedseatscost$100,general admission ticketscost$70,and tickets forlawnseatingcost$50.Ifallthelawnandreservedseatsandhalfthegeneraladmissionseatsaresold,thetotalcollectedwouldbe$49,000.Ifalltheticketsaresold,thetotalwouldbe$63,000.HowmanyseatsofeachtypeareavailableatthePavilion?


Let x=numberofreservedseattickets,y=numberofgeneraladmissiontickets,andz=numberoflawnseatingtickets.Thesystemofequationsthatmustbesolvedis:

800100 70 50 63000

100 70(0.5 ) 50 49000

x y zx y z

x y z

+ + =+ + =

+ + =

Solution:300reserved,400general,and100lawnseats.


Solveusingsubstitution:2 11,

3 2 13.x yx y

− + = − = −


(–1,5).


• FinanceandBusiness




Solveusingmatrices:2 3 2,5 2 11,

2 4 0.

x y zx y zx y z

− + =− − + = − − − =


Writetheaugmentedmatrixfortheabovesystemofequations.UseGaussianeliminationtotransformtheaugmentedmatrixintorow-echelonform(byhand)orusetheGauss-Jordanmethod to transform the augmentedmatrix into reduced row-echelon form (by handorusingacalculator).Theaugmentedmatrixand reduced row-echelon form(RREF)of thesystemofequationsis:

1 2 3 21 5 2 112 1 4 0

A

− = − − − − −

and

1 0 0 3( ) 0 1 0 2 .

0 0 1 1rref A

=

Answer:x =3,y =2,z =1.Solutionshouldbewrittenasanorderedtriple:(3,2,1).


Solvethesystemusingthegraphingmethod:3,5.

x yx y

+ =− + = −

Figure A2-4

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


(4,–1).




a) Forever Bloom Florist is placing an order for roses, lilies, and tulips. At the nearbywholesale flowermarket, 4 dozen lilies, 3 dozen roses, and 2 dozen tulips cost $124;1dozenlilies,2dozenroses,and3dozentulipscost$76;2dozenlilies,4dozenroses,and3dozentulipscost$128.Howmuchdoeseachindividualtypeofflowercostperdozen?

b) Youhave$12,800toinvestina401K.Youplantodiversifyyourmoneybetweenamoneymarketaccount,anincomefund,andagrowthfund.Youhavedecidedtoputtwiceasmuchmoneyinthegrowthfundasthemoneymarketaccountinordertomaximizeyourpotentialearnings.Thegrowthfundearns10%interest,theincomefundearns7%,andthemoneymarketaccountearns5%.Howmuchshouldyouinvest ineachaccounttoearn$1,000annuallyinsimpleinterest?


a) Let x=costperdozenroses,y=costperdozenlilies,andz=costperdozentulips.Thesystemofequationsthatmustbesolvedis:

+ + =+ + =

+ + =

3 4 2 124,2 3 76,4 2 3 128.

x y zx y z

x y z

Thecostperdozenrosesis$20;thecostperdozenliliesis$12,andthecostperdozentulipsis$8.

b) Let x=theamountofmoneyinthemoneymarketaccount,y=theamountofmoneyin the income fund,andz= theamountofmoney in thegrowth fund.Thesystemofequationsthatmustbesolvedis:

12800,2 0,

0.05 0.07 0.10 1000.

x y zx z

x y x

+ + =− =

+ + =

Put$2,600inthemoneymarketaccount,$5,000intheincomefund,and$5,200inthegrowthfund.




Factor polynomials by grouping. (This is a prerequisite skill that is used in Precalculus and Calculus.)

Prior Knowledge:

• ApplytheLawsofExponentstoalgebraicexpressions.• Recognizenumbersasbeingperfectsquaresorperfectcubes.• Recognizebinomialsasthedifferenceoftwosquareterms.• Factoroutacommonmonomialfromapolynomial.

Model Assessments


Factorbygrouping: + + +22 4 3 6.x x x


+ +(2 3)( 2).x x


Factorbygrouping: − + −4 3 1.x x x


+ − − +2( 1)( 1)( 1).x x x x




Factor polynomials by using the sum/difference of cubes pattern.

Prior Knowledge:

• ApplytheLawsofExponentstoalgebraicexpressions.• Recognizenumbersasbeingperfectsquaresorperfectcubes.• Recognizebinomialsasthedifferenceoftwosquareterms.• Factorasecond-degreepolynomialastheproductoftwobinomialsusingstandard

factoringtechniques.• Factoroutacommonmonomialfromapolynomial.• Recognizethatapolynomialisaperfectsquaretrinomial.

Model Assessments


Amanufacturingcompanyorders5ft.x5ft.x5ft.foamcubes.Thecompanycustomizessmallercubesbasedontheircustomer’sorder.However,tosaveonwaste,theywanttoselltheremainingthreefoamblocksandneedtohavethedimensionslistedfortheirWebsite.Whatarethedimensionsofthethreefoamblocks?


Amountremainingis 3 35 ,a− whereaisthesidelengthoftheorderedcube.

Whenfactoredusingthedifferenceofsquarespattern,theresultis 2(5 )(25 5 ).a a a− + +

Bydistributing,thesizeofeachrectangularprismisasfollows:

Thefirstpieceis ( )( ) ( )5 25 or 5 ft. 5 ft. 5 ft.a a− − × ×

Thesecondpieceis ( ) ( )5 5 or 5 ft. 5 ft. ft.a a a a− − × ×

Thethirdpieceis ( ) ( )25 or 5 ft. ft. ft.a a a a a− − × ×


• EngineeringandDesign• ManufacturingandProduct

Development




Factor: +3125 64.x


+ − +2(5 4)(25 20 16).x x x


Factor: −327 8.x


− + +2(3 2)(9 6 4).x x x


Factor: − −3(3 2) 27.x


− − +2(3 5)(9 3 7).x x x




Factor polynomials by extending the difference of squares pattern to polynomials of degree higher than 2. (This a prerequisite skill that is used in Precalculus and Calculus.)

Prior Knowledge:

• Applystandardfactoringtechniquestofactoringseconddegreepolynomials.

Model Assessments


Factor: −4 81.x


+ − +2( 3)( 3)( 9).x x x


Factor: − −2 4( 5) 16 .x y


− − − +2 2( 5 4 )( 5 4 ).x y x y




Solve quadratic equations in the real and complex number systems by factoring.

Prior Knowledge:

• Factorquadraticexpressions.• Simplifyradicals.• Solveaquadraticequationusingfactoring,completingthesquare,orthequadraticformula.• Simplifyaradicalwhentheradicandisnegative.• Findtherootsofanequation.• Findthex-andy-interceptsofanequation.• Findthevertexofaparabola.• Determineifaparabolawillopenupwardordownward.

Model Assessments


Your construction company has just contracted with the owners of a condemned buildingdowntowntodemolishthepresentbuildinganderectanewoneinitsplace.Youhave300metersoffencingavailabletoenclosethebuildingandsurroundingarea,whichis5,000 squaremeters.Findthedimensionsoftherectangularareathatcanbeencompassedbytheavailablefencing.


Writetheperimeterequation2 2 300L W+ = andsolveforL. = −(300 2 ) / 2L W .Substitutethisintotheareaequation

300 25000.

2W

A W− = =

Solvethepolynomialequation 2 150 5000 0.W W− + = Solution:50m.x100m.


Solve: 3 2 1 .x x x− = −


1.x i or x= ± =


• BuildingTradesandConstruction




Solve quadratic equations in the real and complex number systems by completing the square.

Prior Knowledge:

• Factorquadraticexpressions.• Simplifyradicals.• Solveaquadraticequationusingfactoring,completingthesquare,orthequadraticformula.• Simplifyaradicalwhentheradicandisnegative.• Findtherootsofanequation.• Findthex- and y-interceptsofanequation.• Findthevertexofaparabola.• Determineifaparabolawillopenupordown.

Model Assessments


Adesigncompanyhasdecideditisaestheticallypleasingfromanaerialviewiftheareaofapoolisequaltotheareaoftheborderaroundthepool.Givenapoolthatis6meterswideand10meterslongthatissurroundedbyaborderthatisofuniformwidth,determinethewidthoftheborder.


4 31 1.57 m.− + ≅


Solvetheequationbycompletingthesquare: − =2 6 13.x x


= ±3 22.x


Solve: 2 4 20.x x+ = −


2 4 .x i= − ±


• BuildingTradesandConstruction• EngineeringandDesign




Solve quadratic equations in the real and complex number systems by using the quadratic formula.

Prior Knowledge:

• Factorquadraticexpressions.• Simplifyradicals.• Solveaquadraticequationusingfactoring,completingthesquare,orthequadraticformula.• Simplifyaradicalwhentheradicandisnegative.• Findtherootsofanequation.• Findthex- and y-interceptsofanequation.• Findthevertexofaparabola.• Determineifaparabolawillopenupordown.

Model Assessments


Robin, a Calculus student at the local university, throws her Calculus book out of her

dormitorywindowstraightupintotheairwithavelocityof48feetpersecond.Assuming

thatherdormitoryroomis40feetabovetheground,andignoringairresistance,theheight

ofherbookaftertsecondswillbe: = − + +2( ) 16 48 40.s t t t Howlongdoesittakebeforethe

bookhitsthefootofthepersonstandingdirectlybelowherwindow?


t=3.7seconds.


Solve: − − =2 4 1 0.x x


±2 5.


• EngineeringandDesign




Solve: − − =2

1 24 0.

x x


Eventhoughtheequationisarationalequationandnottechnicallyaquadraticequation,once both sides of the equation are multiplied by the least common denominator, theresultingequationisquadraticandcanbesolved.

±1 33.

8


Solve: 22 5 11.x x+ = −


5 3 74

ix

− ±= or 5 3 7

.4

ix

− ±=




Use multiple representations (tables, graphs, and equations) to solve contextualized problems that result in quadratic equations.

Prior Knowledge:

• Factorquadraticexpressions.• Simplifyradicals.• Solveaquadraticequationusingfactoring,completingthesquare,orthequadraticformula.• Simplifyaradicalwhentheradicandisnegative.• Findtherootsofanequation.• Findthex- and y-interceptsofanequation.• Findthevertexofaparabola.• Determineifaparabolawillopenupordown.• Recognizethatthegraphofanyquadraticequationisaparabola.• Recognizewhatthegraphofy=x 2lookslike.

Model Assessments


Atoycompanyisdevelopingawaterballoonlauncher.Whentheballoonisshotstraightup,theheightoftheballoonatanytimet(measuredinseconds)canbedescribedbytheequation

( ) 20 016 ,h t t v t h= − + +

where v0 represents the initial velocity and h0 represents the initial height of the waterballoon.Ifyouknowthattheinitialvelocityis48feetpersecondandtheballoonislaunchedfromapoint64feetofftheground:

a) Writetheequationthatdescribestheheightatanytime.

b) Determinetheheightoftheballoonfromthegroundatt =1second,then2seconds,etc.;createatableshowingthecorrespondingheights.

c) Determinethetimeittakesfortheballoontoreachmaximumheight.

d) Determinethetimeatwhichtheballoonhitstheground.Ifyourfriendistalkingonhisphonedirectlybelow the launchareaand it takeshim5 seconds to realizeyouhavelaunchedawaterballoon,willithithimbeforehehastimetomove?

e) Graphthepathofthewaterballoon’sheightwithrespecttotime.


• ManufacturingandProductDevelopment




a) = − + +2( ) 16 48 64.h t t t

b) Seethetablebelow.

Time (secs)

Height h(t)(ft)

1 0 64

2 0.5 84

3 1 96

4 1.5 100

5 2 96

6 2.5 84

7 3 64

8 3.5 36

9 4 0

10 4.5 -44

11 5 -96

c) Theballoonreachedamaximumheightof100feetat1.5secondsafteritwaslaunched.

d) Itwilltake4secondsfortheballoontohittheground.Thefriendwillgetwet.

e) Graph:

Figure A5-1e

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

10

2030

40

5060

70

80

90100

110

Time (seconds)

Hei

ght (

ft)



C O M P E T E N C Y 1 0

Add and subtract complex numbers.

Prior Knowledge:

• Addandsubtractalgebraicexpressionsinvolvingvariables.• Usethedistributivepropertyinalgebraicexpressionsinvolvingvariables.• Simplifyalgebraicexpressionsinvolvingpositiveintegerexponents.• Simplifysquarerootsofintegers.• Multiplybinomials.• Rationalizeamonomialorbinomialdenominator.• Identifyconjugatesandcalculatetheir

products.• Multiplytwonumberswithexponents

havingthesamebase.

Model Assessments


InanACcircuitwithtwoparallelpathways,thetotalimpedanceZ,inohms,satisfiestheformula

= +1 2

1 1 1,

Z Z Z

whereZ1 is the impedance of the first pathway andZ2 is the impedance of the secondpathway.DeterminethetotalimpedanceiftheimpedancesofthetwopathwaysareZ1=1+i and Z2=2–3i.


+17 713 13

i ohms.


Simplify:(8+2i)+(4–i).


12+i.


Simplify:(2–5i)–(8+6i).


–6–11i.


• EnergyandUtilities• EngineeringandDesign




Multiply complex numbers.

Prior Knowledge:

• Addandsubtractalgebraicexpressionsinvolvingvariables.• Usethedistributivepropertyinalgebraicexpressionsinvolvingvariables.• Simplifyalgebraicexpressionsinvolvingpositiveintegerexponents.• Simplifysquarerootsofintegers.• Multiplybinomials.• Rationalizeamonomialorbinomialdenominator.• Identifyconjugatesandcalculatetheirproducts.• Multiplytwonumberswithexponentshavingthesamebase.

Model Assessments


Inanelectricalcircuit,thevoltage(E)involts,thecurrent(I)inamps,andtheoppositiontotheflowofcurrent—calledimpedance—(Z)inohmsarerelatedbytheequationE=IZ.Acircuithasacurrentof(3+2i)ampsandanimpedanceof(6–4i)ohms.Determinethevoltage.


E=(3+2i)(6–4i)=26volts.


Multiplyandcombineliketerms:(2i+3)(4–5i)+(–2i)5.


22–39i.






Simplify: i(7–3i)(2+10i).


–64+44i.


Simplify:2

2 3.

2 4i

+


− +1 3 2

.16 4

i




Perform division with complex numbers.

Prior Knowledge:


Model Assessments


Electricalcircuitscanbeconnectedinseries,oneafteranother,orinparallelcircuitsthatbranchoffamainline.Ifthecircuitsarehookedupinparallel,thetotalimpedanceisgivenby:

Figure A10-1

R1R R2

1 2

1 2

.

R RR

R R=

+

FindthetotalimpedanceconnectedinparalleliftheimpedanceofthefirstresistorisR1=5+2iohmsandtheimpedanceofthesecondresistorisR2=3–iohms.


−137 9

.65 65

i






Dividetoidentifyrealandimaginaryparts:−3

.2 i


+6 3

.5 5

i


Simplify: 4 2.

7 3ii

+−


+11 13

.29 29

i




Simplify rational expressions with higher order polynomials in the denominator by canceling common factors in the numerator and denominator. (This is a prerequisite skill that is used in Precalculus and Calculus.)

Prior Knowledge:

• Usethepropertiesofexponents.• Add,subtract,multiply,divide,andsimplifyrationalexpressions.• Add,subtract,multiply,anddividepolynomials.• Factorpolynomials.• Identifywhenarationalexpressionisundefined.• Simplifyarationalexpressionbycancelingcommonfactorsinthenumeratorandthe

denominator.• Multiplyanddividerationalexpressionswithmonomialandpolynomialdenominators.• Findthelowestcommondenominatorbetweentwoormoremonomialorpolynomial

denominators.• Addandsubtractrationalexpressions.• Identifycomplexrationalexpressions.

Model Assessments


Simplify:

a) ( )( )

41 3

12 2

2.

4

a b

a b

−

−−

b) 3

3 2

8.

3 10x

x x x+

− −

c) 2

2

6.

7 12x x

x x− −

− +


a) 10

2

64.

ba

b) ( )2 2 4

.5

x xx x

− +−

c) 2.

4xx

+−




Add and subtract rational expressions with higher order polynomials in the denominator. (This is a prerequisite skill that is used in Precalculus and Calculus.)

Prior Knowledge:




Model Assessments


Simplify:

a) 2

1 4 6.

2 2 4x xx x x

−+ +

+ − −

b) 2 2

1 2.

6 4 3x x

x x x x− −

−+ − + +

c) 2

2

3 2 4 7 24 28.

5 3 4 3 11 20y y yy y y y

+ + ++ −

− + − −


a) 3.

2xx

−−

b) ( )( )( )

4 5.

3 1 2x

x x x−

+ + −

c) ( )2 4

.3 4

y

y

+

+




Multiply and divide rational expressions with higher order polynomials in the denominator. (This is a prerequisite skill that is used in Precalculus and Calculus.)

Prior Knowledge:




Model Assessments


Simplify:

a) 2 2 2

2 2

2 2 5 3 5 6.

2 41 2 4x x x x x x

xx x x− − − − − +

÷−− −

b) 2 2 2

2 2 2

5.

3 4 20 2 11 5 6 17 10a b b a a ab

a a a a a a+ +

÷− − + + − −

c) 2 2 2 2

2 2 2 2

2 3 2 3 2.

3 4 6x xy y x xy yx xy y x xy y

− − − −

− + + −


a) ( )2 1

.1

x x

x

+

−

b) ( )

2 1.

2a

b a++

c) ( )( )( )( )2 3

.3 3

x y x y

x y x y

+ +

− +




Divide polynomials by binomials using long division and synthetic division. (This is a prerequisite skill that is used in Precalculus and Calculus.)

Prior Knowledge:

• Addandsubtractpolynomials.• Usethedistributivepropertytomultiplypolynomials.• Applytheappropriateexponentialrulestosimplifyalgebraicexpressions.• Dividepolynomialsbyamonomial.• Dividepolynomialsbybinomialsusinglongdivision.• Translateandsolvemultistepwordproblemsinvolvingpolynomials.

Model Assessments


Usesyntheticdivisiontofindthequotientof

( ) ( )5 32 10 2 1 .y y y y− + − ÷ +


4 3 2 52 2 8 8 7 .

1y y y y

y− − + − +

+


Simplify: ( ) ( )− + ÷ +3 212 4 3 2 .x x x


− +24 3 2.x x


Simplify:− + −

−

3 2 2 312 26 19 15.

3 5x x y xy y

x y


− +2 24 2 3 .x xy y




Simplify a fraction where the numerator, denominator, or both contain a fraction (complex fractions).

Prior Knowledge:




Model Assessments


Simplify:+

+ −

+

1 11 1 .

1

x xx

x


2.1x −


• Transportation




Simplify:− −+

−

1 1

2 2.

x yx y

xy


1.

x y−


Theaveragerateonaround-tripdeliveryroutethathasaone-waydistanceofdisgivenbythecomplexrationalexpression

+1 2

2,

dd dr r

wherer1 and r2aretheratesontheoutgoingandreturntrips,respectively.

a) Simplifythecomplexrationalexpression.

b) Findtheaveragerateiftheoutboundtriphasanaveragerateof30mph.andthereturntriphasanaveragerateof40mph.


a) 1 2

2 1

2.

r rr r+

b) 240034.3 mph.

70≈




Use multiple representations (tables, graphs, and equations) to solve contextualized problems that result in equations involving rational expressions.

Prior Knowledge:


Model Assessments


Electricalcircuitscanbeconnected inseries,oneafteranother,or inparallelcircuitsthatbranchoffamain line. If thecircuitsarehookedup inparallel, thereciprocalof thetotalresistanceinthecircuitisfoundbyaddingthereciprocalofeachresistance,asshownbelow.

Figure A10-1

R1R R2

+ =1 2

1 1 1.

R R R

IfR1=x and R2=x+4,andthetotalresistanceRis1.5Ω(ohms),findthepositivevalueofR1.


Solve: + =+

1 1 1;

4 1.5x xsolutionisx=2Ω.






Simplify: + =1 1 1

.4 5x x


= −16.

5x


Simplify:− −

− =+ + + +2

1 2 8 10.

4 5 9 20x x

x x x x


5, 3.

2x = −




Use the properties of exponents to simplify expressions with rational exponents.

Prior Knowledge:

• Applythepropertiesofexponents.• Usethebasicnotationoffractionalexponents.• Simplifyradicals.

Model Assessments


Thehalf-lifeofaspirininaperson’sbloodstreamisabout15minutes: ( ) 0t

A t A e τ−

= whereA(t)istheamountremainingaftertimet,A0istheinitialamount,andτ isthemeanlifetime.Determine themean lifetimeofaspirin in thebloodstream. (HINT: themean lifetime, ,τ is definedas half-lifedividedby ln 2.) Simplify the rational exponent in the exponentialequationandrewritetheequation.


Themeanlifetime, ,τ is21.64seconds.Theresultingsimplifiedexponentialequationis

( ) 0.046210 .tA t A e−=


Neglecting overlapping regions, howmuch cardboardwould be needed to construct anopencube-shapedboxwithvolumeV?Writeanexpressionfortheamountofmaterial intermsofVusingrationalexponents.


Let xbethelengthofthesideofthecube.IfthevolumeisVcubicunits,then 3.V x= So

13 .x V= Thismeans thateach faceof thecubehasanareaof

232 ,x V= so theopenbox

requires2

35V squareunitsofcardboard.


• HealthScienceandMedicalTechnology• ManufacturingandProduct

Development




Evaluatethefollowingexpressions:

a) ( )2

3.27−

b) 34.16−


a) 9.

b) 18.


Usepropertiesofexponentstosimplify2 25 3

.

5

2

2x y

x y

−


5

13 133 3

2 32.

y y=




Use the properties of exponents to solve equations with rational exponents.

Prior Knowledge:

• Applythepropertiesofexponents.• Usethebasicnotationoffractionalexponents.• Simplifyradicals.

Model Assessments


Solve: + =32 3 19.x


2, 1 3.x i= − ±


Solve: − =45 4 401.x


3, 3 .x i= ± ±


Theamountofmaterialneededforanopencube-shapedboxwithvolumeVisgivenbytheexpression

235 .V Findthecostofproducing144boxes,eachofwhichhasavolumeof3,000

cubiccentimeters,giventhatthecardboardcosts$1.60persquaremeter.


Each box requires ( )22 33 25 5 3000 1040cm.V = ≈ of cardboard, so 144 boxes require

about 150,000 square centimeters, or 15 squaremeters, of cardboard. The total cost is

( )15 $1.60 $24.00.=


• ManufacturingandProductDevelopment• FinanceandBusiness




Simplify radical expressions by removing repeated factors and performing addition and subtraction operations. (This is a prerequisite skill that is used in Precalculus and Calculus.)

Prior Knowledge:

• Usethepropertiesofexponents.• Simplifysquarerootsofintegers.• Simplifyalgebraicexpressions.

Model Assessments


Simplify:

a) − −135 60 15.

b) +3 2 372 50 .x y y x

c) −3 37 1054 250 .x x x

d) −5 56 532 243 .x x


a) 0.

b) 11 2 .xy x

c) − 3 32 2 .x x

d) −52 3 .x x x




Simplify radical expressions by removing repeated factors and performing multiplication operations. (This is a prerequisite skill that is used in Precalculus and Calculus.)

Prior Knowledge:

• Usethepropertiesofexponents.• Simplifythesquarerootsofintegers.• Add,subtract,andmultiplypolynomials.

Model Assessments


Findtheproduct:

a) ( )( )− +4 2 3 5 5 2 .

b) ( )−3 7 2 6 5 .


a) − +7 10.

b) −6 7 18 35.




Rationalize the denominator of radical expressions. (This is a prerequisite skill that is used in Precalculus and Calculus.)

Prior Knowledge:

• Usethepropertiesofexponents.• Simplifysquarerootsofintegers.• Add,subtract,andmultiplypolynomials.• Add,subtract,andmultiplyradicalexpressions.

Model Assessments


Simplify:

a) 4

12.

24

b) 3 2

3.

3x

c) +

−

2 3 5.

7 2


a) 42 54.

b) 3 9

.x

x

c) + + +14 3 2 6 35 5 2.

47




Thetimet(d),inseconds,ittakesforanobjecttofalldfeetcanbemodeledbytheequation

( ) 2.

dt d

g=

Simplifytheright-handsideofthemodel’sequation,i.e.,rationalizethedenominator.


( ) 2.

dgt d

g=


Thedistanced(h),inmiles,tothehorizonatanaltitudeofhfeetabovesealevelisgivenbytheequation

( ) 3.

2h

d h =

Simplifytheright-handsideofthemodel’sequation,i.e.,rationalizethedenominator.


( ) 6.

2h

d h =




Use multiple representations (tables, graphs, and equations) to solve problems that result in radical equations.

Prior Knowledge:

• Applypropertiesofradicalexpressions.• Add,subtract,andmultiplyradicalexpressions.• Applypropertiesofexponents.• Multiplypolynomials.

Model Assessments


Solve:

a) 2 2 6.x x+ = +

b) Usingagraph,explainwhy 3 8x + = − hasnosolution.

Model Assessment Answer(s):a) 31.94.x ≈

b) Therangeofthegraphoftheequation 3y x= + isgreaterthan0;therefore, itwillnevercross.


Body surface area (BSA)mustbe calculated formanymedical reasons, including renalkidneyfunction,cardiacinputandindex,andchemotherapyandglucocorticoiddosing.Theformula,developedin1987byR.D.Mosteller††,is:

2cm kgheight weight

BSA .3600m =

NurseReynoldshasapatientwhoseBSAhasbeen recordedon themedicationchartat2.04squaremeters.Ifthepatient’sheightis157centimeters(5ft.,2in.),whatisthepatient’sweightinkilograms(roundedtothenearesttenth)?


2

cm kg157 weight2.04 .

3600m =

Weight=95.4kg.(roundedtothenearesttenth).

†† Mosteller, R. D., “Simplified Calculation of Body-Surface Area.” The New England Journal of Medicine 1098, no. 317,

1987: 17.


• HealthScienceandMedicalTechnology




Apply and graph transformations of parent functions (quadratic, cubic, square root, absolute value, exponential, and logarithmic).

Prior Knowledge:

• Recognizethegraphof 2.y x=

• Recognizethegraphof 3.y x=

• Recognizethegraphof .y x=

• Recognizethegraphof .y x=

• Recognizethegraphof .xy a=

• Recognizethegraphof ln .y x=

• Findthex-andy-interceptsofanequation.

• Findthevertexofaparabola.

• Determineifthegraphofaparabolaopensupordown.

• Applythepropertiesoflogarithms.

• Applythepropertiesofexponents.

Model Assessments


Identifythetransformationsusedtographthefollowingfromtheirparentgraphs:

a) = − −2( ) ( 3) 2.f x x

b) = − +3( ) 2.f x x

c) =+1

( ) .4

f xx


a) = − −2( ) ( 3) 2;f x x parentgraphshiftedright3unitsanddown2units.

b) = − +3( ) 2;f x x parentgraphshiftedleft2unitsandreflectedoverthex-axis.

c) =+1

( ) ;4

f xx

parentgraphshiftedleft4units.


• AgricultureandNaturalResources• EngineeringandDesign




Solve:

a) = − − +( ) 2 5 4.f x x

b) −= ( 1)( ) 2 .xf x

c) = +2( ) log ( 4).f x x


a) = − − +( ) 2 5 4;f x x parentgraphshiftedright5units,verticallystretchedbyfactorof2,reflectedoverx-axisandshiftedup4units.

b) −= ( 1)( ) 2 ;xf x parentgraphshiftedright1unit.

c) = +2( ) log ( 4);f x x parentgraphshiftedleft4units.


TheaerialviewofMrs.GreenThumb’sflowergardenhasoneboundary identifiedas thegraphofaparabolaandtheotherboundaryisthehouseitself.Theviewusesthecornerofherhouseastheorigin,withthehousesittinginthe3rdquadrant.Theparabolahasbeenreflectedwithaverticalstretchof¼andshifted5feetleftand4feetup.Findtheequationofthegraphinordertomakeasketchofthegarden.Useascalemeasuredinfeet.


Theequationis = − + +21( 5) 4.

4y x

Figure A16-3

House

5

Flower garden

10

4




Identify the parent function (quadratic, cubic, square root, absolute value, exponential, and logarithmic) and transformations for a given transformed function in algebraic or graphical form.

Prior Knowledge:

• Recognizeandidentifythegraphof 2.y x=

• Recognizeandidentifythegraphof 3.y x=

• Recognizeandidentifythegraphof .y x=

• Recognizeandidentifythegraphof .y x=

• Recognizeandidentifythegraphof .xy a=

• Recognizeandidentifythegraphof ln .y x=

• Graphpolynomialfunctions.

• Identifythex-andy-interceptsofagraphofafunction.

Model Assessments


Youaredesigningaheaderforyourcompany’sstationery. Itrequiresaparabolicarcthatbeginsandends1.5inchesbelowthetopedgeofthepaper,andpeaksinthemiddleofthepage1inchbelowthetopofthepaper.Findanequationforthisparabola.


Transformtheparabolay=x2,whichpassesthroughthepoints(-1,1),(0,0),and(1,1).Ifx and y are bothmeasuredininches,thenthisarchasawidthof2inchesandaheightof1inch.Assumingthepagemeasures8½x11inchesandalldistancesaremeasuredfromthebottomleft-handcornerofthepage,thetargetarchasawidthof8½inchesandaheightof½inch.Stretchtheparabolahorizontallybyafactorof4.25andcompressitverticallybyafactorof0.5inordertomatchthetargetmeasurements.Reflectthearcacrossthex-axissothatitopensdownward,andtranslate4.25inchesrightand10inchesupsothatitsvertexisintherightplace.Itsequationwillbe:

− − − =

24.25( 10) 0.5 ,

4.25x

y or = − − +

240.5 1 10,

17x

y or = − + +28 4

9.5.289 17

x xy

Analternativesolutionwouldbetowritey=ax2+bx+c,usethepoints(0,9.5),(4.25,10),and(8.5,9.5)tobuilda3x3linearsystemina,b,andc,andsolvethatsystem.


• Marketing,Sales,andService




Aparabola has an equation of y = –3x 2 + 18x – 11. Complete the square to convert thisequationtovertexform;thenidentifyasequenceoftransformationsthatturnthegraphofy=x 2intothegraphofy=–3x 2+18x–11.


Completethesquaretoobtainvertexformequation: ( )23 3 16.y x= − − +

Sequenceoftransformations:

y=x 2.

Horizontaltranslation3unitsright(replacevariablexwithx–3):y=(x–3)2.

Verticalstretchbyfactorof3(multiplyright-handsideby3or,equivalently,replaceywith3y):

y=3(x–3)2.

Reflectionacrossx-axis(multiplyright-handsideby-1or,equivalently,replaceywith–y):y=–3(x–3)2.

Verticaltranslation16unitsup(add16toright-handsideor,equivalently,replacey withy–16):y=–3(x–3)2+16.


Paireach function in thefirstcolumnwith itsparent function in thesecondcolumn,andidentifyasequenceoftransformationstoturnoneintotheother:

( ) ( )( )

( )

( ) ( )

4

4 1

1. 5 1 3.

2. 3 7.

13. 4 2.

24. 2ln 3 2 .

x

f x x

g x e

h x x

p x x

+

= + −

= −

= − +

= +

4

A. ln .

B. .

C. .

D. .

x

y x

y x

y e

y x

=

=

=

=


1. D.Startwithy=x 4;translate1lefttogety=(x+1)4;verticallystretchbyafactorof5toget y=5(x+1)4;translate3downtogety=5(x+1)4–3.

2. C.Startwithy=ex;horizontallycompress(fixingthey-axis)byafactorof¼togety=e4x;translateleft¼togety=e4x +1;verticallystretch(fixingthex-axis)byafactorof3 to get y=3e4x +1;translate7downtogety=3e4x +1–7.

3. B.Startwith = ;y x translate4lefttoget = +4;y x reflectacrossthey-axistoget

= − + = −4 4 ;y x x compressverticallybyafactorof½toget = −14 ;

2y x translate

2uptoget = − +14 2.

2y x

4. A.Startwithy=lnx;horizontallycompress(fixingthey-axis)byfactor½togety=ln(3x);

translateleft23 to get y=ln3(x+2);verticallystretchbyafactorof2togety=2ln(3x+2).




Namethetransformation(s)shownineachpicture.Giventhatthefunctionshownbythesolidcurveisrepresentedbytheequationf(x)=x3,findanequationforthefunctionrepresentedbythedashedcurve.

a)

Figure A17-4a

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

5

6

b)

Figure A17-4b

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4



c)

Figure A17-4c

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

d)

Figure A17-4d

-2 -1 1 2

-3

-2

-1

1

2

3

e)

Figure A17-4e

-3 -2 -1 1 2 3

-3-2-1

123456789

10




a) Thisisaverticaltranslation,slidingeachpoint3unitsup.Thedashedfunctionhastheequationg(x)=x3+3.Thisisreasonablebecauseg(0)=3andg(1)=4.

b) Thisisahorizontaltranslation,slidingeachpoint2unitstotheright.Thedashedfunctionhastheequationg(x)=(x–2)3.Thisisreasonablebecauseg(2)=0andg(3)=1.

c) Thisisadiagonaltranslation,slidingeachpoint1unitrightand2unitsdown.(Equivalently,itisthecompositionoftwotranslations:oneofthemslidingallpoints1unittotherightandtheotherslidingallpoints2unitsdown.)Thedashedfunctionhastheequationg(x)=(x–1)3–2.Thisisreasonablebecauseg(1)=–2andg(2)=–1.

d) Thisisahorizontalcompressionbyafactorof½.Thedashedfunctionhastheequationg(x)=(2x)3.Thisisreasonablebecauseg(0)=0andg(½)=1.

e) Thisisaverticalstretchbyafactorof8.Thedashedfunctionhastheequationg(x)=8x3.Thisisreasonablebecauseg(0)=0andg(1)=8.Notethat(2x)3=8x3,sothisfigureandthepreviousonehavethesamedashedfunction,eventhoughthetransformationsandviewingwindowsaredifferent.




Compose two linear, quadratic, cubic, square root, absolute value, exponential, or logarithmic functions.

Prior Knowledge:

• Defineafunction.• Usefunctionnotation.• Add,subtract,multiply,anddividepolynomialstographfunctions.• Recognizetheconceptofreflectioningraphicalterms.

Model Assessments


If = −2( ) 6 2f x x and = −( ) 5,g x x

a) find ( )( )6 .f g

b) find .f g

�


a) ( (6)) 4.f g =

b) ( )( ) ( ( )) 6 32.f g x f g x x= = −


Let −

=2 1

( )3

xh x and = −( ) 3 7 .g x x Find ( (4)).g h


( )( )4 0.g h =


• AgricultureandNaturalResources• EngineeringandDesign• PublicServices• Transportation




Anoiltankerstruckasubmergedrockastheresultofanavigationerror.Therockrippedaholeinthebottomofthetanker,spillingoilfromtheoilstoragebunkers.Thespreadofoilleakingfromthetankerisintheshapeofacircle.Theradiusoftheleakagezone(infeet)afterthoursis ( ) = 200 .r t t FindtheareaAoftheleakagezoneasafunctionoftime.


Use π= 2( )A r r torepresenttheareaofthecircleandgiven =( ) 200 .r t t

( )( ) ( ( )) 40000 .A r t A r t tπ= =




Find the inverse of linear, quadratic, cubic, square root, absolute value, exponential, and logarithmic functions algebraically and graphically.

Prior Knowledge:

• Usefunctionnotation.• Add,subtract,multiply,anddividepolynomials.• Identifythegraphsoftheparentfunctionslistedinthecompetency.• Applytheconceptofareflectionacrosstheliney=x.• Applypropertiesofexponents.

Model Assessments


Findtheinverseofthefunction:

−= +2( 4)

( ) 1.5

xg x


− +=1 5 3

( ) .2

xg x


Findtheinverseofthefunction: = − −3( ) 2( 7) 32.f x x


− + = +

131 32

( ) 7.2

xf x






Given: ( ) −= 3

7 48

xp x ,find: ( )−1 .p x


( )3

1 8 4.

7x

p x− +=


Theprofitmadebyacompanymanufacturingshoesisgivenby = + −2( ) 0.01 0.5 100,P x x x wherexrepresentsthenumberofpairsofshoesproduced.

a) FindtheinverseofP(x), −1( )P x for ≥ 0.x

b) Approximatelyhowmanypairsofshoeswouldthecompanyhavetoproducetomakeaprofitof$10,000?


a) Tofindtheinverse,let = + −20.01 0.5 100.y x x

−

= + −

+ = +

+ + = + +

+ = +

+ = +

= + −

= + −

2

2

2

2

1

100 50 10000,

100 10000 50 ,

100 10000 625 50 625,

100 10625 ( 25) ,

100 10625 25,

100 10625 25,

( ) 100 10625 25.

x y y

x y y

x y y

x y

x y

y x

P x x

b) Tomakeaprofitof$10,000,findP-1(10000),whichisapproximately981pairsofshoes.




Find the equation of an exponential function given two points on the graph.

Prior Knowledge:

• Applythepropertiesofexponents.• Usethebasicnotationoffractional

exponents.• Simplifyradicals.

Model Assessments


Findtheequationofanexponentialfunctionthatpassesthroughthepoints(2,264)and(6,4044)andhasahorizontalasymptoteaty=12.


= + .xy ab c

Sincey=12 isanasymptote,y=abx+12.

Sincethegraphpassesthrough(2,264),2

2

2

264 12,

252 ,252

,

ab

ab

ab

= +

=

=

2

252So, 12.xy b

b= +

Sincethegraphpassesthrough(6,4044),

62

4

4

2

2524044 12,

4032 252 ,

16 ,2 ,

252,

263 .

bb

b

bb

a

a

= +

=

==

=

=

Sothesolutionis (63)2 12.xy = +


• FinanceandBusiness• PublicServices• HealthScienceandMedical Technology




Findtheexponentialfunctionthatpassesthrough(2,252)and(6,4032).


= .xy ab

Thesolutionis (63)2 .xy =


IfthepopulationofMathvillewas3,020in1926and5,955in1939,determinethepopulationofMathvillein1984givensteadyexponentialgrowth.


(1926,3020)and(1939,5955)→ (0,3020)and(13,5955).Using = ,xy ab weknowthat = 3020.a

=

=

=

= =

=

13

13

113

,

5955 3020( ) ,

1.9719 ,

(1.9719 ) 1.054,

3020(1.054) .

x

x

y ab

b

b

b

y

Theyear1984is58yearsafter1926,whichiswhywewillusex=58.

So,y=3020(1.054)58=63791.

ThepopulationinMathvillein1984was63,791.


Fiveyearsago,Sam’scarwasworth$15,400.Fiveyearsfromnowitwillbeworth$4,600.Assumingthedepreciationisexponential,whatisitworthnow?(Roundyouranswertothenearestdollar).


-5 15400

5 4600

,0.8862,8416,$8,416.

xy abbay

=≈==

Sam’scariscurrentlyworth$8,416.




Use multiple representations (tables, graphs, and equations) to solve contextualized problems that result in one- and two-step exponential and logarithmic equations.

Prior Knowledge:

• Recognizeinversefunctionsandtheirproperties.

• Usepropertiesofexponents.• Usepropertiesofrealnumbers.• Applythepropertiesofexponents.• Usethebasicnotationoffractional

exponents.• Simplifyradicals.• Applythepropertiesoflogarithms.• Applythepropertiesofexponents.• Applythedefinition/propertiesoflogarithms.• Applythelawsofexponents.• Factornumbersintotheirprimefactorizationswithappropriateexponentiation.• Recognizethatasinglelogarithmicexpressioncanbeexpandedintoanequivalentexpression.• Simplifyorexpandlogarithmicexpressions.

Model Assessments


Findtheparticularexponentialequationforthegraphshown.

Figure A30-1

-4 -3 -2 -1 1 2 3 4 5

-60

-50

-40

-30

-20

-10

10

(-1, -2)

(4, -64)


• AgricultureandNaturalResources• FinanceandBusiness• HealthScienceandMedicalTechnology• PublicServices




Theequationofthisgraphisoftheform ( ) ,xy c b= − whichpassesthroughthetwopoints(–1,–2)and(4,–64).

Theequationofthegraphis 4(2) .xy = −


Onanostrichfarm,assumethenumberofostrichesinapopulationdoublesapproximatelyevery10months,andthefarmbeginswith6ostriches.

a) Writeanequationtomodelthenumberofostrichesatagiventime.

b) Howmanyostricheswilltherebeafter5years?

c) Whenwillthenumberofostrichesreach10,000?


a) Equationis:P=6(2)xwherexrepresents10-monthperiods.

b) 5years=60months⇒ 6ten-monthperiods,sox=6andP=384ostriches.

c) P=10,000, 10.7,x ≈ so107monthsor8years,11months.


Ateacherhasjustcontractedabaldingbacterialvirus.Hecurrentlyhassixbacteriapresentinhissystemandthebacteriadoubleevery4.5hours.Whenthebacteriareach1billion,theteacherwillbeforcedtofacebaldness.

a) Howlongdoestheteacherhavebeforethebacteriahitthecriticallevel?

b) Afterthreedays,theteachershowsthefirstsignsofthedreadeddisease.Hestartstakingantibioticstokilloffthevirus.Theantibioticswillstoptheproductionofanynewbacteriaandwillkill20%oftheviruseveryhour.Whenthenumberofbacteriadropsbelowone,heiscured.Whenwillhebecured?


a) Findthegrowthrateofthebacteriabyusingtheexponentialfunction = 0ktP P e whereP

isthepopulationatagiventime,t;P0istheinitialpopulation;kisthegrowthrate;andt

isthetime(inhours).Weknowthat =0 6P andthatwhen 4.5t = hours,Pwillbe12.122.9t ≈ hours,orabout5days.



b) Thepopulationofthebacteriaafterthreedayswillbewhen 3(24) 72t = = hours.

= =ln2 (72)4.56 393,216P e bacteriaafter3days.

Adifferentexponentialmodelisneededtoaccountforthedeathof20%oftheviruseveryhour.After1hour,thebacteriacountwillbe:

393216 (0.2)(393216) (0.8)(393216).− =

Usethistofindthedecayrate,k: ( )1(0.8)(393216) 393216 ,ke=

ln0.8 ln ln .ke k e k= = =

Tofindoutwhenhewillbecured,let 1P = andsolvefort: (ln0.8)1 383216 .te=

57.6t ≈ hours,orabout2.4days.


Theequation120.06

112

t

A P = +

modelstheamountofmoneyinabankaccountbalanceafter

tyears,withinterestcompoundedmonthly.

a) DoesP represent thepopulation, interest rate,finalbalance, startingbalance,or timeelapsed?

b) Whatdoes0.06represent?


a) Prepresentsthestartingbalance.

b) 0.06representsthe6%annualinterestrate.


ThevalueofMr.Investor’sstocksisplummeting.Eachmonthhisportfolio(thesumofhisstocks)loses4%ofitsvalue.Currentlytheportfolioisworth$512,342.

a) Atthisrate,whatwillhisportfoliobeworthinayear?

b) Whatwashisportfolioworth8monthsago?




Thegeneralformoftheequationis

( )( ) ,toA t A r=

whereA0istheinitialamountintheportfolioandristhepercentagerate.Sincethestockloses

4%ofitsvalueeverymonth,thepercentagerateremainingeachperiodis96%.Thespecific

formoftheequationforthisproblemis:

( )( ) 512342 0.96 ,tA t =

whereA(t)istheamounthisportfolioisworthaftertmonths.

a) Tofindoutwhattheportfoliowillbeworthinayear, 12t = (sincetisinmonths)and

( ) ( )1212 512342 0.96 $313,916.94.A = ≈

b) Todeterminewhathisportfoliowasworth8monthsago,

( ) 8( 8) 512342 0.96 $710,215.42.A −− = ≈


Jenniferhad$1,250inasavingsaccount3yearsago.Theannual interestrateatthetimeofdepositwas6%,compoundedquarterly.

a) Writeanequationtorepresenttheamountofmoneyintheaccount.

b) Howmuchmoneyisintheaccounttoday?

c) Howmanyyearswillittakeforthemoneyintheaccounttodouble?


a) ( ) 1250(1.015) ,tA t = t=#ofquarters.b) 3years=12quarters,t=12and $1494.52.A ≈

c) 46.56t = quarters,sotheamountwilldoublein11.6years.




a) Writeinexponentialform: =2log 64 .y

b) Writeinlogarithmicform: + =2 1 59 3 .x x

c) Application:Buildatableofvaluestofindfourpointsonthegraphof ( ) 3 .xf x =

d) Usethistabletographthefunction.

e) Usethetabletobuildacorrespondingtablefor ( )−1 .f x

f) Graphthatinverse.Writethedomainandrangeofeach.

g) Explainhowthedomainandrangeofexponentialand logarithmic functions relate toeachother.

h) Betweenwhichtwointegersmust 5log 21lie?


a) =2 64.y

b) = +59log 3 2 1.x x

c)

x 0 1 2 3

y 1 3 9 27

d) Graph = 3xy (below);domain ( ), ;−∞ ∞ range ( )0, .∞

Figure A21-8d

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-5-4-3-2-1

12345678

e)

x 1 3 9 27

y 0 1 2 3



f) Graphoftheinverseof = 3xy (below);domain: ( )0, ;∞ range: ( ), .−∞ ∞

Figure A21-8f

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

-5

-4

-3

-2

-1

1

2

3

4

5

g) Foranyexponentialequationanditsinverselogarithmicequation,thedomainandrangewilltradeplaces.

h) Between1and2.


Solveforx:

a) ( )4.2 3.7 2 26.46.x − =

b) =log (5,280) 8.x

c) ( )( )+= 318 2 3 .x

d) ( ) ( )+ + − =3 3log 3 log 5 2.x x

e) ( )− − =2 2 2log log 6 log 21.x x

f) ( )+ + =2 2log 10 log 8 1.x




a) 28.46

log4.2

,log3.7

1.4625.

x

x

=

≈

b) 2.9196.x ≈

c) ( )( )

( )( )

( )( )

+

+

+

=

=

=

= +− =

3

3

32

18 2 3 ,

9 3 ,

3 3 ,

2 3,1 .

x

x

x

xx

d) 6.x =

e) 63/10.

f) -39/5.


Choosetheexpressionthatisnotequivalentto: + −log log log :x y z

a)

log .xyz

b) ( )+ +log .x y z

c) −log log .xy z

d) +

log log .y

xz


b) isnotequivalent.




Ascientistlefta32-gramsampleofaradioactivesubstanceoutanditwasexposedtotheair.Thesubstancedecaysatanexponentialrate;after30minutes,only26gramsremained.

a) Howmuchofthesubstancewasleftwhenthejanitorfoundit8hoursafterthescientistleft?

b) Whatisthehalf-lifeofthesubstance?


( ) = 0 ,ktA t A e

a) 1.15gramsleftafter8hours.

b) Half-lifeisabout1.670hours.


Inanursinghome,thestaffmustmakesurethatthecleaningsolutionwillnotbeharmfultothepatients.IfthepHislowerthan5,itwillbetooacidicandcouldbedangerous.TheconcentratedsolutionavailablehasapHof4andneedstobediluteduntilthepHis5.ThepHscaledefines thepHofa solutionas = −

+pH log H where

+H is theconcentrationofhydrogenionsinthesolutioninunitsofmolesperliter.If1literofcleaningsolutionofpH4isaddedto9litersofwater,whatwillbetheresultingpHofthesolution?


ThepHofthestartingsolutionis4,sowehave:+4 log H . = −

Multiplybothsidesby-1:+4 log H , − =

thenrewriteinexponentialform:4 +10 H ,− =

sotheconcentrationofhydrogenionsis10-4moles/liter.

Thisneedstobemultipliedbythe1literofstartingsolutiontogive:

− −=4 4(10 moles/liter)(1liter) 10 moles.

When9litersofwaterareadded,the10-4molesofhydrogenionsarenowin10litersofsolution,sotheconcentrationis:

−−=

4510 moles

10 moles/liter.10liters

TofindtheresultingpH,pH=–log(10–5)=–(–5)=5,sothesolutionwillbesafe.




Carbon-14isusedindatingorganismsthathavedied.Thehalf-lifeforCarbon-14todecaytoNitrogen-14is5,730years.FindtheequationthatcanbeusedtodeterminetheamountofCarbon-14,C(t),remainingaftertyears.


Thehalf-lifeofaradioactivedecaycanbedeterminedfromtheexponentialdecayequation

0( ) .rtC t C e=

Theequationis 0.000120968( ) 100 .tC t e−=




Graph simple conics, including:

• a circle given its equation (centered anywhere)

• an ellipse given its equation (centered at the origin)

• a hyperbola given its equation (centered at the origin)

Prior Knowledge:

• Performarithmeticcomputationswithrationalnumbers.

• Graphalinearequation.• Graphaquadraticequation.• Convertbetweengeneralquadratic

equationandstandardform.• Interprettheconceptofasymptote.• Drawarectanglethroughfourgivenorderedpairs.• UsethePythagoreanTheorem.• Usethedistanceformula.

Model Assessments


Graph:x2+(y –1)2=4.


center=(0,1),radius=2

Figure A22-1

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456


• EngineeringandDesign• Transportation




Graph:x2+y2+4x–6y+12=0.


(x+2)2+(y–3)2=1.

center=(-2,3),radius=1

Figure A22-2

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456


Thereisanexplosioninapipelineinanurbanneighborhood.Theareaofthedamageisroughlycircular,andtheouterlimitisdescribedbytheequationx 2+y 2+2x+4y–20=0,withthepoint(0,0) locatedat the localfire station.Find the locationof theexplosion (locatedat thecenterofthecircle)andtheblastradius.


(x+1)2+(y+2)2=25.

Thecenterofthecircle(thesiteoftheexplosion)is1unitwestand2unitssouthofthefirestation.Theblastradiusis5units.




Findthecoordinatesoftheverticesandfocioftheellipse,thengraphtheellipse: + =2 2

1.9 4x y


Figure A22-4

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

Vertex (3, 0)

Vertex (-3, 0)

Foci

center=(0,0)

vertices=(3,0)(-3,0)

foci= ( )5, 0±


Anarchintheshapeoftheupperhalfofanellipseisusedtosupportabridgespanningariverthatis20meterswide.Thecenterofthearchis6metersabovethecenteroftheriver.Writeanequationfortheellipseinwhichthex-axiscoincideswiththewaterlevelandthey-axispassesthroughthecenterofthearch.Makeagraphtorepresentthissituation.


Theriveris20meterswide,soa=10meters;thebridgeis6metersoverthewater,sob=6meters:

= − → = − → = → = ±2 2 2 2 236 100 64 8.b a c c c c Equationis: + =2 2

2 21.

10 6x y

Figure A22-5

-12-10 -8 -6 -4 -2 2 4 6 8 10 12-1

12

3

4

5

6

78

9

10



Assessment Example 6 :

Graphthehyperbolaandshowtheasymptotes:

− =2 2

1.16 4x y


asymptotes: =12

y x and = −1.

2y x

Figure A22-6

-10 -8 -6 -4 -2 2 4 6 8 10

-8

-6

-4

-2

2

4

6

8


TheLORANsystem(LongRangeNavigation)useshyperbolastodetermineaship’spositionatsea.Twostationstransmitradiopulsestotheship.Thedifferenceinthetimesofarrivalofthesepulsestotheshipisaconstantonthehyperbolawiththestationsasthefoci.

Supposetwostations(AtowestofcenterandBtoeast)are300milesapartalongastraightsectionofshore.Ashipapproachingtheshoredeterminesthatitis80milesfartherfromstationAthanitisfromstationB.

a) Findtheequationofthehyperbolaonwhichtheshipislocated.

b) Iftheshipis50milesoffshore,findthecoordinatesforitslocation.




Graph:

Figure A22-7

-200 -150 -100 -50 50 100 150 200

-200

-150

-100

-50

50

100

150

200

Station B (150, 0)

Station A (-150, 0)

Ship

a) Ifstationsare300milesapart,thecoordinatesofthefociare(-150,0)and(150,0)andc=150.Bydefinitionofahyperbola,thedifferenceindistancesfromapointtothefociisequalto 2a,so2a =80anda=40.

Forahyperbola,b2=c2–a2,sob2=22500–1600,or20900.

Theequationofthehyperbolais: − =2 2

1.1600 20900

x y

b) Iftheshipis50milesfromshore,theny=50.

2

2

25001,

1600 209002500

1600 1 ,20900

1791.1388,42.32.

x

x

xx

− =

= +

≈ ±≈ ±

BecausetheshipisclosertostationB,wewouldusethepositivexvalue.

Thecoordinatesoftheshipare(42.32,50).




Solve statistical problems:

• Computepermutationsusingfundamentalcountingprinciple.• Computecombinations.• Computeprobabilitiesusingcombinations.• Computeprobabilitiesusingpermutations.• Expandbinomialexpressionsusingthebinomialtheorem.

Prior Knowledge:

• Defineanduseafactorial.• Applypropertiesofexponents.

Model Assessments


Abankmanagerassignspasswordstoeachemployee.Howmanypasswordscanbeassignedusingthreelettersandfivedigits:

a) withrepetitions?

b) withoutrepetitions?


Permutationsusethefundamentalcountingprinciple.Abankmanagercanassign:

a) withrepetitions,3 3 3 5 5 5 5 16,875passwords.=

b) withoutrepetitions,3 2 1 5 4 3 2 720passwords.=


Arestaurantmanageroffersavaluecombomealwithachoiceof12varietiesofsandwiches,6varietiesofdrinks,and5varietiesofchips.Howmanypossiblecombomealsareavailabletocustomers?


12 6 5 360= possiblecombomeals.


• FinanceandBusiness• Hospitality,Tourism,andRecreation• AgricultureandNaturalResources




a) Theteammanagerofthelocalcoffeeshophastopickfouremployeestorepresenttheirstoreattheregionalsalesmeeting.Howmanycombinationsofemployeesarepossibleifthereare15employeesfromwhichtochoose?

b) Therearethreepositionsopenat the localnewspaper. If thereare 12applicants,howmanystaffingassignmentsarepossible?


Combinationsusethefundamentalcountingprinciple:

a) 15 415

1,3654

C

= =

combinations.

b) 12 312

2203

C

= =

staffingassignmentsifthesethreepositionsarethesame,and

12 3 12 11 10 1,320P = = staffingassignmentsifthesethreepositionsarealldifferent.


Thereare12peopleatXYZCompanywhowanttoattendaconference.WhatistheprobabilitythatJoseandAliciawillbethetwochosen?


12 212

662

C

= =

waystochoose2peoplefrom12.

2 22

12

C

= =

wayforJoseandAliciatobechosentogether.

Theprobabilityis1/66=0.01515.


Amediafirmishavingacompany-widecontesttoseewhocancreatethebestcommercialtomarkettheirservices.Theywillbeawardingprizesforfirst,second,andthirdplacewinners.There are five departments in the firm and each department has 10 employees. If eachemployeesubmitsacommercial,whatistheprobabilitythatthethreewinningcommercialswillcomefromonedepartment?




Thereare(5)(10)=50totalemployees.

Thereare 50 3 117,600P = waystoawardthreeprizes.

Thereare 8 3 336P = waysforaspecificdepartmenttowinallthreeprizes.

Theprobabilityforaspecificdepartmenttowinallthreeprizesis:

= =

8 7 6 3360.002857.

50 49 48 117600

Anyoneofthefivedepartmentscouldbethewinningdepartment,sotheprobabilitythatthethreewinningcommercialscomefromanyonedepartmentis:

( ) =

3365 0.01429.

117600


Adogbreedermatesablacklab(b)andagoldenretriever(g).Theprobabilityofhavingblackfuris0.6.Writethetermintheexpansionof(b+g)6 foreachofthefollowingoutcomes:

a) exactlytwoblack-furredpuppies.

b) exactlythreegolden-furredpuppies.


( ) ( )= = − =0.6and 1 0.6 0.4.P b P g

Intheexpansionof(b+g)6foratotalof6puppies,theprobabilityofgetting

a) exactlytwoblack-furredpuppiesis:2 4 2 4

6 2[ ( )] [ ( )] 15(0.6) (0.4) 0.1382;C P b P g = =

b) exactlythreegolden-furredpuppiesis:3 3 3 3

6 3[ ( )] [ ( )] 20(0.6) (0.4) 0.2765.C P b P g = =


Usethebinomialtheoremtoexpandtheexpression 5(2 3) .x +


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )5 0 4 3 2 2 3 4 0 5

5 4 3 2

5 5 5 5 5 52 3 2 3 2 3 2 3 2 3 2 3

0 1 2 3 4 5

32 240 720 1080 810 243.

x x x x x x

x x x x x

= + + + + +

= + + + + +




Compute the general term and sums of arithmetic series, finite geometric series, and infinite geometric series.

Prior Knowledge:

• Evaluatealgebraicexpressions.• Simplifyexpressionspriortosolvinglinearequations.• Solvemultistepproblemsincludingwordproblemsinvolvinglinearandnonlinear

equations.• Usenotationinvolvingsubscriptsandexponents.• Differentiatebetweenafiniteset

ofnumbersvs.aninfinitesetofnumbers.

Model Assessments


Aclientofacontractorwantsabrickpatiofloorinthebackofhishouse.Hewants36brickstobethefirstrowofhispatiofloorandeachsuccessiverowtohavetwofewerbricks.

a) Writearuleforthenumberofbricksinthenthrow.

b) Whatisthetotalnumberofbricksneededforthispatiodeck,ifthelastrowhas10bricks?


a) an=36–2(n–1).

b) S14=322bricks.


In2000,about680homesweresoldinalocalcommunity.From2000to2010,thenumberofhomessolddecreasedby7%eachyear.

a) Writearuleforthetotalnumberofhomessoldintermsoftheyear(n=1fortheyear2000).

b) Whatwasthetotalnumberofhomessoldfrom2000to2010(inclusive)?


a) an=680(0.93)n–1.

b) S11=5,342homes.


• BuildingTradesandConstruction• FinanceandBusiness• Marketing,Sales,andService




Anofficesupplystorehadaprofitof$84,000thefirstyearitopened.Withthedepressionoftoday’seconomy,theirprofithasdecreasedby15%peryear.

a) Writearuleforeachyear’sprofitfortheofficesupplystore.

b) Withthedepressionoftheeconomy,whatisthetotalprofittheofficesupplystorecanmakeoverthecourseofitslifetime?


a) an=84000(0.85)n–1.

b) S=$560,000.


P r e c a l c u l u s C o m p e t e n c i e s 87

Precalculus Competencies


P r e c a l c u l u s C o m p e t e n c i e s88


Perform matrix operations, including addition and multiplication, and calculate the determinants of 2x2 and 3x3 matrices.

Prior Knowledge:

• Solvesystemsofequationsintwoandthreevariablesusingsubstitutionandaddition/eliminationmethods.

• Writeasystemoflinearequationsinmatrixform.

• Performbasicmatrixrowoperations.

• Identifythedimensionsofamatrix.

• Identifytheelementsofamatrix(a11, b23,etc.).

Model Assessments


Addthefollowingmatrices,ifpossible.Ifnotpossible,explainwhynot.

Giventhesematrices,

2 16 4 10 3 5 2, , 5 3 ,5 2 10 1 4 7

0 43 3 1 9 3 0A B C

− − = = = −− − −

−− − − −

addthefollowing:

a) A+C

b) B+A


a) A+CcannotbeperformedbecausethedimensionsofAandCaredifferent.

b)9 1 8

.4 2 312 0 1

B A−

+ = − − − − −


• BuildingTradesandConstruction• FinanceandBusiness




Applymatrixoperationstosolveproblemsinvolvingmultiplication.Multiplythefollowingmatrices,ifpossible.Ifnotpossible,explainwhynot.

a)12

23

43 2 5.

5 1 63

− − −

b)12

23

4 5 3 2.

5 163

− − −


a)

9 323

2 3 .11 62

312 3

− −

b) Matricescannotbemultipliedbecausetheinnerdimensionsdonotmatch.


Acontractor isdevelopingabudget forhercurrentconstructionprojects.Hercompanyisbuildingsingle-familyhomes,duplexes,andcondominiumcomplexes.Thecostsforthevarious categoriesofbuilding supplies are as follows: lumber costs $5,000 for a single-familyhome,$8,000 foraduplex, and$125,000 foracondominiumcomplex.Thecostsforwindowsare$3,000forasingle-familyhome,$5,000foraduplex,and$75,000foracondominiumcomplex.Concretecosts$7,500forasingle-familyhome,$12,000foraduplex,and$170,000foracondominiumcomplex.Plumbingsuppliesforeachrun$2,500,$3,500,and$50,000forthesingle-familyhome,duplex,andcondominiumcomplex,respectively.Roofingmaterialsforeachrun$5,000,$8,000,and$100,000forthesingle-familyhome,duplex,andcondominiumcomplex,respectively.

Thisyearsheisplanningtobuild10single-familyhomes,4duplexes,and7condominiumcomplexes. Use matrices to determine her total costs this year for each category ofconstructionmaterials.


5000 8000 1250003000 5000 75000 107500 12000 170000 4 ,

72500 3500 500005000 8000 100000

Lumber 957000Windows 575000Concrete 1313000 .Plumbing 389000Roofing 782000

=




Applymatrixoperationstosolveproblemsinvolvingdeterminants.Findthedeterminantofthefollowingmatrices,iftheyexist:

a) 3 2.

5 1−

b)12

23

4 5.

63

− −

c)−

− −

4 2 12 1 2 .5 10 0.5


a) -13.

b) Determinantdoesnotexist;notasquarematrix.

c) 75.


Clarisseinvestedatotalof$60,000inmoneymarketaccountsatthreedifferentbanks.BankApays2%interestperyear,BankBpays2.5%,andBankCpays3%.ShedecidedtoinvesttwiceasmuchinBankBasintheothertwobankscombined.After1year,Clarissehadearned$1,575ininterest.Howmuchdidsheinvestineachbank?UseCramer’sRuletosolvethissystemofequations.


+ + = 60000,A B C

2 2 0,A B C− + =

0.02 0.025 0.03 1575.A B C+ + =

BankA=$2,500

BankB=$40,000

BankC=$17,500




Describe different types of functions (polynomial, exponential, logarithmic, and trigonometric) using a table, a graph, an equation, and a verbal description.

Prior Knowledge:

• Determinethezeros(x-intercepts)ofpolynomialfunctions.

• Factorpolynomialsusingstandardfactoringtechniques.

• Apply appropriate function vocabulary (domain, range, asymptotes, etc.) todescribepropertiesoffunctions.

Model Assessments


ACCESSFoodServiceispreparingpiestoservefordessertinthediningcommons.Theywanttoservethepiesassoonaspossible,butneedthemtocoolafterremovingthemfromtheoven.Toavoidinjuries,thepiesneedtohaveaninternaltemperatureof120°Forlower.Therateofcoolingwillfollowadecreasingexponentialfunctionthatapproachestheambienttemperatureoftheroom,whichis75°F.Apieistakenoutoftheovenanditstemperatureismeasured.

• After2minutes,thepie’stemperatureis311°F.

• After5minutes,thepie’stemperaturehasdroppedto276°F.

a) Sketchagraphthatshowshoweachpie’stemperaturedecreasesovertimeafterbeingtakenoutoftheoven.Includeallimportantfeaturesofthegraph.

b) Temperature can be modeled by the function ( ) .tT t am k−= + The constant a is thetemperaturedifferencebetweenthepieandtheambientroomtemperatureatt=0andmisthetemperaturedecayrate.Whatisthevalueofk?

c) Usingthedatafromabove,findthefunctionforthepietemperature.

d) Whattemperatureisthepiewhenitfirstcomesoutoftheoven(time=0)?

e) Whenisitsafetoservethepies?


• AgricultureandNaturalResources




• Hospitality,Tourism,andRecreation




Figure P2-1

10 20 30 40 50 60 70 80 90

255075

100125150175200225250275300325350

Time (min)

Tem

p (o F)

a) Thegraphstartsatthey-interceptofthetemperatureatwhichitisremovedfromtheoven.Thehorizontalasymptoteisattheroomtemperature,whichis75°.

b) Thefunctionis ( ) ,tT t am k−= + sothevalueofkisroomtemperature,whichis75°.

c) Theequationis ( ) (262.7)(1.055) 75.tT t −= +

d) (0) 337.7;T = thepiecomesoutoftheovenat337.7°F.

e) Tofindwhenthepiecanbeserved,setthetemperatureto120°andsolvefort.

Thepiecanbeservedafter33minutes.


Acabinetmakerneedstomakearectangulardrawerthatwouldhaveavolumeof4,500cubiccentimeters.Becauseofwhatthedrawerwillhold,thelengthofthedrawerneedstobetwicethewidth.

a) Determinetheformulaforthesurfaceareaasafunctionofthewidthofthedrawer.

b) Makeatableofthesurfaceareaasafunctionofthewidthofthedrawer.

c) Drawagraphofthesurfaceareaasafunctionofthewidthofthedrawer.

d) Useagraphingcalculatortofindtheheightsothatthesurfacearea isassmallaspossibletousetheleastamountofwood.




Let L =lengthofthedrawer,W =widthofthedrawer,H =heightofthedrawer,andV =volumeofthedrawer.

V=LWH=4,500cubiccentimeters.

a) Surfacearea= ( ) 22 13500 / .SA W W W= +

b) Table:

Width (W) Surface Area = 2W2 + 13,500/W5 2,750

10 1,550

15 1,350

20 1,475

25 1,790

c) Graph

Figure P2-2

5 10 15 20 25

Width (cm)

Volu

me

(cm

3 )

4000

3000

2000

1000

0 0

d) ThegraphshowsthattheminimumsurfaceareaappearstooccurwhenW =15.WhenW =15cm.,H=10cm.




Ifthecost,indollars,ofcleaningupanoilspillisgivenbythefunction 2,000,000( ) ,

100f x

x=

−

wherexisthepercentofthespillthathasbeencleanedup:

a) Howmuchwillitcosttocleanup50%ofthespill?80%?90%?

b) Drawagraphofthecostfunction.Describethebehaviorofthegraph.

c) Ifthecompanyisgoingtospend$4,000,000cleaningupthespill,whatpercentwillbecleanedup?

d) Will 100% of the spill be cleaned up?What feature of the graph leads you to thisconclusion?


a) Findthevaluesofthefunctionwherex=50,then80,then90.

(50) 40000, (80) 100000, (90) 200000.f f f= = =

Itwillcost$40,000tocleanup50%ofthespill,$100,000tocleanup80%,and$200,000tocleanup90%.

b) Graph

Figure P2-3

10 20 30 40 50 60 70 80 90 100

Percent of spill (%)

Cos

t of c

lean

up ($

)

800K

600K

400K

200K

0K 0

1000K

c) Setf(x)=4,000,000andsolveforx.

Soifyouspend$4,000,000,youwillbeabletocleanup99.5%ofthespill.

d) It’simpossibletocleanuptheentirespill.Theamountofmoneyapproachesinfinityasx approaches100%.




As sand isbeingunloaded fromahopper at a construction site, the risingpile is in theshapeofacone.Asmoreandmoresandpoursontothepile,thepilegrowsbuttheconesatdifferentstagesarealwayssimilar.After1minute,theheightoftheconeis5feetandtheradiusofthebaseis9feet.After3minutes,theheightis12feet.

a) Findanequationfortheareaofthebaseasafunctionoftime.

b) Howmuchareaonthegrounddoesthepilecoverafter3minutes?


a) Findtheareaofthebaseasafunctionoftime.

Volumeformulaforarightcircularcone: 213 ,V r hπ= wherer=radiusofthecircularbase

andh=heightofthecone.Assumingthatthesandispouringoutataconstantrate,the

radiusofthebasewillchangeatalinearrate.

Areaofacircle: ( ) ( )22 6.3 2.7 .A r t A tπ π= = + =

b) Areacoveredatt=3.

( ) ( ) ( )2 23 6.3 3 2.7 21.6A π π = + = sq.ft.




Graph piecewise-defined functions.

Prior Knowledge:

• Graphlinear,quadratic,andradicalfunctions.

• Identifythedomainandrangeofafunction.

Model Assessments


a) Graph: − ≤=

+

214 for 2< 2,

( )for >2.2

x xf x

xx

b) Graph: ( )−

= − ≤ ≤

2ifx<0,

1 if0 x 2,1 ifx>2.

x

f x x


a)

Figure P3-1a

-5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

-5

-4

-3

-2

-1

1

2

3

4

5


• EnergyandUtilities• FinanceandBusiness



b)

Figure P3-1b

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-5

-4

-3-2

-1

1

2

34

5

6


HonestAbeFinancehad to revalue theirportfolio inmid-2008.Theirmonthlyprofit formonthtof2008isgivenby:

+= − ≥

25000 500000, 0< <6,( )

75000 50000 , 6.t t

P tt t

Graphthisfunctionfor2008.


Figure P3-2

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

Month in 2008

Mon

thly

Pro

fit ($

100K

)

0 100

300

500

700

-200

-400

-600




SanCalpassiaGas & Electric charges its residential customers a base rate of $7.00 permonthplus$0.06perkilowatt-hour(kWh)forthefirst491kWh,and$0.111perkWhforallusageabovethatamount.Thereisanadditionalsurchargeof$0.02perkWhforelectricusageduringpeak-demandhours.

a) Graphtheenergychargefunction(amountcustomershavetopay)fornormalusage,notincludingthepeak-demandsurcharge,upto1,000kWhofelectricityusage.

b) DuringthemonthofAugust2008,yourenergyusagewas753kWh.Ofthat753kWh,75kWhwereusedduringthepeak-demandtimeperiod.DeterminethetotalcostoftheenergybillforthemonthofAugust2008,excludingtaxesandadditionalsurcharges.


( ) ( )7.00 0.06 for 491 ,36.46 0.111 491 for 491 .

x x kWhC x

x x kWh+ ≤= + − >

a) EnergychargefunctionforSanCalpassiaGas&Electric:

Figure P3-3

200 400 600 800 1000

102030405060708090

100110120

0 Energy usage (kWh)

Ener

gy c

harg

e ($

)

b) ( ) ( )+ =753 0.02 75 $67.04.C




Find all real zeros (roots) of polynomial functions exactly using the rational root theorem.

Prior Knowledge:

• Applystandardfactoringtechniquestopolynomials.

• Dividepolynomialsusinglongdivisionandsyntheticdivision.

• Solvepolynomialequations.

Model Assessments


Listthepossiblerationalzeros;findallrealzeros:

= + − −3 2( ) 4 10.f x x x x


Possiblerationalzeros: 1, 2, 5, 10.± ± ± ±

Realzeros:1

2, 6.2

− − ±


Listthepossiblerationalzeros;findallrealzeros:

= − + −3 2( ) 3 3 9.f x x x x


Possiblerationalzeros: 1, 3, 9.± ± ±

Realzeros:3.


• ManufacturingandProductDevelopment




Inmetalshop,youhavebeentaskedwithcreatingaprototypemetaltraythatisopenontop.Yourstartingmaterialisarectangularsheetoftinthatis22centimetersby28centimeters.Youaretocutsquaresofequalsizesfromeachcornerandfoldtheresultingflapsuponeachsidetoformthetray.

a) FindaformulaforthevolumeV(x).

b) Whatisthedomainofthisfunction?

c) SketchthegraphofV(x).

d) Forwhichvalue(s)ofxdoesthetrayhaveavolumeof1,080squaredcentimeters?


a) ( ) ( )( )= − − = − +3 228 2 22 2 4 100 616 .V x x x x x x x

b) ∈ (0,11).x

c) GraphofV(x):

Figure P4-3c

5 10 15 20-200

0

200

400

600

800

1000

1200

1400

d) Twosolutions: 5 cm. or 3.2 cm.x x= ≈




Find all the zeros (roots) of a polynomial function.

Prior Knowledge:

• Dividepolynomialsusinglongdivisionandsyntheticdivision.

• Applystandardfactoringtechniquestopolynomials.


• Identifyrealandcomplexsolutionstopolynomialequations.

Model Assessments


Findallthezeros(roots)ofthefunctionsbelow.

a) ( ) = − + −3 23 3 9.f x x x x

b) = − − − −4 3 2( ) 2 2 2 3.g x x x x x

c) = − + − + −5 4 3 2( ) 2 2 4 2.h x x x x x x


a) f(x)are 3, 3.x i= ±

b) g(x)are = − ±1, 3, .x i

c) h(x)arex=2,+i(multiplicityof2),-i(multiplicityof2).


• EngineeringandDesign• FinanceandBusiness• ManufacturingandProductDevelopment• Marketing,Sales,andService




Youhavebeentaskedtodesignaboxforshippinganewproduct.Theboxistobemadefromasinglesheetofcardboardthatis30centimetersby60centimeters.Theboxistobeformedbycuttingoutequalsizesquarepiecesfromeachcornerandfoldingupthesides.Theresultingboxformedhastohaveavolumeof2,400cubiccentimeters.Whatarethedimensionsofthecutoutsectionsofthecardboardtothenearestcentimeter?


Therearetwosolutionstotheproblem: ( ) ( )( )= − −60 2 30 2V x x x x wherexisthedimensionofasideofthecutoutsquare.

Solution1:1.57cm.

Solution2:12.24cm.


Youandyour friendshavedecided thatall thecurrentenergydrinkson themarketarelackinginaspecialingredientthatyouhavefoundmakesasignificantdifferenceintermsofperformanceandlackof“crashing”astheeffectswearoff.YouhavedonesomepreliminarymarketinganalysisanddeterminedthatthedemandfunctionD(x),whichrelatesthenumberofcasesconsumerswillpurchaseataspecificpricex,isgivenbythefollowingpolynomial:

= − +2( ) 200 10000D x x x for0 100.x≤ ≤ ThesupplyfunctionS(x)givesthenumberofcasesyoucanmakeifyouaresellingthematapriceofxdollarsandisgivenbythefollowing:

( ) = + + −3 23 500 5000S x x x x for 0 100.x≤ ≤ Thebusinessexpertssaythatyoushouldsetthepriceofyourproductsothatthesupplyequalsdemand,i.e.,S(x)=D(x).

a) Whatpricedoyouhavetochargejusttomakethefirstcase?(HINT:S(x)=0.)

b) Atthatprice,howmanycasesdoyouexpecttosell?

c) Whatpriceshouldyouchargeforthebusinessexpertstobehappy?


a) $8.39.

b) 8,392cases.

c) $15.46.




Solve polynomial inequalities.

Prior Knowledge:

• Solvelinearinequalitiesandgraphthesolutionspace.


Model Assessments


Solveforx: − + <3 25 6 0.x x x


( )− + <2 5 6 0,x x x

( )( )− − <3 2 0.x x x

Thiswillbetrueforx <0or2<x<3or ( ) ( ), 0 2, 3−∞ inintervalnotation.


Solveforx: − − >3 22 3 0.x x x


( )− − ≥2 2 3 0,x x x

( )( )− + ≥3 1 0.x x x

Thiswillbetruefor − ≤ ≤1 0x andfor ≥ 3x or [ 1, 0] [3, )− ∞ inintervalnotation.






Arectangularenclosureforstoringconstructionequipmentmusthaveanareaofatleast3,200squareyards.Ifthereare240yardsoffencingavailable,andthewidthcannotexceedthelength,withinwhatlimitsmustthewidthoftheenclosurelie?


Thewidthmustbebetween40and60yards,inclusive.




Solve rational inequalities.

Prior Knowledge:

• Solvelinearinequalities.

• Solverationalequations.

Model Assessments


Solveforx:+

>+

40.

2xx


( ) ( ), 4 2, .−∞ − − ∞


Solveforx: + ≥ −−5 4

2.1x x


5 42 0

1x x+ + ≥

− simplifies to

( )( )( )

2 1 40.

1

x x

x x

− +≥

−

Criticalnumbersare:½,-4,0,1.

Thesolutionsetis ( ( ( )12, 4 0, 1, .−∞ − ∞


• FinanceandBusiness• ManufacturingandProduct

Development• Marketing,Sales,andService




The set-up cost for manufacturing a roller for a large printer is $2,000. The cost tomanufactureeachroller is$1,050.Writeaformulaforthecost,C(x),asafunctionofthenumberofrollersproduced.Theaveragecostisgivenby ( ) .C x x Ifthecustomeriswillingtopaynomorethan$1,200foreachroller,whatwouldtheminimumorderneedtobe?


Solution:

C(x)=2000+1050x,

>40

.3

x

However,because theanswermustbeanatural number (thequestion is aboutdiscreteobjects),theminimumorderwouldbe14rollers.




Find the composition of two or more functions, each containing two or more operations (such as linear, quadratic, rational, cubic, square root, cube root, and trigonometric functions).

Prior Knowledge:

• Evaluatefunctions.

• Identifythedomainandrangeoffunctions.

• Identifyanalgebraicexpressionaseitherafunctionorarelation.

Model Assessments


a) Letf(x)=4x2+6andg(x)=3x–7.Findf(g(x))andf(f(x)).

b) Let ( ) = 3f x

xand

4( ) .

5x

g xx+

=−

Findf(g(x)).

c) Let 2( ) 4.h x x= − Findtwofunctionsfandgsuchthath(x)=f(g(x)).


a) 2 2( ( )) (3 7) 4(3 7) 6 36 168 202,f g x f x x x x= − = − + = − +

2 2 2 4 2( ( )) (4 6) 4(4 6) 6 64 192 150.f f x f x x x x= + = + + = + +

b) 4 3( 5) 3 15( ( )) or .

5 4 4x x x

f g x fx x x+ − − = = − + +

c) Letgbethe“inside”function: 2( ) 4.g x x= −

Let f bethe“outside”function: ( ) .f x x=


• BuildingTradesandConstruction• FashionandInteriorDesign




Afloorrefinishingcompanychargespersquarefoottorefinishahardwoodfloor.TheareaofacertainsquarefloorisgivenbyA(s),wheresisthelengthofasideofthefloor.Agallonofvarnishcosts$45andwillcover300squarefeetofhardwoodfloor.

a) ExpressthecostC(A)asafunctionofs.

b) Findthecostofrefinishingaroomwithlength25feet.


a) TheareaofasquarefloorisA(s)=s2,wheresisthelengthofasideofthefloor.

Thevarnishcosts$45per300squarefeet,so ( ) 45 .300

AC A =

So2

2( ( )) ( ) 45 ( ).300s

C A s C s C s

= = =

b)

= =

225(25) 45 $93.75,

300C

or,assumingthatthecompanychargesbythegallonofvarnishused:

A(25)=252=625.

1625 300 2 ,

12÷ = so3gallonsofvarnishareneeded.

( )45 3 $135.=




Given a function, determine if an inverse function exists. If so:

• Findtheinverseofthefunctionalgebraicallyandgraphically.

• Identifythedomainandrangeoftheoriginalandinversefunctions.

Prior Knowledge:

• Identifyifarelationisafunction.

• Graphlinear,polynomial,andradicalfunctions.

• Identifythedomainandrangeoffunctionsandrelations.

• Applytransformationstothegraphsoffunctions,includingreflection.

Model Assessments


Giventhefollowingfunctions,determineiftheinversefunctionexists.Ifso,findtheinversefunctionandidentifythedomainandrangeoftheoriginalandinversefunctions.

a) += +12 4.xy

b) =( ) 3sin2 .g x x

c) −= 11( ) tan .

2h x x

d) = +( ) 2 3.f x x

Sketcheachfunctionanditsinverse:

e)−

=+

5( ) .

2x

f xx

f) ( ) = 4 .xh x






a) For > 4x , ( ) ( )− = − −12log 4 1,f x x

domainof ( ) { }: ,f x x x ∈ rangeof ( ) { }>: 4 ;f x y y

domainof ( ) { }1 : 4 ,f x x x− > rangeof ( ) { }− ∈1 : .f x y y

b) ( )1 11sin ,

2 3x

g x− − =

domainof ( ) : ,4 4

g x x xπ π

− ≤ ≤

rangeof ( ) { }− ≤ ≤: 3 3 ;g x y y

domainof ( ) { }1 : 3 3 ,g x x x− − ≤ ≤ rangeof ( ) π π− − ≤ ≤

1 : .

4 4g x y y

c) ( )1 tan2 ,h x x− =

domainof ( ) { }: ,h x x x ∈ rangeof ( ) π π − < <

: ;

4 4h x y y

domainof ( )1 : ,4 4

h x x xπ π−

− < <

rangeof ( ) { }− ∈1 : .h x y y

d) ( )2

1 3,2x

f x− = −

domainof ( ) { }: 3 ,f x x x ≥ − rangeof ( ) { }≥: 0 ;f x y y

domainof ( ) { }1 : 0 ,f x x x− ≥ rangeof ( ) { }− ≥ −1 : 3 .f x y y



e) Graphof −=

+5.

2x

yx

Figure P9-1e

-12 -9 -6 -3 3 6 9 12

-10

-5

5

10

f(x)

f-1(x)

Findtheinverseofthefunction: 5,

2x

yx−

=+

( )1 2 5.

1x

f xx

− +=

−

f) Graphof ( ) 4xh x = and ( )14

loglog .

log4x

h x x− = =

Figure P9-1f

-3 -2 -1 1 2 3 4 5

-6

-4

-2

2

4

6

8

1012

14

16

h(x)

h-1(x)




Youhavebeenasked toprovide thegroundbeef for abarbecueat a local festival.Youneedatotalof50poundsoftwotypesofgroundbeefcosting$1.25and$1.60perpound,respectively.AmodelforthetotalcostC(x)ofthetwotypesofbeefis:

C(x)=1.25x+1.60(50–x), ≤ ≤0 50,x

wherexisthenumberofpoundsofthelessexpensivegroundbeef.

a) Findtheinverseofthecostfunction.Whatdoeseachvariablerepresentintheinversefunction?

b) Usethecontextoftheproblemtodeterminethedomainoftheinversefunction.


a) ( )1 20 1600.

7 7C x x− = − + Inthisfunction,xisthetotalcost.Ifyouknowthetotalcost,you

cancalculatehowmanypoundsofthelessexpensivegroundbeefyouneed.

b) Thedomainoftheinversefunctionwouldbe ≤ ≤62.50 80x (dollars).




Simplify expressions involving exponents. (This is a prerequisite skill that is used in Calculus.)

Prior Knowledge:

• Applythepropertiesofexponents.

• Convertbetweenrationalexponentsandtheircorrespondingradicalforms.

Model Assessments


Simplify:

a)−

−

3 5 6

5 20 15

5.

a b ca b c

b)

253

447 .32

c)− −

− −

+

−

1 1

2 2.

x yx y

d) + +− −7 5 5 2 9[(3 2) ] (3 2) .m nx x


a)8

15 21

5.

ab c

b)6

72 .

c) .xy

y x−

d) ( )35 2 343 2 .m nx + +−




Expand logarithmic expressions. (This is a prerequisite skill that is used in Calculus.)

Prior Knowledge:

• Applythelawsofexponents.

• Expandsimplelogarithmicexpressionsintoequivalentexpressions.

• Convertexpressionsbetweenrationalexponentsandtheircorrespondingradicalforms.

Model Assessments


Write the following expression as a sumand/ordifferenceof logarithmsormultiples oflogarithms:

( )( )

+ +

+ +

523

8

2 3 3ln .

2 3 5

x x

x x


( ) ( ) ( ) ( )+ + + − + − +21 1ln 2 3 5ln 3 ln 2 8ln 3 5 .

3 2x x x x


Writetheexpressionasasumand/ordifferenceoflogarithmsormultiplesoflogarithms:

23

5log .25x y


5 52 1 2log log .

3 3 3x y+ −




Writetheexpressionasasumand/ordifferenceoflogarithmsormultiplesoflogarithms:

( ) − 41 2log 1 .x x


( )1log 4log 1 .

2x x+ −




Condense logarithmic expressions. (This is a prerequisite skill that is used in Calculus.)

Prior Knowledge:

• Applythelawsofexponents.

• Expandsimplelogarithmicexpressionsintoequivalentexpressions.

• Convertexpressionsbetweenrationalexponentsandtheircorrespondingradicalforms.

Model Assessments


Writethefollowingexpressionsasasinglelogarithmicexpression:

a) −

− − +

2 1log 1 log .

1x

xx

b) + + + 3 3ln ln 2 ln 2ln .x y y z

c) − −1

4log 2log 6 log .2b b bx y


a)1

log .1x

−

b) ( )9 5 4ln .x y z

c)4

log .36

bx

y




Solve multistep exponential equations.

Prior Knowledge:

• Applypropertiesofexponentsinsimplifyingexpressions.

• Applypropertiesoflogarithmsinsimplifyingexpressions.

• Convertbetweenexponentialandlogarithmicformsofanequation.

Model Assessments


Solvethefollowingexactly: + =2 1 5 .x xe


( )=

−1

.ln 5 2

x


At7%interest,compoundedannually,howmanyyearswouldittakeaninitialinvestmentof$6,000toreach$11,802.91?(Roundtothenearestyear.)


= ≈

11802.91ln6000

10years.ln1.07

x


• FinanceandBusiness• Marketing,Sales,andService• AgricultureandNaturalResources




YouhavebeengivenabonusatworkandyouwanttoinvestitinaCDormoneymarketfund.Youhavetwooptions:amoneymarketfundthatpays7%interest,compoundedmonthly,andaCDthatpays5%,compoundedcontinuously.Inordertoevaluatetheeffectivenessofeachoption,youwanttocomparethedoublingtimeforeach.Calculatethedoublingtimeofeachinvestment(roundtothenearestyear):

a) $14,500at7%interest,compoundedmonthly.

b) $14,500at5%interest,compoundedcontinuously.


a) = +

120.0729000 14500 1 ,

12

x

= ≈ +

29000ln14500

10years.0.0712ln 1

12

x

b) = 0.0529000 14500 ,xe

= ≈

29000ln14500

14years.0.05

x


Youareconductingresearchintothedeclineofaverylargeforestedareathatconsistsofabout2.6milliontrees.Somekindofdiseaseisattackingthetreescausingthemtodieattherateof3%peryear.Howmanyyearsfromnowwilltheforesthaveshrunkto1.5milliontrees?


( )= −1.5 2.6 1 0.03 ,x

( )

= ≈

1.5ln

2.618yearsfromnow.

ln 0.97x




Solve multistep logarithmic equations.

Prior Knowledge:

• Convertexponentialequationstologarithmicequationsandviceversa.

• Applythepropertiesoflogarithmsinsimplifyingexpressionsandequations.

• Applythepropertiesofexponentsinsimplifyingexpressionsandequations.

Model Assessments


Solve:

a) + + − =2 2log ( 2) log ( 5) 3.x x

b) − + + = +3 3 3log ( 1) log ( 1) log ( 5).x x x


a) Solution:x=6.

b) Solution:x=3.


Aspart of a research team investigating the effects of a newdrug,Calpassia, you are

chargedwithcreatingamathematicalmodelthatdescribeshowmuchofthedrugremains

inthebodyatanygiventime.Itisknownthat15%ofCalpassiaismetabolized(usedup)by

thebodyeachhour.Ifadoseof45milligramsisgivenattimet=0,writeanexponential

function thatgives theamount remainingafter thours.Atwhat timewill 12milligrams

remain?Roundyouranswertothenearesthundredth.


Thegeneralformofthefunctionis ( ) 0 ,ktA t A e= whereA(t)istheamountremainingafter

t hours,A0 is the initial amount,k is thedecay constant, and t is time in hours. For this

particularproblem,thefunctionis ( ) (ln0.85) 0.162545 45 .t tA t e e−= =

At t=8.13hours,12milligramsofCalpassiawillremaininthebody.


• HealthScienceandMedicalTechnology




Sketch the graph of a conic, identifying center, vertices, foci, and asymptotes (if present).

Prior Knowledge:

• Graphlinear,polynomial,andradicalequations.

• Usecompletingthesquaretosolvequadraticequations.

Model Assessments


Sketchthegraphof4(x +1)2 –9(y –5)2 =36andidentifythevertices,foci,andasymptotes.


Equationinstandardformis:( ) ( )2 21 5

1,9 4

x y+ −− = hyperbola,centerat(–1,5).

Verticesare:(-4,5)and(2,5);fociare: ( 1 13, 5)− − and ( 1 13, 5);− + asymptotesare:

2 17 2 13, .

3 3 3 3y x y x= + = − +

Figure P15-1

-10 -8 -6 -4 -2 2 4 6 8

-6

-4

-2

2

4

6

8

10

Foci

Vertices






AcometisduetoreturntoEarthspaceinthenearfuture.Youaretaskedwithpresentingadescriptionofthecomet’spath.Thepathofthecometinitsellipticalorbitaroundthesuncanbemodeledbytheequation:9x2 + 16y2=144.

a) Sketchthegraphofthecomet’sorbit.

b) Thesunislocatedatoneofthefocioftheellipse.Identifyoneofthepossiblelocationsofthesunonyourgraph.


a)

Figure P15-2

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

Foci

b) Thesuncouldbelocatedatoneoftwofoci: ( 7, 0)− or ( 7, 0).




Graph trigonometric functions of the form ( ) .y A f Bx C k= + +

• Determinetheamplitudeofasineorcosinefunctionfromitsequation.

• Determinetheperiodofasine,cosine,ortangentfunctionfromitsequation.

• Determinetheverticalshiftofasine,cosine,ortangentfunctionfromitsequation.

• Determinethephaseshiftofasine,cosine,ortangentfunctionfromitsequation.

• Sketchbyhandthegraphofasine,cosine,ortangentfunction.

Prior Knowledge:

• Graphlinear,polynomial,andradicalequations.

• Describegraphsoftransformedfunctionsintermsofthetransformationsappliedtoparentfunctions.

Model Assessments


Giventheequation ( )3sin 2 4 :y x π= − + −

a) Determine the amplitude, period, vertical shift, and phase shift without using a

calculator;thensketchthegraphbyhandinthecoordinateplane.

b) Nowwritetheequationofthesamegraphintermsofcosineinsteadofsine.






a) Reflectiononx-axis:amplitude=3,period= ,π verticalshift=–4,phaseshift= .2π

−

Figure P16-1

-8

-7

-6

-5

-4

-3

-2

-1

1

2

b) π = − + −

3cos 2 4.2

y x


TheUSAirForcerecentlylaunchedasatellitetokeeptabsonotherspacecraftandalsoto

keepaneyeonthegrowingproblemoforbitingspacejunk.ThissatelliteisorbitingtheEarth

sothatitsdisplacementDnorthoftheequatorisgivenbytheequation sin( ).D A tω α= +

SketchtwocyclesofDasafunctionoftifA=500km., 3.60ω = rad./hr.and 0.α =


D=500sin(3.6t).

Figure P16-2

-600-500-400-300-200-100

100200300400500600




Graphthefollowingequations,aandb,onseparateaxes.

a)π = −

1sin 2 .

2 4y x

b)π = + −

3 tan 2 .

4y x


a)

Figure P16-3a

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

b)

Figure P16-3b

-10

-8

-6

-4

-2

2

4

6

8

10




Given the graph of a trigonometric function, write the equation (sine, cosine, tangent).

Prior Knowledge:

• Identifygraphsofperiodicfunctionsasgraphsofsine,cosine,ortangentfunctions.

• Identifytheperiod,amplitude,andfrequencyofsine,cosine,andtangentfunctions.

• Identifytransformationsappliedtoparentfunctions.

Model Assessments


ThetableshowsthemaximumdailyhightemperaturesforDaytonaBeachformontht,witht=1correspondingtoJanuary.

tDaytona Beach high

temperatures1 64.8

2 68.4

3 75.0

4 81.0

5 87.5

6 91.9

7 93.0

8 92.5

9 89.5

10 82.2

11 73.9

12 66.8

a) Plotthepointsonagraph.

b) Usingthedataandthegraph,findanequationforthehightemperatureatmontht.

c) Mostpeopleplantheirvacationswhenthehightemperaturesarebetween76°and82°.Thetourismindustryneedstobeginmarketingonemonthbeforethisperiod.Duringwhatmonthsshouldtheyimplementtheirmarketingcampaign?






a)

Figure P17-1a

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

10

20

30

40

50

60

70

80

90

100

Month

Hig

h Te

mpe

ratu

re (o F)

0 0

b) Usingagraphingcalculator,thedataisenteredintolists.Sincethedataisfor12months,thedataisrepeatedinthelistscreatingaperiodof24monthsforanalysis.UsingtheSinRegL1,L2,12,Y1,theresultingequationis:

( ) 14.193sin(0.520 2.065) 80.447T t t= − + for 1 12t≤ ≤ months.

Figure P17-1b

1 2 3 4 5 6 7 8 9 10 11 1250

55

60

65

70

75

80

85

90

95

100

Month

Hig

h Te

mpe

ratu

re (o F)

0

Regression function

76 oF

82 oF

Data points

c) The tourism industry needs to runmarketing campaigns beginning in February andSeptember.




Giventhegraphbelow,writetheequationforthegraphasasinefunctionandasacosinefunction.

Figure P17-2

-8

-7

-6

-5

-4

-3

-2

-1

1

2


( )π= − + −3sin 2 4,y x

or π = − + −

3cos 2 4.

2y x




Graph inverse trigonometric functions. (This is a prerequisite skill that is used in Calculus.)

Prior Knowledge:

• Graphsine,cosine,andtangentfunctions.

• Identify amplitude, period, andphase shift of common trigonometric functions fromtheirequationandtheirgraphs.

• Determinetheinverseoffunctions.


Model Assessments


Graphthefollowingfunctions:

a) =( ) arcsin2 .f x x

b) −= 1( ) tan .f x x

c) ( ) 1cos .2x

f x − =




Figure P18-1a

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1



b) −= 1( ) tan .f x x

Figure P18-1b

-10 -8 -6 -4 -2 2 4 6 8 10

c) ( ) 1arccos cos .2 2x x

f x − = =

Figure P18-1c

-2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5




Prove trigonometric identities. (This is a prerequisite skill that is used in Calculus.)

Prior Knowledge:• Definebasictrigonometricfunctions.

• Solvetrigonometricequations.

• Applypolynomialfactoringtechniques.

• ApplythePythagoreanTheorem.

Model Assessments


Showthat:sin(x + y)+sin(x – y)=2sinx cos y.


sin(x + y)+sin(x – y)=2sinx cos y,

sin x cos y + cos x sin y + sin x cos y – cos x sin y = 2 sin x cos y.


Showthat:cos

1 sin .sec tan

xx

x x= −

+


( )( )

= −+

= −+

= −+

= −+

−= −

+

− += −

+

− = −

2

2

cos1 sin ,

sec tan

cos1 sin ,

1 sincos cos

cos1 sin ,

1 sincos

cos1 sin ,

1 sin

1 sin1 sin ,

1 sin

1 sin 1 sin1 sin ,

1 sin

1 sin 1 sin .

xx

x x

xx

xx x

xx

xx

xx

x

xx

x

x xx

x

x x




Showthat:−

=+sin 1 cos

.1 cos sin

x xx x


−=

+

− −=

+ −

− −=

−

− −=

− −=

2

2

sin 1 cos,

1 cos sin

sin (1 cos ) 1 cos,

(1 cos )(1 cos ) sin

sin (1 cos ) 1 cos,

sin(1 cos )

sin (1 cos ) 1 cos,

sinsin

1 cos 1 cos.

sin sin

x xx x

x x xx x x

x x xxx

x x xxx

x xx x




Solve trigonometry equations.

Prior Knowledge:

• Identifythevaluesoftrigonometricfunctionsatstandardpointsaroundtheunitcircle.

• Determinethevaluesofinversetrigonometricfunctionsgivenvaluescorrespondingtostandardpointsontheunitcircle.


• Identifythedomainandrangeofinversetrigonometricfunctions.

• Identifytheperiodoftrigonometricfunctions.

• Convertbetweenanglemeasuresinradiansanddegrees.

Model Assessments


a) Solvetheequation =3

cos2 2x

forxontheinterval π≤ <0 2 .x

b) Findallsolutionsfortheequation =3

sin2 .2

x

c) Solvetheequationsinx + sin3x=2sinx cos2 x forxontheinterval π≤ <0 2 .x


a) π= .

3x

b) Therearetwofamiliesofsolutions:π π= +6

x k and ,3

x kπ π= + wherekisaninteger.

c)5 7 11, , , .

6 6 6 6x

π π π π=






Solvethefollowingequations(roundedtotwodecimalplaces,answersindegrees):

a) = ° ≤ < °sin 0.3for0 360 .x x

b) = ° ≤ < °tan 1.5for0 360 .x x


a) −= ≈ ° °1sin 0.3 17.46 or162.54 .x

b) −= ≈ ° °1tan 1.5 56.31 or236.31 .x


Asnowboardmanufacturerfindsthatsalesoftheirnewestmodelofsnowboardfollowsaseasonalpattern,withsalesgivenby:

6320 2850cos ,6t

Sπ = +

whereS is thenumberofboards soldduringagivenmonth t, and t = 1corresponds toJanuary.Aslocalstoresdeveloptheirinventory,theyneedtoestimatesalesbymonth.Forwhichmonthsoftheyeararesalesgreaterthan5,000boards?


Solvetheinequality6320 2850cos 5000.6tπ + >

2850cos 1320,6tπ > −

π > −

cos 0.46316.6t

Findcriticalvalues(i.e.,whenπ = −

cos 0.46316

6t

).

Thesewillbewhen 1cos ( 0.46316),6tπ −= −

orwhenπ

−= −16cos ( 0.46316).t

Thisgivesustwovaluesoft, ≈ 3.9t and ≈ 8.1.t Whentisbetweenthesetwovalues(forexample,whent =6)wegetafalsestatement,sowemustwantthevaluesoftthatfalloutsideoftheinterval(3.9,8.1).Therefore,weknowthatsalesaregreaterthan5,000boardswhen < 3.9t andwhen > 8.1.t

That wouldmean that in themonths of January, February, March, September, October,November,andDecember,saleswillbegreaterthan5,000boards.




Solve for all unknown parts of a given triangle:

a) by using right triangle trigonometry

b) by using the Law of Sines

c) by using the Law of Cosines

Prior Knowledge:

• Definesine,cosine,andtangentintermsofthesidesofarighttriangle.


• UsethePythagoreanTheoremtofindthemissinglengthsofasideofarighttriangle.

• Determinethereferenceanglegivenitssineorcosinevalue.

Model Assessments


Findthemeasureof∠A ,thelengthofAC,andthelengthofBC.

Figure PC21-1

30°

b

a5

B

CA


UsingTriangleSum:∠ = °60 .A UsingSpecialTriangles: 5 3; 10.a b= =

Usingtrigonometry:5 5

tan30 ; 8.6603andsin30 ; 10.a ba b

° = ≈ ° = =


• Hospitality,Tourism,andRecreation• PublicServices




Findthemeasuresofalloftheanglesinthetrianglebelow.

Figure PC21-2

B

1517

28A C


UsingLawofCosines: 31.00 .A ≈ ° UsingLawofCosinesagain: 121.97 .B ≈ °

UsingTriangleSum: 27.03 .C ≈ °


Findthemeasureof∠ ∠, C B ,andthelengthofBC.

a

Figure P21-3

21

30 A

B

C 42o


UsingLawofCosines: 20.12.a ≈ UsingLawofCosinesagain: 93.69 .B ≈ °

UsingTriangleSum: 66.19 .C ≈ °




Asearchandrescueteamneedstoestimatetheheightofacliffinordertoplantheirrescueattempt. To estimate the height, two sightingswere taken 50 feet apart. The angles ofelevationwere40degreesand31degrees.Findtheheightofthecliff,h(seefigure).


UsingTriangleSum:Theanglesoftheobtusetriangleare9°and140°.

UsingLawofSines: 205.45feet.a ≈

UsingRightTriangleTrig.: 105.81feet.h ≈

Figure PC21-4

40°

h

9°

31°

140° a

50 ft.




Convert between polar and rectangular coordinates.

Prior Knowledge:

• Convertbetweenanglemeasuresindegreesandradians.

• UsethePythagoreanTheoremtofindamissingsideofarighttriangle.

• Usetrigonometrytodetermineunknownsidesoranglesinarighttriangle.

• Computethevaluesoftrigonometricfunctionsandinversetrigonometricfunctionsatvariousstandardpoints.

Model Assessments


ConvertthefollowingCartesian(rectangular)pointstopolarcoordinates ( , ) :r θ

a) ( 3, 3).−

b) ( )2, 2 3 .−

ConvertthefollowingpolarcoordinatestoCartesian(rectangular)coordinates:

c)5

8, .3π

d) ( )4, 30 .− °






Figure P22-1

1 2 3 4

1

2

3

4

x

r

y

θ

Givenanoverlapofthepolargridandtherectangulargrid,fourequationsarisetodotheseconversions:

θ

θ θ

+ = =

= =

2 2 2 cos

tan sin

x y r x ry

y rx

Rectangulartopolarcoordinates:

a) (-3,3)wouldbeinQII.Twopossiblepolaranswersfor(-3,3)are

( ) 3, 3 2 , 3 2 , .

4 4r or

π πθ = − −

b) ( )2, 2 3− wouldbeinQIV.Twopossiblepolaranswersfor ( )2, 2 3− are

( ) ( ) ( ), 4, 60 4, 120 .r orθ = − ° − °

Polartorectangularcoordinates:

c) 58, :

3π

( ) ( ), 4, 4 3 .x y = −

d) ( )4, 30 :− ° ( ) ( ), 2 3, 2 .x y = − −




Aforestsurveytechnicianiscompletingasurveyofdiseasedtrees.Heplanstoplotthelocationofallinfectedtreesonarectangulargrid.Standingattheorigin,hetakesabearing(40°EofN)onthenearestinfectedtree.Hemeasuresthedistancefromhislocationtothetreetobe45meters.AtwhatlocationshouldhemarkthetreeinCartesiancoordinatesonhisgrid?


Figure P22-2

Origin

45 m

40o

N

E

Usingtheangleof50°madewiththex-axis: ( ) ( ), 28.9 m.,34.5m. .x y =




Graph equations written in polar form.

Prior Knowledge:

• Convertbetweenanglemeasuresindegreesandradians.

• Computethevaluesoftrigonometricfunctionsandinversetrigonometricfunctionsatvariousstandardpoints.

• Graphlinear,polynomial,radical,andtrigonometricfunctions.

Model Assessments


Graphthefollowingequations:

a) θ= −2cos .r

b) θ= −2 4cos .r

c) θ= +4 4sin .r


θ= cosr hassymmetryoverthepolaraxis,and θ= sinr hassymmetryoverthe 2π

axis.

Foraandb,usethistableofvalues:

θ 0 4π

2π

3

4π π

θcos 1 0.707 0 -0.707 -1θ= −2cosr -2 -1.414 0 1.414 2θ= −2 4cosr -2 -0.828 2 4.828 6


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a)

Figure P23-1a

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

b)

Figure P23-1b

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7-6-5-4-3-2-1

1234567



Forc,usethistableofvalues:

θ 2π− 4

π− 0 4π

2π

θsin -1 -0.707 0 0.707 1θ= +4 4sinr 0 1.2 4 6.8 8

c)

Figure P23-1c

-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

-8

-6

-4

-2

2

4

6

8


Afireinvestigatorisanalyzingarecentbrushfireinwhichaperpetratorpropelledafieryobjectintoafield.Thepathoftheobjectisgivenbytheequation:

θθ

=+20

( ) ,1 sin

r

whererismeasuredinmeters.Tohelptheinvestigatorlocatethelaunchpoint,graphthepathoftheobjectfor θ π≤ ≤0 .




θ 0 4π

2π 3

4π π

θsin 0 0.707 1 0.707 0

θ=

+20

1 sinr 20 11.7 10 11.7 20

Figure P23-2

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14 16 18 20

123456789

101112




Graph equations written in parametric form.

Prior Knowledge:

• Graphlinear,polynomial,radical,andtrigonometricfunctions.

• Graphpolarequations.

• Solveproblemsinvolvingcompositionoffunctions.

• Identifygeometricpropertiesofconicsectionsfromtheirquadraticequations.

• Graphconicsectionsincludingtranslations.

• Simplifyequationsusingtrigonometricidentities.

• Evaluatetrigonometricfunctions.

Model Assessments


Graph;showorientation:

a) = =cos , sin .x t y t

b) = −2 3x t for ≤ <0 10t and = − +5.y t


a) Pointsareplottedandarrows indicate thedirection inwhich thepointsarebeingsweptout.

Figure P24-1a

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3





b)

Figure P24-1b

-4 -2 2 4 6 8 10 12 14 16 18 20

-10

-8

-6

-4

-2

2

4

6

8

10


Graphequationsinparametricform.

Acombineharvestscornfromafieldandthekernelsareremovedfromthecobandejectedoutofthebackinastream.Thestreamexitsthecombineataheightof10feetabovethegroundwithaninitialspeedof25feetpersecondatanangleofelevationof20relativetotheground.Thestreamofcornhastolandina14-footwagontowedbehindthecombine.Thetopofthewagonis8feetabovethegroundandthebedofthewagonis2feetabovetheground.

a) Writeasetofparametricequationsthatmodelthepathofthestream.

b) Graphthepathofthestream.

c) Howfarbehindthecombinemustthewagonfollowinordertocatchthecorninthemiddleofthetrailer?




Thedrawingillustrateswhatisoccurring(notnecessarilydrawntoscale):

Figure P24-2a

0

10 ft20o

V0 = 25 ft/sec

7 ft

d

a) Inthex-direction,ignoringfriction,D=rt.Soxisequaltothespeedinthex-directionmultipliedbytime;x =25cos20°t,measuredinfeet.

In they-direction,wemust take intoconsideration the forceofgravitypullingon the

corn aswell as the speed acting in they direction and the elevation fromwhich the

cornisbeingpropelled.So ( )2125 sin20 10

2y gt t= − + ° + measuredinfeet,wheregisthe

accelerationduetogravity(32.2ft./sec.2).

b)

Figure P24-2b

5 10 15 20 25 30

5

10

15

Distance from combine (ft)

Hei

ght a

bove

the

grou

nd (f

t)

Bed of wagon = 2 ft

0

c) Solvingtheequationinthey-directionfortheheightofthebedofthewagon,thecornwouldhit thebedof the trailer about23.9 feetbehind thecombine.Therefore, thetrailershouldbeabout16.5feetbehindthewagonforthecorntoclearthesidesofthewagonandlandnearthemiddle.




Given the components of a two-dimensional vector, determine the magnitude and direction of the vector.

Prior Knowledge:

• Solveproblemsinvolvingexponentsandsquareroots.

• ApplythePythagoreanTheorem.

• Computeinversetrigonometricfunctions.

Model Assessments


Findthemagnitudeanddirectionofthefollowingvectors:

7, 7 3− or − +7 7 3i j or ˆ ˆ7 7 3 ,i j− + where ,i j or ˆ ˆ,i j aretheunitvectorsinthexandy

directions,respectively.


Magnitude:14.

Direction: 60rθ = − ° instandardposition.However,sincethevectorisgoingintheoppositedirection,thedirectionangleinstandardpositionisθ = °120 .


Let point PbeP=(–2,5)andletpointQbeQ=(3,2).

ExpressPQincomponentform.


5 3 5, 3 .= − = −PQ i j


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WhatisthemagnitudeanddirectionofthevectorPQabove?


Magnitude: 34.=PQ

Direction:θ − − = ≈ − °

1 3tan 30.96

5instandardposition.


Acropduster(smallairplane)istakingoff.Thereisa179-foot-tallgrainstoragesilothatis355feetawayfromthetake-offpoint.Findtheangleofelevationthepilotneedsinordertoclearthesiloby10feet,andwriteanexpressionforthedeparturevector.Howfarhasthecropdustertraveledwhenitisdirectlyabovethesilo?


Cropdusterpathvector: = +355 189 .v i j

Distancetraveled: 2 2355 189 402.18ft.= + ≈v

Angleofelevation:θ − = ≈ °

1 189tan 28.03 .

355r




Perform the vector operations of addition, subtraction, and scalar multiplication.

Prior Knowledge:

• Usematrixnotation.

• Identifythedimensionsofamatrix.

• Identifythepositionsofeachentryinamatrix.

Model Assessments


Given: 2, 5 , 3, 7 , 1, 4 ,= − = = − −a b c find:

a) a + b.

b) 2a – 3c.

c) 7c + b – 5a.


a) 1, 12 .+ =a b

b) 2 3 1, 22 .− = −a c

c) 7 5 6, 46 .+ − = −c b a


Aplaneflies250milesperhouronacompassheadingorbearingof300degrees.Theairismovingwithawindspeedof50milesperhouronabearingof195degrees.Findtheplane’sresultantvelocity(speedandbearing)byaddingthesetwovelocityvectors.


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Bearing is fromnorth, going clockwise. The twovectors aredrawnbelow.Pwill be thevectoroftheplanewithanglemeasuredinstandardposition;wwillbethewindvectorwithangleinstandardposition.

Figure P26-2

P

w

N

Resultantvector:

229.45, 76.70 mph.= + = −R P w Actualspeed: 241.93 mph.=P

Headinginstandardposition: 18.5 .rθ ≈ °

Figure P26-2(2)

N

R

w P

Bearing: 270 18.5 288.5° + ° = ° fromnorth,clockwise.




Use factorial notation.

Prior Knowledge:

• Usesimplemathematicalconceptssuchasmultiplicationandorderofoperations.

Model Assessments


Fivefriendswenttothemovies.Inhowmanydifferentarrangementscouldtheybeseatedinarowtogether?


5!=120differentarrangementsarepossible.


Thereare100songsonyouriPod.Ifthesongsareshuffledwithnorepeats,inhowmanydifferentorderscouldtheybeplayed?


100!differentarrangements(thatwouldbeapproximately 1579.333 10 ).


Aconcertpromoterhas20bandsparticipating in aweekendconcert event.Howmanydifferentordersarepossibleforthebandsduringtheconcert?


Thereare 1820! 2.43 10≈ differentorderspossible.


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Expand binomial expressions. (This is a prerequisite skill that is used in Calculus.)

Prior Knowledge:

• Multiplybinomialsandpolynomials.

• Simplifypolynomialexpressions.

• Applypropertiesofexponents.

• Computecombinations.

Model Assessments


Expand ( )− 52 .a b


UsingtheBinomialTheorem,weget:

( )5 5 4 3 2 2 3 4 52 32 80 80 40 10 .a b a a b a b a b ab b− = − + − + −


Intheexpansionof ( )+ 16 ,x y whatisthetermcontainingx 7?


11440x 7y 9.


Simplify ( )+ −4 4x h x

husingtheBinomialTheorem.


3 2 2 34 6 4 .x x h xh h+ + +


S t a t i s t i c s C o m p e t e n c i e s 153

Statistics(Non-STEM)

Competencies


S t a t i s t i c s C o m p e t e n c i e s154


Determine whether a variable is categorical or quantitative, given a situation.

Prior Knowledge:

• Identifythevariableinanexperiment.

• Organize,represent,andinterpretnumericalandcategoricaldata.

Model Assessments


Asamemberof themathdepartment at the local community college, youneed tomakeapresentationonthedatacollectedaboutthestudentsinthestatisticsclassesattheschool.Inordertoperformanykindofanalysisonthedata,youhavetoidentifywhattypeofdatayouhave;thisdictatesthemethodsofanalysisthatcanbeemployed.Determinewhetherthefollowingvariablesarecategoricalorquantitativeifstudentsinastatisticsclassaretheobservationalunits:

a) howmanyclassesastudenttakes

b) thestudents’eyecolor

c) averageweeklysleeptime


a) quantitative

b) categorical

c) quantitative


Astheheadofnursingforthehospital,youhavecollecteddataregardingthenursesandtheirfunctions.Youareanalyzingthisdata,andinordertoperformtheanalysescorrectly,you have to know the types of variables that are represented.Determinewhether eachvariableiscategoricalorquantitativeifnursesinahospitalaretheobservationalunits:

a) thenumberofpatientsunderanurse’scare

b) theamountsofantibioticsadministeredtopatients

c) theyearanursebeganworkingatthehospital


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a) quantitative

b) quantitative

c) categorical


Yourcityisseekingareasinthebudgetwherecostscanbereduced.Oneareaidentifiedforpotentialcutsinthebudgetisthefiredepartment.Datahavebeencollectedtoanalyzethispossibility.Inordertodeterminethepropermethodofanalysis,you(astheanalyst)havetoidentifythevariabletypes.Determinewhethereachvariableiscategoricalorquantitativeiffirefightersatafirestationaretheobservationalunits:

a) thenumberoffirestowhichafirefighterhasbeencalled

b) thedaysoftheweekafirefighterreportstoafirestation

c) thestreetnumbersofbuildingstowhichafirefighterhasbeencalled


a) quantitative

b) categorical

c) categorical




Identify the sampling method used (random, systematic, convenience, stratified, cluster, or voluntary response), given a scenario in which data were collected from a population.

Prior Knowledge:

• Readandinterpretpolls.• Interpretsurveysandtheirresults.• Definerandomsamples.

Model Assessments


Identifywhichtypeofsamplingisused:random,systematic,convenience,stratified,cluster,orvoluntaryresponse:

a) Alocalcollegeconductsastudyonstudentcheatingandplagiarismbyrandomlyselecting30differentclassesandgivingallofthestudentsineachofthoseclassesasurveyformtocomplete.

b) Atelevisionstationwantstosolicititsviewers’opiniononthecurrentstateoftheeconomybyaskingthemtogotoitsWebsitetocompleteasurvey.

c) Amarketingexecutiveforatelevisionnetworkisplanningtosurvey1,500ofitsyoungviewers.The1,500viewerswillberandomlyselectedfromeachagegroupof8–12,13–17,and18–22.


a) cluster

b) voluntaryresponse

c) stratified


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a) Acellphoneproviderconductsacustomersatisfactionsurveyandusesacomputertorandomlygenerate1,000telephonenumberstocall.

b) Amilitarybasesetsupasobrietycheckpointinwhichevery10thdriverisstoppedandinterviewed.

c) Astatisticsstudentconductsresearchontheamountoftimepeoplespendtextingperdaybypollingherfamily,friends,andclassmates.


a) random

b) systematic

c) convenience



a) Auniversitymedicalcenterresearchersurveysallcancerpatientsineachof25randomlyselectedhospitals.

b) A radio station conducts a popularity survey and asks its listeners to call in statingwhethertheyareonTeamXorTeamY.


a) cluster

b) voluntaryresponse




Identify the basic terminology of experimental design used in a given scenario.

Prior Knowledge:

• Identifythedifferentmethodsofsamplingdata.• Identifythevariablethatismeasuredinanexperiment.

Model Assessments


Asvideogamingbecomesmorepopular, researchers are interested in the effectof lowlightingoneyesight.Onesuchresearchgroupconductsanexperimentonlabratsinwhichextendedexposuretolowlightingwillbeadministered.Fiftywhitelabratsareavailablefortesting.Researchersnumbertheratsfrom01to50andusearandomdigittabletoselect25rats.These25willreceivethetreatment;theremaining25willbethecontrolgroup.Resultswillbecomparedattheendoftheexperiment.Whichexperimentaldesigndidtheresearchersuse?


completelyrandomizeddesign


Amanufacturerofgolfclubsfieldtestsanewproduct.Thecompanywouldliketoadvertisethatusingthisproductwillincreasetheeffectivenessofagolfer’sswing,butmustobtainsuchresultsfirst.Onehundredvolunteersareavailabletotestthenewproduct.Aresearcherdecidestohaveeachvolunteertryacoursetwice,oncewiththenewproductandoncewithout.Theorderinwhicheachgolferreceivesthetreatmentisdecidedrandomly.Whichexperimentaldesigndoestheresearcheruse?


matched-pairsdesign


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Anewpesticideistobetestedonalargesampleofgreenbeans.Frompastexperience,researchers have reason to believe that a new hybrid type of green bean might reactdifferentlytopesticidesthanthetraditionalgreenbean.Therefore,allplantsofthehybridtypeofgreenbeanaregrowntogether inoneplotof land;theregularplantsaregrowntogetherinaseparateplot.Thepesticideisappliedtohalfofeachplot;theotherhalf isleftpesticide-free(forcontrolpurposes).Afterafewweeks,resultsarecompared.Whichexperimentaldesigndidtheresearchersuse?


blockdesign




Identify biases in experimental designs or in the creation of a sample.

Prior Knowledge:

• Identifythedifferentmethodsofsampling.• Recognizethetypeofexperimentaldesignemployedinagivensituation.• Identifythevariablethatismeasuredintheexperiment.• Identifydatathatrepresentsamplingerrors.• Explainwhyasamplemightbebiased.

Model Assessments


In order to determinewhether regular exercise reduces the risk of a heart attack, twomethodsareemployed.Whichdesignwillproducemoreaccuratedataandwhy?

Design 1:Aresearcherfinds2,000middle-agedmenwhoexerciseregularlyandhavenothadheartattacks.Shematcheseachwithasimilarmanwhodoesnotexerciseregularly,andshefollowsbothgroupsfor5years.After5years,shecomparesthenumberofheartattacksinbothgroups.

Design 2:Anotherresearcherfinds4,000middle-agedmenwhohavenothadheartattacksandarewillingtoparticipateinastudy.Sherandomlyassigns2,000ofthementoaregularprogramofsupervisedexercise.Theother2,000continuetheirusualhabits.Theresearcherfollowsbothgroupsfor5years.


Design2willproducemoreaccuratedatabecause it isadesignedexperimentandcanestablish causal relationships, whereas Design 1 is an observational study and can beaffectedbyotherfactors.


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Aquestionnairewassentto10,000peopleaskingabouticecream.Fivethousandpeopleresponded to thequestionnaire.Three thousandof the respondents indicated that they“lovechocolateicecream.”Itisconcludedthat60%ofpeoplelovechocolateicecream.Inthissituation,whatiswrongwiththesurveymethodandconclusion?


Thisisavoluntaryresponsesample.Thesurveyisbasedonvoluntary,self-selectedresponsesand,therefore,haspotentialforbias. Inaddition,the5,000peoplewhodidnotrespondmaynotlikechocolateicecream.


Aresearcherwishedtogaugepublicopiniononguncontrol.Herandomlyselected1,000people fromamong registeredvotersandasked them the followingquestion: “DoyoubelievethatguncontrollawsthatrestricttheabilityofAmericanstoprotecttheirfamiliesshouldbeeliminated?”Identifyhowtheresearcher’smethodscouldbeimproved.


Thereisresponsebiasbecausethequestionleadsrespondentstoanswerinacertainway.Amoreneutralwaytophrasethequestionwouldbe:“Doyoubelievethatguncontrollawsshouldbestrengthened,weakened,orleftintheircurrentform?”




Determine an appropriate experimental design for a given scenario.

Prior Knowledge:

• Determinewhetheravariableiscategoricalorquantitative.• Identifydifferentmethodsofsampling.• Identifythebasicterminologyofexperimentaldesign.

Model Assessments


Asvideogamingbecomesmorepopular, researchers are interested in the effectof lowlightingoneyesight.Onesuchresearchgroupconductsanexperimentonlabratsinwhichextendedexposuretolowlightingwillbeadministered.Fiftywhitelabratsareavailablefortesting.Determinetheappropriateexperimentaldesignforthisscenario.


Researchersshoulduseacompletelyrandomizeddesign.Numbertheratsfrom01to50andusearandomdigittabletoselect25rats.These25willreceivethetreatment;theremaining25willbethecontrolgroup.Resultsshouldbecomparedattheendoftheexperiment.


Amanufacturerofgolfclubs isfield-testinganewproduct.Thecompanywould liketoadvertisethatusingthisproductwillincreasetheeffectivenessofagolfer’sswing,butmustobtainsuch resultsfirst.Onehundredvolunteersareavailable to test thenewproduct.Determinetheappropriateexperimentaldesignforthisscenario.


Researchersshoulduseamatched-pairsdesignatadrivingrange.Haveeachvolunteerhit10golfballswithhis/herownequipmentand10golfballswiththenewproduct,measuringtheball’sdistanceforeachhit.Theorderinwhicheachgolferreceivesthetreatmentshouldbedecided randomly.Thispairingallows researchers tocollectdataon theproduct’seffectiveness.


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Anewpesticideistobetestedonalargesampleofgreenbeans.Frompastexperience,researchers have reason to believe that a new hybrid type of green bean might reactdifferently to pesticides than the traditional green bean. Determine the appropriateexperimentaldesign.


Since therehasbeenacharacteristic identifiedaheadof time that isexpected toaffectresults,researchersshoulduseablockdesign.Therefore,alltheplantsofthehybridtypeofgreenbeanshouldbegrowntogetherinoneplotoflandandtheregularplantsgrowntogetherinaseparateplot.Thepesticideshouldbeappliedtohalfofeachplot;theotherhalfshouldbeleftpesticide-free(forcontrolpurposes).Afterafewweeks,resultsshouldbecompared.




Construct and describe an appropriate visual display given a data set (histogram, bar graph, stem plot, scatter plot, box and whisker plot, and pie chart).

Prior Knowledge:

• Calculatemeanandmedian.• Calculatepercentage.• Graphpointsonacoordinateplane.

Model Assessments


Thenumberofyearsofteachingexperienceofteachersatalocalhighschoolislistedbelow. Constructanappropriatevisualdisplayofthedata:

0,3,8,14,26,0,4,9,17,23,4,9,1,18,21,2,21,7,32,9.


Oneofthefollowingwouldbeappropriate:

a) boxandwhiskerplot

Figure S6-1a

Years






b) histogram

Number of years of experience

Frequency

0–5 7

6–11 5

12–17 2

18–23 4

24–29 1

30–35 1

Teaching experience at local high school

Figure S6-1b

0

1

2

3

4

5

6

7

8

Number of years of teaching experience

Freq

uenc

y

5.5 11.5 17.5 23.5 29.5 34.5




Suppose200seniorsatthelocalhighschoolhaveafter-schooljobs.Thetypeofjobsandrespectivenumberofstudentswhoholdthemaredetailedbelow.Constructanappropriatevisualdisplayofthedata.

retail,65

food,72

manuallabor,8

healthandhumanservices,38

business/office,5

other,12


Oneofthefollowingwouldbeappropriate:

a) bargraph

Jobs held by seniors at the local high school

0

10

20

30

40

50

60

70

80

Retail Food Manuallabor

Health andhuman

services

Business/office Other

Num

ber o

f stu

dent

s




b) piechart



Datahasbeengatheredatabookstoreformanyyears.Analystswanttostudyhowanauthor’spopularityhaschangedovertheyears.Belowisatablethatprovidesdataofinterest.Createagraphtoshowtheauthor’spopularityovertime.

Number of years since author’s debut book

Number of fans at author’s book signing

0 3981 2842 1893 1524 1025 776 527 398 259 1810 10911 1312 7

Retail32.5%

Food36%

Manuallabor4%

Health andhuman

services19%

Business/office2.5%

Other6%





Author’s popularity over time

Figure S6-3

0

100

200

300

400

2 0 6 4 8 12 10 Number of years since the author’s debut book

Num

ber o

f fan

s at

an

auth

or’s

bo

ok s

igni

ng




Identify which measure of the center (mean, median, or mode) is most appropriate for a given situation and what the relationship between the mean, median, and mode indicates about the distribution of the data.

Prior Knowledge:

• Calculatethemean,median,andmodeofadataset.• Identifytheshapeofdifferentdatadistributionsfromtheirgraph(i.e.,histogram,

curve,scatterplot,etc.).

Model Assessments


Thecoachofalittleleagueteamissuspiciousoftheagesofhisrivalteambasedontheplayers’heights.Thecoachobtainstherosteroftheopposingteamandfindsthesereportedheights(roundedtothenearestinch):

50,69,50,55,50,54,49,47,50,49,70,56,58,50,57,50.

a) Whichmeasureofcentersupportsthecoach’ssuspicion?Explainandsupportyourreasoningwithnumericalproof.

b) Whichmeasure(s)ofcenterdefend(s)theopposingteam’slegitimacyoftheirplayers’agesbasedontheirheights?Explainyourreasoningwithnumericalevidence.


a) mean=54.0inches,median=50.0inches,mode=50inches

Themeansupportsthecoach’ssuspicion.Themeanisanonresistantmeasureofcenterandisbeingpulledafull4incheshigherthanboththemedianandmode.Thisresultsfrom twounusually tall kids; thedistributionofheights is skewed right,orpositivelyskewed. Four inches in height difference could make an impact on players’ speed,strength,andathleticability.



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b) mean=54.0inches,median=50.0inches,mode=50inches

Boththemedianandthemodehelpdefendtheopposingteam’splayers.

Themeanheightisfairlyhigh,butthemediandemonstratesamoretypicalheight.Thefactthatthemedian is50.0 inches—afull4 inchesshorterthanthemean—showsthatthetypicalplayerontheteamisaboutthatheight. Inaddition,thefactthatthemodealsois50inchesemphasizestheheightoftheteamasnotbeingunusuallytall.Moreplayers on the team are 50 inches tall than any other height. Since themostfrequentlyoccurring(6players)heightis50inches,therewouldbelittledifferenceinspeed,strength,orathleticabilityinthesechildrenbasedonheight.


Adistrictattorneyisrunningforre-electioninyourcity.Hiscampaignstrategyistofocusonhis“toughoncrime”approachbyprovidingdataabouttheamountoftimetheconvictsheprosecutedhavespentinjail.Jailtimesfromlastyear’scourtcaseswere(inmonths):

4,18,7,2,0.5,12,3,6,6,6,9,6,10,84,6,1,2,8,6,8.

Whichmeasureofcentershouldthedistrictattorneychoosetohelphis“toughoncrime”appearancethemost?Supportyouranswerwithnumericalevidence.


mean=10.23months,median=6.00months,mode=6months

Themeanwouldgive theappearance thatcriminalsprosecutedby thisattorney receivelongerjailsentences.Themeanisanonresistantmeasureofcenter,andis,therefore,pulledhigherthanthemedian(themore“typical”jailtime)bythe18-and84-monthsentences.Todemonstratejusthoweasilyinfluencedthemeanis,onemightlookattheaveragejailsentencewiththe84-monthvalueremovedfromthedata. Insodoing, themean isnowapproximately6.34months,whichisquiteadifference!Comparingthemedianandmodewiththemeanvaluealsoindicatesthatthedistributionofthedataisskewedpositively(orskewedright),whichalsoshowstheeffectofthosetwosentencesonthemeanvalue.




Acompanythatmanufacturesenergy-savinglightbulbsforlampsguarantees3,000hoursofuseforeachbulb.Everymajorrunofbulbsfromtheprevious6-monthperiodwassampletested,andthebulbfailuresarelistedinthetablebelow:

Date Batch number Number of bulbs tested

Number of bulb failures

1/15 2381 20,000 501/30 2549 20,000 952/15 2711 20,000 552/28 2908 20,000 613/15 3175 20,000 573/30 3318 20,000 554/15 3560 20,000 634/30 3795 20,000 795/15 3953 20,000 555/30 4150 20,000 726/15 4308 20,000 546/30 4511 20,000 57

Considerthenumberofbulbfailures.Calculatethemean,median,andmodeofthenumberoffailuresperrun.Explainwhythemeanissignificantlyhigherthantheothermeasures.Asamanagerofmanufacturing,doesthedatasupportamajoroverhauloftheprocessinordertoreducefailures?


Themeanis753/12=62.8failures.

Themedianis57.0.

Themodeis55(themostfrequentlyoccurringnumber).

The mean is significantly higher because of one outstanding run of failures. On 1/30,therewerealmosttwiceasmanyfailuresasthepreviousrun.Therewerealsotwootherrunswithhighnumbersoffailures,whichoccurredon4/30and5/30.Itseemsthattheproblemsoccurinthesecondhalfofthemonth(in5outof6months).Amajoroverhauldoesnotseemwarranted,yetthereshouldbesomeinvestigationintoanyprocessthatseemstogeneratemorefailuresattheendofthemonthwithfrequency.Lookingatthedistributionofthedata,whichisskewedpositively(orskewedright),alsoindicatesthatthereissomethingaffectingthedatafromthesecondhalfofthemonth.




TheadministrationofalocalhighschooldistrictisconsideringapushtoincreaseenrollmentinAP(AdvancedPlacement)coursesatallschoolsites.TheadministrationisbasingtheirinitiativeonthecurrentaverageAPtestpassingrateoftheirstudents.Theadministratorshireyouasastatisticaladvisortohelpthemmaketherightdecision.Whatwouldyouadvisethemtoconsiderintermsofmeasuresofcenterbeforemakingtheirdecision?


AdvisetheadministratortolookatthemedianAPtestscoreaswellasthemode.Thesemeasures of center are both resistant to outliers and, therefore, can provide a more“honest”lookatAPtestscoresaroundthedistrict.Forexample,ifthemedianscorewasa1(ona1–5scale)andsomestudentshadscoreda5,themeanwouldbepulledhigherthan1.Thiswouldgiveafalsesenseofachievement.Theeffectofthefew5sonthemeanvaluemaynotbethatgreat,butitissomethingtoconsider.Bylookingatthemode,theadministratorwouldfindthescorereceivedbythemajorityofthestudentstakingthetest.Themodecanberevealingandmightshowthatthereshouldbemoreofapushtoimprovethescoresforthestudentscurrentlyenrolledbeforepushingtoaddmorestudents.Attheveryleast,showingtheadministratorthedifferentmeasuresofcentercanhelphim/hertomakeaneducateddecision.Acomparisonofthemean,median,andmodealsowouldrevealtheshapeofthedistributionofthedata,indicatingwhetherornotasmallgroupofhighscoresmaybeadverselyaffectingtheaverage,givingtheadministratorsafalsepictureofthetrueresults.




Determine standard deviation from multiple representations of data (table, graph, and word problems) and explain standard deviation in the context of a given situation.

Prior Knowledge:

• Calculatethemeanofadataset.• Recognizeandapplysummation

notation.

Model Assessments


Astudentisconcernedthathercarisnotrunningoptimallyanddecidestocollectsomedata.Shemeasuresthemilespergallonshegetseachmonthfor12months.Belowisthegathereddata.

29,28,29,37,27,30,30,31,31,32,30,28.

Afterdoingsomeresearch,shefindsthatanoptimallyrunningcarhasastandarddeviationof1.6milespergallon.Ishercarrunningoptimallybasedonthestandarddeviation?Whatwouldthestandarddeviationbeifsheremovedtheoutlierinherdataset?Explain.


Accordingtothedata,thesamplestandarddeviationis2.6mpg.(whichindicateshercarisnotrunningoptimally).However,sheseesthatbyremovingthe37mpg.datapoint,s =1.5,whichcouldchangeherconclusion.Therefore,shedecidestolookmorecloselyatthe37mpg.outlier.


NorthHighSchoolisinterestedinthespreadofNorthfansfromgametogameforjuniorvarsity(JV)andvarsity(V)football.Theteammanagercountsthenumberoffansfor10at-homegames.Thecountsaregivenasfollows.

JVfans:95,75,42,61,79,87,95,108,109,129;

Varsityfans:215,201,186,175,192,184,199,213,220,232.

Determine if standarddeviation isanappropriatechoiceofmeasurementof spreadandcomparethespreadofthetwodistributionsoffans.Whichteamappearstohavealargerstandarddeviationoffansattendingthegames?


Ahistogramorboxplotforeachdatasetrevealsthedataisroughlynormal.Oncenormalityisverified,thecalculatedsamplestandarddeviationsares(JV)=25.2ands(V)=18.1.Thevarsityteamhasatighterspreadoffansattendingthegame.


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CompanyA andCompany Bmanufacture tractor brakes. Both companies recorded thehoursofbrakeusageuntilthebrakesfailed.Thedataisasfollows:

A:998,1023,974,911,928,981,974,1108,1064,959;

B:1005,1000,1001,979,1004,1014,1003,966.

Youarethepurchasingagentforyourfarm.WithconsistencyasyourNo.1priority,fromwhich company would you buy brakes for your tractors and why? Demonstrate yourreasoningwithanappropriategraphortable.


Agraph(suchasaboxplot)canbeconstructedandthestandarddeviationscalculated.Itcanbesuggestedthatthepurchasingagentrecognizethedifferencesinvariation.Thepurchasingagentcouldshowtheresultstoher/hispurchasingteam,butfurthercomparetheresults:

CompanyA’sstandarddeviationis59.7;itsmeanis991.9.

CompanyB’sstandarddeviationis15.7;itsmeanis996.4.

CompanyBshowsmoreconsistencybecauseofitssmallstandarddeviation.

Figure S8-3

900 950 1000 1050 1100 1150

Company B

Company A




Use the normal distribution to solve for the probability of an event or solve for the boundaries for a particular event.

Prior Knowledge:

• Translateprobabilitiestopercentagesandpercentagestoprobabilities.• Solveequationsinvolvingrationalnumbers.

Model Assessments


Studentslivingawayfromhomeforthefirsttimeoftenfacemanystressesand,therefore,change their dietary and exercise habits. It is reported that freshmen living away fromhomeforthefirsttimeandeatingatypicalcollegedietusuallygainweightwithameanof15poundsandastandarddeviationof10pounds.Assumethechangeinstudentweightisnormallydistributed.

a) Findtheapproximateprobabilitythatagivenstudentwillnotgainweight.

b) Inwhatrangedothemiddle95%ofallweightchangeslie?

c) For 25 randomly selected students, find the probability that the change in studentweightwillhaveameangreaterthan10pounds.


a) Wewanttofindtheprobabilitythatastudentdoesnotgainweight.

( )1.50 0.0668.P z < − =

6.68%ofalltypicalfreshmancollegestudentsdonotgainweight.


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Figure S9-1b

b) Themiddle95%fallwithin2standarddeviationsofthemean,

so 2 15 2 10 5 and35.µ σ± = ± = −

Themiddle95%ofallweightchangesarefromlosing5poundstogaining35pounds.

c) ( ) ( )10 2.50 0.9938.P x P z≥ = ≥ − =

Theprobabilityfor25randomlyselectedstudentstohavemeanweightchangegreaterthan10poundsis99.38%.


TheStanford-BinetIntelligenceScale(atypeofIQtest)wasoriginallydevelopedtohelpplacestudentsinappropriateeducationalsettings.Scoresarenormallyandapproximatelydistributedwithamean µ = 100 andastandarddeviationσ = 15.

a) WhatpercentofstudentshaveIQscoresabove115?

b) Whatpercenthavescoresbelow93?

c) TheTripleNineSocietyisamembershipassociationofpeoplewhohavescoredatthe99.9thpercentile.WhatscoreontheStanford-Binet IntelligenceScalewouldqualifyapersonformembership?




a) ( 115) ( 1.00) 1 0.8413 0.1587,P x P z> = > = − =

so15.87%ofstudentshaveIQscoresabove115.

b) ( 93) ( 0.47) 0.3192,P x P z< = < − =

so31.92%ofstudentshaveIQscoresbelow93.

c) Thez-scorecorrespondingtothe99.9thpercentileis 3.1.z =

Ascoreof146.5(or147)wouldqualifyapersonformembershipintheTripleNineSociety.


One June day at a large nursery, the temperature was taken at 100 locations. Thetemperaturesarenormallydistributedwithanaveragetemperatureof62degreesandastandarddeviationof2degrees.Whatistheprobabilitythatarandomsite’stemperatureismorethan65degrees?


p =0.0668.




Determine the size of a sample space using counting principles, permutations, and combinations.

Prior Knowledge:

• ApplytheFundamentalCountingPrinciple.• Applytheformulasforcombinationsandpermutations.

Model Assessments


Adaycare currently has three different activities offered in themorning, afternoon, andevening.Achildcanengageinoneactivitypertimeperiod.Becausethefacilityisincreasinginsize,thedirectorhiresadditionalemployeeswhohelpcreateanadditionalfiveactivitiesper time period. The director then wishes to advertise the total number of possiblearrangementsofdailyactivitiesachildcanexperience.Howmanyoptionsdoeseachchildhave,assumingthatthereisnoduplicationofactivities?


Thereare8 7 6 336= options.


a) Astatisticsclasshas6nursingmajors,7environmentalsciencemajors,35mathmajors,and1statisticsmajor.Howmanywayscouldadelegationof4studentsbechosen?

b) Ifeachmajorhastoberepresented,howmanywayscouldadelegationbechosen?


a) 211,876.

b) 1,470.


Yourolladiethreetimes.Howmanypossibleoutcomescanyouhave?


216differentoutcomes.






Compute basic probability of independent events and solve problems involving the binomial distribution:

• Determine situations that are Bernoulli trials.

• Calculate the probability of an event (success) and its complementary probability (failure).

• Discern the difference between geometric probability and binomial probability of

Bernoulli trials.

• Apply the formula for binomial probability for x successes in n trials ( ) .x n xn rP x C p q −=

• Use the normal model µ σN( , ), where µ = np and σ = npq to approximate a binomial

probability when the number of trials and desired successes is inordinately large.

Prior Knowledge:

• Applythepropertiesofexponents.• Applytheformulasforcombinationsandpermutations.• Usenotationtorepresenttheprobabilityofanevent.

Model Assessments


Pop-upadsareanannoyancethatmostInternetsurfershaveencountered.Aninternetadfirmreportsthattheprobabilitythatapersonclicksanadis0.02.Assumethatclicksareindependent.

a) AparticularWebsitewithpop-upadsgets100visitors.Whatistheprobabilitythatnoonewillclickonanad?

b) Whatistheprobabilitythatfivepeoplewillclickonanad?


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a) Usethebinomialprobability ( ) ( )1 .n kknP x k p p

k−

= = −

( ) 0 1001000 0.02 0.98 0.1326.

0P

= =

Theprobabilitythatnoonewillclickonanadis13.3%.

b) Usethebinomialprobability ( ) ( )1 .n kknP x k p p

k−

= = −

( ) 5 951005 0.02 0.98 0.0353.

5P

= =

Theprobabilitythatfivepeoplewillclickonanadis3.53%.


Giventhefollowingtablefromapreliminaryscreeningtestoncollegeathletesandsteroiduse,findtheindicatedprobabilities.

Positive Negative

Usedsteroids 0.998 0.002

Didnotusesteroids 0.004 0.996

Supposethat1.1%ofalargecollegepopulationhasusedsteroidsinthepast.

a) Whatistheprobabilityofselectinganathletefromthispopulationwhohas

i. usedsteroidsandtestedpositive

ii. usedsteroidsandtestednegative

iii. notusedsteroidsandtestedpositive

iv. notusedsteroidsandtestednegative

b) Whatistheprobabilitythatthetestispositiveforsteroidsforarandomlychosenathletefromthispopulation?

c) Whatistheprobabilitythatanathletehastakensteroids,giventhatthetestispositiveforsteroids?




a) Theprobabilityofselectinganathletefromthispopulationwhohas:

i. usedsteroidsandtestedpositiveis0.010978

ii. usedsteroidsandtestednegativeis0.00002

iii. notusedsteroidsandtestedpositiveis0.003956

iv. notusedsteroidsandtestednegativeis0.985044

b) P (testedpositive)=P (usedandtestedpositive)+P (didnotuseandtestedpositive)=

(0.011 0.998) (0.989 0.004) 0.010978 0.003956 0.014934.+ = + =

c) P(usedandtestedpositive)

=(usedandtestedpositive)

(testedpositive)P

P

= 0.0109780.014934

=0.74.


Testsbyanationallaboratoryshowthatthequalityofwidgetsfromalargemanufacturerhasimprovedfrom30%defectivewidgetsin2006to13%in2010.Giventhattheprobabilitythatanyrandomlyselectedwidgetwillbedefectiveis13%,whatistheprobabilitythatasampleof6widgetswillNOTcontainadefectivewidget?


0.4336,or43.36%.


a) DetermineifthefollowingsituationsdescribeaBernoullitrial:

i. Pick15cardsfromadeckof52cardswithoutreplacement.Letxbethenumberofblackcardspicked.

ii. Pickacardfromadeckof52,replacethecardandpickanothercard.Repeatthisprocess15times.Letxbethenumberofheartspicked.

b) Inquestion iiabove,determinetheprobabilityofselecting8hearts ifyourepeattheprocess15times.Inthisexample,whateventisconsidereda“success”andwhateventisconsidereda“failure”?Identifythenumberoftrials,theprobabilityofsuccess,andtheprobabilityoffailure.




a) DotheabovesituationsdescribeaBernoullitrial?

i. In this example there is sampling without replacement, so the selections aredependent.This isnotaBernoulli trial.Note that theprobabilityof thefirstcardbeingblackis0.5,butitkeepschangingwitheachcardthatisselected.

ii. ThisexampleisaBernoullitrialbecausethenumberoftrialsis15,p=0.25,andeachtrialisindependentoftheothers.

b) Asuccessisselectingaheart;afailureisselectinganycardthatisnotaheart,i.e.,clubs,spades,ordiamonds.Thenumberofindependenttrialsis15;theprobabilityofsuccessis0.25andtheprobabilityoffailureis1–0.25,or0.75.

( ) 8 7158 hearts 0.25 0.75 0.0131.

8P

= =

Theprobabilityofselecting8heartsin15trialsis1.31%.


a) Supposeinagameyourollasingledieuntilyougeta6.Whatistheprobabilitythatittakesfourrollsbeforeyourolla6?

b) Whatistheprobabilitythatyouwillrolla6withinthefirstfourrolls?

c) Supposeinadifferentgameyourollasingledie10timesandcountthenumberoftimesyourolla6.Whatistheprobabilitythatyouwillrollexactly4sixes?


a) ThefirstquestionisaBernoullitrialwithageometricdistribution.

( )( )4 15 1

4 0.096, or9.6%.6 6

P x−

= = =

b) Theprobabilitythatyouseea6withinthefirstfourrollsis:

0 1 2 35 1 5 1 5 1 5 10.518, or5.2%.

6 6 6 6 6 6 6 6 + + + =

c) ThisexampleisaBernoullitrialwithabinomialdistribution,wheren=10,x=4,

p=16andq = (1–p)=

56.

Solution:0.054,or5.4%.




Acompanymanufactures250,000packagesofjellybeanseachweek.Onaverage,3%ofthepackagesdonotsealproperly.Arandomsampleof400packagesisselectedattheendoftheweek.Whatistheprobabilitythatinthissamplebetween8and15packagesarenotproperlysealed?


ItisappropriatetouseaNormalmodeltoapproximatethebinomialdistributionbecause

thesamplesize(n=400)isverylarge.Inthiscase,p=0.03,q=(1–p)=0.97,np=12,

3.41.npqσ = = Correct for continuity by subtracting0.5 from the lower limit of 8 and

adding0.5totheupperlimitof15.

Solution:0.754,or75.4%.


Anairlinehasapolicyofbookingasmanyas15peopleonanairplanethatcanseatonly14.Theairlinehasdeterminedthat85%ofthebookedpassengersactuallyarrivefortheflight.Findtheprobabilitythatiftheairlinebooks15passengers,therewon’tbeenoughseats.


n =15,p =0.85,x =15.

Ifallthebookedpersonsarrive,therewon’tbeenoughseats,thusx =15inthiscase.

p(15)=0.0873,sothereisan8.73%chancethattherewon’tbeenoughseats.


Adepartmentstorehasexperienceda3.2%rateofcustomercomplaintsandwantstolowerthisratewithanemployeetrainingprogram.Aftertheprogram,850customersaresurveyedand7filedcomplaints.

a) Findthemeanandstandarddeviationforthenumberofcomplaintsin850customers.

b) Is it unusual to have 7 complaints out of 850 customers?Was the trainingprogrameffective?


a) 850(0.032) 27.2,

850(0.032)(0.968) 5.13.

np

npq

µ

σ

= = =

= = =

b) Theminimumusualvalueis2standarddeviationsbelowthemean:

µ σ− = − =2 27.2 2(5.13) 16.94.

Sincethesevencomplaints is lessthan2standarddeviationsbelowthemean,sevencomplaintsoutof850customersisunusuallylow.Thusthetrainingwaseffective.




Apply the Central Limit Theorem to sampling distributions.

Prior Knowledge:

• Identifythesamplemeanandthesamplestandarddeviation.• Identifythepopulationmeanandthepopulationstandarddeviation.

Model Assessments


A bank wants to apply the Central Limit Theorem to estimate the total income fromtheirATMcharges.Theydecidetotakeasimplerandomsample.WhichofthefollowingstatementsaretrueaccordingtotheCentralLimitTheorem?Selectallthatapply.

a) Anincreaseinsamplesizefromn=16ton=25willproduceasamplingdistributionwithasmallerstandarddeviation.

b) Themeanofasamplingdistributionofsamplemeansisequaltothepopulationmeandividedbythesquarerootofthesamplesize.

c) The larger the sample size, the more the sampling distribution of sample meansresemblestheshapeofthepopulation.

d) Themeanofthesamplingdistributionofsamplemeansforsamplesofsizen=15willbethesameasthemeanofthesamplingdistributionforsamplesofsizen =100.

e) The larger the sample size, themore the samplingdistributionof samplemeanswillresembleanormaldistribution.


a) Thestandarddeviation(SD)ofthesamplingdistributionofsamplemeansisthestandarddeviationofthepopulationSDdividedbythesquarerootofthesamplesize.Sowhensamplesizesincrease,theSDwilldecrease.

d) Themeanofthesamplingdistributionofsamplemeansisthemeanofthepopulation(regardlessofthesamplesize).

e) ThisisbythedefinitionoftheCentralLimitTheorem.


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Adesignerofexerciseequipmentforwomenneedstoestimatethemeanheightofwomentomanufactureanewpieceofequipment.TheheightofAmericanadultwomenisdistributedalmostexactlyasanormaldistribution,withameanof63.5inchesandastandarddeviationof 2.5 inches. Imagine that all possible random samples of size 25 are taken from thepopulationofAmericanadultwomen’sheights.Themeansfromeachsamplewouldthenbegraphedtoformthesamplingdistributionofsamplemeans.Drawandlabelthesamplingdistribution.Indicatethemeanandstandarddeviation.


PopulationdistributionofAmericanadultwomen’sheights;meanis63.5inchesandSDis2.5inches.

Figure S12-2(1)

54 56 58 60 62 64 66 68 70 72 74

0.05

0.1

0.15

0.2

Height (inches)

Prob

abilit

y

µ = 63.5

SD = 2.5

Samplingdistributionof samplemeans froma randomsampleof25adultAmericanwomen’sheights;meanis63.5inchesandSDis0.5inches.

Figure S12-2(2)

54 56 58 60 62 64 66 68 70 72 74

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Height (inches)

Prob

abilit

y

µ = 63.5

SD = 0.5




Atalocalretailstore,therehasbeenanincreaseincomplaintsaboutemployees’inattentionto customer service due to the use of electronic devices. The number of personal textmessagesreceivedinthelast3minutesbyemployeeswasthree,two,andfive.Assumethatsamplesofsizetwoarerandomlyselected,withreplacementfromthispopulationofthreevalues.Listthedifferentpossiblesamplesandfindthemeanofeachofthem.Describethemeanandstandarddeviationofsamplingdistributionofsamplemeans.


Thepossiblesamplesare3-3;3-2;3-5;2-3;2-2;2-5;5-3;5-2;5-5.Theircorrespondingmeansare3,2.5,4,2.5,2,3.5,4,3.5,5,givingadistributionof

Mean Probability2.0 1/9

2.5 2/9

3.0 1/9

3.5 2/9

4.0 2/9

5.0 1/9

withameanof3.33andastandarddeviationof0.88.




Estimate population parameters using confidence intervals for means and proportions.

Prior Knowledge:

• Identifythesamplemeanandthesamplestandarddeviation.

• Identifythepopulationmeanandthepopulationstandarddeviation.

Model Assessments


Aboatmanufacturerhasaproblemwithfiberglasscrackingwhenitsboatsencounterroughwaters.Theytryanewmaterialandtest50boats.Theresultisthat5boatscrack.

a) Whatisthe95%confidenceintervalestimatingthetrueproportionofboatsthatwillcrack?

b) Boatsusingthecurrentmaterialcrack15%ofthetime.Isthisnewmaterialbetter?


a) (0.01685, 0.18315). We are 95% confident that the interval from 0.01685 to 0.18315containsthetrueproportionofboatsthatwillcrack.

b) No,thenewmaterialisnotbetterbecausethecurrentmaterialcracks15%ofthetime,whichisbetween1.7%and18.3%.


Toperformaz-confidenceinterval,whatthreesimpleconditionsmustbemet?


SimpleRandomSample(SRS),normallydistributed,knownpopulationstandarddeviation.


Forastatisticsprojectatyourhighschool,yourecordthenumberofhoursofsleepthat10randomstudentsgetonagivennight:7.6,6.7,7.5,6.7,7.1,6.5,7.7,6.7,7.5,7.0.

Find the 95% confidence interval formean amounts of sleep the students at your highschoolgetpernight.Interpretthisintervalinthecontextoftheproblem.


Weare95%confidentthattheintervalbetween6.78and7.42hourscontainsthetrueamountofsleepahighschoolstudentreceives.


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Perform appropriate hypothesis test (state the hypotheses, determine the significance level, check conditions/criteria, calculate the test statistic, determine the p-value, make a decision and interpret it in the context of the problem) for a given situation.

Prior Knowledge:

• Calculateaproportiongivenpercentagesandvalues.• Calculatenormalizedvalues(z-ort-values).• Calculatetheprobabilityofaparticulareventusinganormalizeddistribution.

Model Assessments


Asoftdrinkcompanycomesoutwithanewflavor.Theygivetheiroriginalsoftdrinktoagroupof23consumersandthenewsoftdrinktoanothergroupof23consumers.Theythenaskeachgrouptoratehowmuchtheylikethesoftdrinkonascaleof1–10.Determinetheappropriatehypothesistest.


Two-samplet-test.


• HealthScienceandMedicalTechnology• ManufacturingandProductDevelopment• PublicServices• Transportation




Anexperimentwasperformedonthesideeffectsofanewcosmeticsurgeryprocedure.Ofthe445patientswhounderwenttheprocedure,25sufferedan“adversesymptom.”Doesthisexperimentprovidestrongevidencethatfewerthan10%ofpatientswhounderwentthisprocedurehaveadversesymptoms?Performanappropriatetestatthe 0.01α = significancelevel.


Hypotheses:Wewanttotestaclaimaboutp,thetrueproportionofpatientswhosufferedanadversesymptom.Ourhypothesesare:

H0:p=0.10,

Ha:p<0.10.

Conditions: Weshoulduseaone-proportionz-testiftheconditionsaremet.

• SRS:Weareassumingthata“randomsample”ofpatientsundergoinganewcosmeticsurgeryprocedurewasanalyzed.IfthesamplingdesignwasnotanSRS,ourcalculationsmaynotbeaccurate.

• Normality: Theexpectednumberof“yes”and“no”responsesare(445)(0.10)=44.5and(445)(0.90)=400.5,respectively.Bothareatleast10,sowecanusethez-test.

• Independence: Sincewearesamplingwithoutreplacement,thishospitalmusthaveatleast(10)(445)=4,450patientsundergoingthenewprocedure.

Test-statistic: Theone-proportionz-statisticis

(1 ) 0.0562(0.9488)445

ˆ 0.0562 0.103.02.

p pn

p pz

−

− −= = = −

P-value: ( 3.02) 0.0010.P z ≤ − =

Interpretation: There is a 0.10% chance of obtaining a sample result as unusual as orevenmoreunusual thanwedid ( )=ˆ 0.0562p whenthenullhypothesis is true.Sinceour

p-valueof0.0010islessthan 0.01,α = rejectH0.Wehavesignificantevidencetosaythattheproportionofpatientsundergoingthisnewcosmeticsurgeryprocedurewhosufferanadversesymptomislessthan10%.




A school newspaper reported on a study of bacteria found on teachers’ keyboards in ahighschool.Atotalof50keyboardsweresampledandanalyzedforbacteria.Thebacterialconcentrationofeachspecimenwasdeterminedusinganinfraredmicroscopicmethod.Thesampleresultedinameanbacteriallevelof84ug/mLandastandarddeviationof80ug/mL.Testthehypothesisthatthetruemeanbacterialevelinteachers’keyboardsfallsbelow100ug/mL.Use 0.01.α =


Hypotheses: Wewanttotestaclaimaboutthetruepopulationmeanbacterialevelfoundonteachers’keyboards.Ourhypothesesare

0: 100,H µ =

: 100.aH µ <

Conditions:Sincewedonotknowthestandarddeviationofthebacterialevelfoundonallteachers’keyboards,wemustuseaone-samplet-test.

• SRS: Wemustbewillingtotreatour50keyboardsasanSRSfromthepopulationofall teachers’keyboards ifwewant todrawconclusionsabout teachers’keyboards ingeneral.Thisisajudgmentcall.

• Normality: Useofthet-procedureisjustifiedbecausethesamplesizeisreasonablylarge

(n=50)andthusthedistributionof x willbeapproximatelyNormalbytheCentralLimitTheorem.

• Independence:Wearesamplingwithoutreplacementandthepopulationsizeismuchlargerthan 10 50 500.=

Test statistic: Thet-teststatisticis

8084

84 1001.412.

sn

xt

µ− −= = = −

P-value:Althoughthedegreesoffreedomaredf=n–1,sodf=50–1=49,weusethemoreconservativedf=40fromastandardt-table.Fort=–1.412,theuppertailprobabilityisbetween0.10and0.05.Usingthecalculator,p=0.0818.

Interpretation:Sinceourp-value(0.0818)isnotlessthanα = 0.01, wefailtorejectH0.Wedonothavesufficientevidencetosaythatthemeanlevelofbacteriafoundonteachers’keyboardsislessthan100ug/mL.




Accordingtoareport,therateoffirearmdeathsper100,000peoplefortheUnitedStateswas10.34.Therateoffirearmdeathsper100,000forKansaswas10.51.Testthehypothesisthattherateoffirearmdeathsper100,000inKansasissignificantlydifferentfromtheratefortheUnitedStates.Useα = 0.05.


Hypotheses:Wewanttotestaclaimaboutp1,thetrueproportionoffirearmdeathsinKansas,andp2,thetrueproportionoffirearmdeathsintheUnitedStates.Ourhypothesesare

0 1 2: ,H p p= 1 2: .aH p p≠

Test-statistic:Thetwo-proportionz-statisticisz=-0.37.

P-value: p=0.7097.

Interpretation:Thereisa71.0%chanceofobtainingasampleresultasunusualasorevenmoreunusualthanwedid,sowedonothaveenoughevidencetosaythatthefirearmdeathratesaredifferent.


AlocalmunicipalairportconductedastudyoverseveralyearsofpiloterrorsandthenumberofDWIs(drivingwhileintoxicated).Thefollowingtableshowsthedatacollected.

No DWIs 1 or more DWIsPilotswithno

pilot-erroraccidents 21,588 608

Pilotswithpilot-erroraccidents 199 11

Test the hypothesis that there is no relationship between the number of DWIs and thenumberofpilot-erroraccidents.Useα = 0.05.


H0:ThereisnoassociationbetweenthenumberofDWIsandthenumberofpilot-erroraccidents.

H1:ThereisanassociationbetweenthenumberofDWIsandthenumberofpilot-erroraccidents.

Test-statistic: 2 4.836.χ =

P-value:p=0.0279.

Interpretation: Thereisa2.79%chanceofobtainingasampleresultasunusualasorevenmoreunusualthanwedid,sowerejectthenullhypothesis.TheevidenceappearstosuggestthatthereisanassociationbetweenthenumberofDWIsandthenumberofpilot-erroraccidents.




Supposethat fourcoughsyrupsarecompared inanexperiment,withthehoursof relieffromcoughingasthemeasuredvariable.Dothesecoughsyrupshavethesamemeanhoursofrelief?

Cough Syrup A Cough Syrup B Cough Syrup C Cough Syrup D

Hoursofrelief

0.53 9.80 5.79 19.19

0.77 1.90 1.91 7.17

12.99 11.45 1.97 8.96

7.41 4.97 8.73 5.28

8.82 1.76 11.13 17.18

4.40 11.18 13.01 0.20

16.35 9.49 3.86 4.23

- 17.55 8.77 10.85

- 2.34 - 6.73

- 12.99 - 2.49

- 17.86 - 9.06

- - - 9.99

- - - 3.48


0 1 2 3 4: .H µ µ µ µ= = =

H1:Thepopulationmeansarenotallequal.

Test statistic:F=0.3269.

P-value:p =0.8059.

We fail to reject thenull hypothesis. Themean timeof relief for the four syrupsdonotappeartobestatisticallydifferent.




Explain the relationship between parameters and statistics.

Prior Knowledge:

• Usebasicstatisticsterminology.• Calculatethemeanandstandarddeviationforagivendataset.• Identifythepopulationandsampleinagivenscenario.

Model Assessments


Anovicebusinesspersonhasabusinessideaandisinterestedinobtainingaloan.Afriendwhoworksatabanktellsherthattheaveragebusinessloanrateinthisareais8.25%.Shedoessomeresearchandfindsfivebanksthatgiveanaveragerateof 10.25%.Determinewhichpercentageisaparameterandwhichisastatistic.


8.25%isaparameterand10.25%isastatistic.


The public health agency worker has been concerned about the ever-increasing foodcontaminationissuesinhercity.Afterinspecting55,000kilogramsofmeatstoredatalocalsausagecompany,shefoundthat45,000kilogramsofthemeatwasspoiled.Identifythesample,population,statistic,andparameter(ifpresent).


Thesampleis55,000kilogramsofmeatstoredatthesausagecompany,andthestatisticisthe45,000kilogramsofspoiledmeat.Inthisquestion,thepopulationisconsideredtobeallpossiblemeatproducedinthiscity,andtheparameteristhetotalkilogramsofspoiledmeatfromthecity.


• FinanceandBusiness• HealthSciencesandMedicalTechnology• Hospitality,Tourism,andRecreation• PublicServices




Ahealthandfitnessclubmanagersurveys40randomlyselectedmembersregardingtheidealdimensionsofthenewswimmingpool.Hefoundthattheaverageidealdimensionsofthosequestionedtobe100yardsby25yards.Identifythesample,population,statistic,andparameter(ifpresent).


Thesampleis40randomlyselectedmembersandthestatisticistheaverageidealdimensionsof100yardsby25yards.Thepopulationisconsideredtobeallpossiblemembersofthisclub,andtheparameteristhemeanofallidealdimensionsfromallclubmembers.


Apollwasdoneforanelectionandarandomselectionof1,000citizenswasaskediftheywereinfavorofanationaluniversalhealthcarebill.Ofthosepolled,175citizensindicatedthat they believed the health care bill was a necessity. Identify the sample, population,statistic,andparameter(ifpresent).


Thesampleisthe1,000citizensrandomlyselectedandthestatisticistheproportion(0.175)ofcitizensinfavorofthebill.Thepopulationisconsideredtobeallcitizensofthisnation,andtheparameteristheproportionofallthecitizensofthisnationinfavorofthebill.




Given a p-value, interpret its meaning in the context of the variables of a problem:

• Define the p-value.• Identify the appropriate null and alternative hypotheses, and use the proper notation of

H0 and Ha .• Interpret a given p-value in the context of a hypothesis testing situation as it relates to

the rejection of or failure to reject H0 .

Prior Knowledge:

• Identifytheprobabilityofaneventusingastandardizednormaldistribution.• Identifytheparameterandstatisticforagivenscenario.• Calculatez-andt-valuesforagivenscenario.

Model Assessments


AtourismservicewishestodetermineifmorepeoplecontactthemtheweekbeforeNewYear’sDayortheweekbeforeLaborDay.Astudentinterngathersdatafromtheprevious35monthsanddeterminesthesampleaveragenumberofpeoplecontactingthemaweekbeforeeachholiday.Theythenrunatwo-samplet-testusingthesampleaverages.Theyobtainap-valueof0.1325.Whatcanbeconcluded?


ThereisnostatisticaldifferencebetweentheaveragenumberofpeoplewhocontactthetourismservicetheweekbeforeNewYear’sDayandtheweekbeforeLaborDay.


• FashionandInteriorDesign• HealthScienceandMedicalTechnology• Hospitality,Tourism,andRecreation




A fashionmerchandiser wants to know if increasing the price on theirmost purchaseddress increasedthetotalsales.Theyrandomlychoose30storesandperformamatchedpaircomparison,comparingtheaveragetotalsalesofdressesbeforeandafterthepriceincrease.Theyobtainap-valueof.003556.Whatcanbeconcluded?


The fashion merchandiser should increase the price in all of her/his stores. There is astatisticallysignificantincreaseintheaveragetotalsalesofdressesafterthepriceincrease.


AnewdrugisbeingtestedbytheFDAthatreducesthedurationofmigrainestolessthan2hours.Thisobservationhasap-value=0.032.Interpretthemeaningofthep-valueinthecontextoftheproblem.


Thep-value=0.032 indicates that thenewdrug, indeed, has an effecton reducing thedurationofmigrainestolessthan2hours.Thereisastatisticallysignificantreductioninthedurationofmigraines.




Find and apply the equation of the regression line when linear regression is appropriate:

• Determine when linear regression is appropriate for a set of bivariate data. • Find the linear correlation coefficient r and the coefficient of determination r 2.• Use the regression line to make appropriate predictions.• Interpret r, r 2, and slope in the context of a given situation.

Prior Knowledge:

• Graphlinearequations.• Findtheslopeandy-interceptofaline.

Model Assessments


Yourfriendrecentlybecamethenewmanagerofapizzarestaurant.Hewantstodetermineifthecurrentpricesforcheesepizzasareconsistentwiththeirsize.Pricesforcheesepizzaareshowninthetablebelow.

Size Diameter (inches) CostPersonal 6.5 $4.69

Small 9.5 $9.00

Medium 12.0 $13.25

Large 14.0 $15.99

ExtraLarge 15.0 $20.25

a) Makeascatterplotofthesedata.

b) Explainhowyouwoulddetermineifthereisalinearrelationshipbetweenthediameterandcost.



• EnergyandUtilities• FinanceandBusiness• HealthScienceandMedicalTechnology• InformationTechnology




a) scatterplot

Figure S17-1a

0

5

10

15

20

25

4 6 8 10 12 14 16 18 20

Cos

t (do

llars

)

Diameter (inches)

b) Thereisacurveintheresidualplot,soalinearrelationshipisnotappropriate.Apowermodelwouldbebetter.Accordingtothenewpowermodel −= 0.697 1.684ˆ (10) ( ) ,y x thepredictedcostof thefivedifferent sizesofpizzaare$4.70,$8.91, $13.20,$17.12, and$19.22.Itappearsthataslightadjustmentinthechargesforlargeandextralargepizzasmightbewarranted.


Typically,thetemperatureinPhoenix,AZ,ishighinJuly.ThetemperatureismoremoderateinJanuary.PlannersatthePhoenixElectricCompanywouldliketouseJanuary’saveragetemperaturestopredictJuly’saveragetemperature.ThefollowingtableshowstheaveragetemperaturesinFahrenheitforJanuaryandJulyinPhoenixfrom2001to2009(dataavailableatwww.noaa.gov).

YearJanuary temperatures in degrees Fahrenheit

July temperatures in degrees Fahrenheit

2001 54.1 94.4

2002 56.0 96.0

2003 62.0 97.7

2004 57.5 94.5

2005 57.8 97.3

2006 57.7 96.5

2007 52.9 95.8

2008 54.7 94.9

2009 58.7 98.3



a) Createascatterplotwith the January temperatures on the x-axis and the July temperaturesonthey-axis.

b) Findtheequationoftheleast-squaresregressionline.

c) WhatisthecorrelationbetweenJulyweatherandJanuaryweather?


a) scatterplot

Figure S17-2a

94.0

94.5

95.0

95.5

96.0

96.5

97.0

97.5

98.0

98.5

52 54 56 58 60 62 64

July

tem

pera

ture

s (d

egre

es F

)

January temperatures (degrees F)

b) = +ˆ 76.159 0.352 .y xc) r=0.685.(NOTE:Thecorrelationisstatisticallysignificantat 0.05,α = butitisnotat

0.01.α = .)


Thewinningtimes(inseconds)inthemen’s100-meterdashintheOlympicsfrom1972to2008aregiveninthefollowingtable:

Year Athlete Country Winning times in seconds1972 ValeryBorzov URS 10.14

1976 HaselyCrawford TRI 10.06

1980 AllanWells GBR 10.25

1984 CarlLewis USA 9.99

1988 CarlLewis USA 9.92

1992 LinfordChristie GBR 9.96

1996 DonovanBailey CAN 9.84

2000 MauriceGreene USA 9.87

2004 JustinGatlin USA 9.85

2008 UsainBolt JAM 9.69



a) Makeascatterplotofthedatawherex representsthenumberofyearssince1972andyrepresentsthewinningtimes.

b) Findtheequationfortheleastsquareslineforthisdata.

c) Interprettheslopeincontextofthequestion.

d) Howcanyoudetermineiftheleastsquareslineisanappropriatemodelforthedata?


a) scatterplot

Figure S17-3a

9.6

9.7

9.8

9.9

10.0

10.1

10.2

10.3

0 10 20 30 40

Win

ning

tim

es in

sec

onds

Years since 1972

b) = −ˆ 10.1705 0.0119 .y x

c) Foreachincreaseofoneyear,theOlympicrecorddecreasesby0.0119seconds.

d) Aresidualplotrevealsarandomscatteringofpoints.Theresidualsvaryfrom–0.06to0.17,soaleastsquareslineisanappropriatemodel.




Sincebonestructurevariesinsizefrompersontoperson,researchershaveaddedframesizeasafactorinhelpingtodetermineaperson’sidealweight.Knowingone’sframesizecanhelpapersonsetreasonableweightgoals.

The students in a statistics class measured their wrist circumference and height incentimeters.Thefollowingdatawerefound.

NameWrist circumference

(in cm.) Height (in cm.)Karen 15.0 163

Chris 17.0 191

Colleen 15.5 168

Cayley 15.5 173

Ethan 16.0 183

Alex 16.5 175

Krizza 14.5 156

Julia 16.0 172

Carrie 17.5 176

Gissel 17.0 163

Natalie 15.5 162

Jeff 17.0 188

Ms.Statsky 14.0 168

a) Makeascatterplotofthedata.Doestherelationshipappeartobeapproximatelylinear?

b) Determinetheequationoftheleastsquaresregressionline.

c) Computetheresidualsandmakeaplotoftheresidualsagainstx.Describethepatterndisplayedbytheresiduals.

d) What is r 2? Interpret this value in context.What does this value indicate about thesuccessofthelinearregressioninpredictingastudent’sheight?




a) scatterplot

Figure S17-4a

150155160165170175180185190195200

12.0 14.0 16.0 18.0 20.0

Hei

ght

(cm

)

Wrist circumference (cm)

Yes,itlooksapproximatelylinear.

b) ˆ 76.89 6.05 .y x= +

c) scatterplot

Figure S17-4c

-20

-15

-10

-5

0

5

10

15

20

12 13 14 15 16 17 18

x

Res

idua

ls

Theyarescatteredabouttheliney=0,withsomepointsfarfromthisline.Thereisnodistinctpattern.



Residuals-3.573

12.335

-1.596

3.404

10.381

-0.642

-7.55

-0.619

-5.688

-15.66

-7.596

9.335

7.473


Twentystudentsarerandomlyselectedfromacollegestatisticsclassof80students.Theirmostrecenthomeworkandexamscoresarecompared.

Homework scores (points out of 10 possible points)

Exam scores (points out of 100 possible points)

10.0 79.0

7.5 79.0

9.5 91.0

9.0 78.5

8.0 75.5

9.0 64.5

10.0 75.0

8.0 80.0

8.5 85.5

7.5 68.0

10.0 89.0

10.0 73.5

8.5 55.0

9.0 76.0

6.5 81.0

10.0 85.5

8.0 71.5

9.5 64.0

9.0 78.0

8.5 89.5

d) r 2=0.3735,so37.35%ofthevariationinheightisexplainedbytheleastsquaresregressionlineof height onwrist circumference. This value isnotverygood; fromthetablewecanseethatthis regression linewasoffby 15.7centimetersforonestudent.



a) Identifytheexplanatoryandresponsevariables.

b) Createascatterplotandfindtheequationfortheleastsquaresline.

c) State the slopeof the least squares regression lineand interpret themeaningof theslopeintermsofthepredictedresponsevariableandtheexplanatoryvariable.

d) Ifastudentscoresa7onhomework,whatisthepredictedexamscoreforthatstudent?


a) Theexplanatoryvariableishomeworkscoresandtheresponsevariableisexamscores.

b) scatterplot

Figure S17-5b

50

55

60

65

70

75

80

85

90

95

5 6 7 8 9 10 11

Exam

sco

res

(poi

nts)

Homework scores (points)

Predictedexamscore 66.73 1.16 homeworkscore.= +

c) Theslopeis1.1615,anditcanbeinterpretedinthefollowingway:Forevery1pointgaininhomeworkscore,thepredictedexamscoreis1.1615pointshigheronaverage.

d) Thepredictedexamscoreis77.0.Sincer=0.127,thelinearcorrelationisnotstatisticallysignificant.Hence,thebestpredictedexamscoreistheaverageexamscore,76.95.


Solarpanelsimproveenergyefficiency.Iftheybecomedirty,efficiencycandropby10–15%.Hamatwantstoseeifthereisarelationshipbetweenthenumberoftimeshecleanshissolarpanelseachyearandthetotalpoweroutput.Afterperforminga linearregressiononthedata,heobtainsacorrelationcoefficientof0.345.Interpretthissolutioninthecontextofthesituation.


Thervalueislow,suggestingthatthereisaweaklinearrelationshipbetweenthenumberoftimeshecleanshissolarpanelseachyearandthetotalpoweroutput.




Thedirectorofaqualitydepartmentinanetworkcommunicationscompanywantstoknowhowmanybugseachpersonfixesperday.Hisnetworkdoesnotinformhimofthenumberofbugseachpersonfixes,butratherthetotalnumberofbugsfixedperdayofallemployees.However,hedoeshaveahistoryofdataandisabletoplotthetotalnumberofbugsfixedperdayfordifferenttotalnumbersofemployees.Thegraphhasaslopeof1.32,acorrelationcoefficientof0.9,andacoefficientofdeterminationof0.81.Interpreteachofthesenumbers.


Foreachadditionalemployee,1.32morebugsarefixedonaverageperday.Theveryhighrvaluesuggeststhattherelationshipisstronglylinear.Ther 2valuestatesthat81%ofthevariabilityinthebugsfixedperdaycanbeexplainedbythenumberofemployees.


Listedbelowarethebudgets(inmillionsofdollars)andthegrossreceipts(inmillionsofdollars)forrandomlyselectedmovies.

Budget 62 90 50 35 200 100 90

Gross receipts 65 64 48 57 601 146 47

a) Makeascatterplotofthisdata.

b) Istherealinearrelationshipbetweenbudgetandgrossreceipts?


a) scatterplot

Figure S17-8a

0

100

200

300

400

500

600

700

0 50 100 150 200

Gro

ss re

ceip

ts

(milli

ons

of d

olla

rs)

Budget (millions of dollars)

b) Eventhoughthecorrelationcoefficientisr=0.926,whichisindicativeofastronglinearrelationship,theplotoftheresiduals isnotrandom.This indicatesthatthere isnotalinearrelationshipbetweenbudgetandgrossreceipts.Eventhoughthereisnotalinearrelationship,thereissomeotherkindofrelationship(exponential,quadratic,orother)asshownbythedata.


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A c k n o w l e d g e m e n t s / C o n t r i b u t o r s 207

A C K N O W L E D G E M E N T S / C O N T R I B U T O R S

Math faculty from 13 Cal-PASS Professional Learning Councils collaborated toward the common

goal of aligning curricula from high school through transfer-level coursework in postsecondary

education in the mathematics subject areas of Algebra II/Intermediate Algebra, Precalculus, and

(non-STEM) Statistics. The 13 Cal-PASS math Professional Learning Councils that participated in

the ACCESS initiative were from the following regions:

Contra CostaEast San Diego County

Los AngelesMerced

North BayNorth Coast

Placer-NevadaSan Francisco

San MateoSanta Barbara

Siskiyou CountyWest Fresno Region

West San Bernardino County

Several Cal-PASS Professional Learning Councils began this type of alignment work in their

local regions prior to the ACCESS Initiative, and their experience helped Cal-PASS develop a model

for aligning exit and entrance competencies for sequential coursework. This initiative is unique in

that it brought together a collaboration of high school and postsecondary faculty from around the

state to discuss curriculum and build a clearly articulated curricula guide chronicling what students

are expected to know upon completing one course and to be prepared for subsequent courses. The

names of Cal-PASS Professional Learning Council faculty participants who graciously volunteered

their time and expertise for this project are listed on the following pages.

The ACCESS initiative was made possible through partnerships between Cal-PASS and The

William and Flora Hewlett Foundation, The James Irvine Foundation, The Evelyn and Walter Haas

Jr. Fund, and The Girard Foundation. This far-reaching project could not have come to fruition

without the keen vision and generous funding of these partners.


A c k n o w l e d g e m e n t s / C o n t r i b u t o r s208

The coordination and management of this statewide endeavor was the accomplishment

of Eden Dahlstrom, ACCESS project director, who traveled around the state over a 2-year

period to meet with the Cal-PASS Professional Learning Councils and keep the project

relevant and on track. She also headed up the summer workshops, where faculty from

around the state came together to specify competencies and assessments. Cal-PASS

leaders Michelle Kalina and Shelly Valdez also traveled to Cal-PASS Professional Learning

Council meetings and offered collegial guidance and assistance to keep this project

energized and relevant.

After hundreds of pages of math assessments and equations were compiled, the guide

was handed over to math instructors Debbie Hill and Kentaro Iwasaki to handle the initial

editing process. Following several rounds of edits, Cal-PASS’s Katheryn Horton stepped up

to oversee the editing process. Math instructor Michael Orr took over the reins as content

editor and created most graphs and figures. Finally, the math team of Michael Orr, Tina

Shinsato, and Diane Murillo edited and honed the content into its final form. Throughout

the editing process, freelance copyeditor Cindi Patton (www.thefinaledit.com) handled

copyediting and formatting as well as production and print manangement. Patton Brothers

Illustration & Design, Inc., (www.pattonbros.com) created the design. Math instructors

Michelle McGinity, Sherry Wallin, Vanson Nguyen, and Gary Remiker also contributed to the

editing and proofing of this guide.

For more information, please contact Dr. Shelly Valdez, EdD, at [email protected].


A c k n o w l e d g e m e n t s / C o n t r i b u t o r s 209

Math Faculty Participants

Name: Region:

Adam, Brianna Santa Barbara

Ahlgren, Joyce San Bernardino - West

Althaus-Cressman, Alana Siskiyou

Ambrozich, Jo Anna San Bernardino - West

Ashlock, James Santa Barbara

Asp, Steve San Mateo

Bader, Sandy Sonoma (North Bay)

Ballinger, Brad North Coast

Barbour, Renee Los Angeles

Barelka, Bridget San Diego - East

Barron, Maile Placer–Nevada

Beemer, Beverly San Bernardino - West

Biesecker, Luke North Coast

Birrell, Jameson Fresno - West

Boyd, Linda Los Angeles

Bridge, Jan San Bernardino - West

Brown, Anna Sonoma (North Bay)

Brubaker, Donny San Bernardino - West

Bruketta, Sandy Contra Costa

Calhoun, Bernadette Sonoma (North Bay)

Cavagnaro, Deann San Mateo

Chamberlain, Paula San Bernardino - West

Chao, Jack Placer–Nevada

Chen, Eileen Contra Costa

Cherrington, April San Mateo

Clark, Julie Merced

Cole, Wayne Santa Barbara

Collins, Bei San Bernardino - West

Conard, Roger San Bernardino - West

Connich, Brian Sonoma (North Bay)

Conway, Teri Santa Barbara

Coppin, Cheryl Siskiyou



Name: Region:

Creech, Teresa North Coast

Crosby, Bauea Contra Costa

Cubillo, Judy Contra Costa

Davis, Lloyd San Mateo

De La Rosa, Angelica Los Angeles

De Palma, Rose Merced

De Primo, Guy San Francisco

De Torre-Ozeki, Jeanne San Mateo

Delcroix, Stefaan Fresno - West

Derksen, Jared San Bernardino - West

Derochers, Debbie Siskiyou

DiJay, Carol San Bernardino - West

Dirksen, Jenn San Mateo

Drake, Lisa San Diego - East

Easton, Paula San Bernardino - West

Eramya, Heddy San Bernardino - West

Fauble, Patty Placer–Nevada

Ferrel, Galen Siskiyou

Francis, Steve Los Angeles

Frenken, Kathryn San Bernardino - West

Fuentes, Iyara Merced

Fuste, Beth Santa Barbara

Garcia, Cathy Placer–Nevada

Gazarian, Kaspar Los Angeles

Gentile, Scott San Francisco

Gipson, AnnMaree Fresno - West

Goodrich-Jones, Mariah Siskiyou

Griffith, Leah Los Angeles

Grybinas, Dalia San Mateo

Guenther, Pamela Santa Barbara

Hampton, Joel San Bernardino - West

Hapeman, Natlee Santa Barbara

Harvey, Derril San Bernardino - West


211A c k n o w l e d g e m e n t s / C o n t r i b u t o r s

Name: Region:

Hasson, Daniel San Mateo

Hayden, Davis Santa Barbara

Hays, Christine Fresno - West

Heinrich, Paul Los Angeles

Heitz, Judi San Diego - East

Helterbran, Stacey Sonoma (North Bay)

Hemphill, Vickie Los Angeles

Hill, Debbie Placer–Nevada

Hoffman, Michael San Mateo

Holmes, Jim Santa Barbara

Hull, Amy San Diego - East

Hum, Denise San Mateo

Iacopetti, Laura San Bernardino - West

Iniguez, Vanessa Los Angeles

Iwasaki, Kentaro San Francisco

Jenkins, Ted San Bernardino - West

Johnson, Ken Placer–Nevada

Johnson, Sue Santa Barbara

Kalbag, Anaroopa Santa Barbara

Kato, Kim Los Angeles

Koos, Karen Los Angeles

Krebs, Michael Los Angeles

Laabs, Kim Sonoma (North Bay)

Lenaburg, Lubi Santa Barbara

Lu, Peter San Francisco

Lum, Lily San Francisco

Lyon, Ann San Francisco

Mantel, Ava Los Angeles

Margala, Julie San Bernardino - West

Marshall, Rob North Coast

Mathiesen, Lauren San Bernardino - West

Mathieson, Christina San Bernardino - West

Matsumoto, Tami North Coast



Name: Region:

McClung, John Placer–Nevada

McGinity, Michelle Sonoma (North Bay)

Metlitzky, Lilian A. San Bernardino - West

Mikhail, Maggie Santa Barbara

Miner, Jason Santa Barbara

Montgomery, John Placer–Nevada

Moore, Bronwen Santa Barbara

Morales, Manuel Los Angeles

Morris, Kristi Santa Barbara

Munton, Dan Sonoma (North Bay)

Murillo, Diane San Bernardino - West

Nguyen, Ho Contra Costa and San Francisco

Nguyen, Vanson San Francisco and San Mateo

Nordman, Jennifer Merced

Odegard, Barbara Siskiyou

Okelberrry, Lydia Los Angeles

Olivieri Llewellyn, Tami San Bernardino - West

Ormseth, Tor Los Angeles

Orr, Michael San Diego - East

Pahl-Martinez, Sabrina San Bernardino - West

Patterson, Kathleen Placer–Nevada

Pender, Karen San Bernardino - West

Perry, Catharine San Bernardino - West

Piontkowski, Dennis San Francisco

Pompa, Jerry Siskiyou

Poulsen, Chrissy Placer–Nevada

Prapavessi, Despina Contra Costa

Protti, Toni Los Angeles

Randall, David Los Angeles

Raychaudhuri, Debasree Los Angeles

Rebman, Josephine Fresno - West

Ritchie-Reese, Patricia Placer-Nevada

Rivera, Jose San Bernardino - West

213


A c k n o w l e d g e m e n t s / C o n t r i b u t o r s

Name: Region:

Roberts, Kristen San Bernardino - West

Romero, Marty Los Angeles

Rust, Tue Contra Costa

Schlinger, Charles Merced

Seeley, Susan Contra Costa

Sheffield, Kevin Santa Barbara

Simon, Debra San Bernardino - West

Simutis, Jean Contra Costa

Smith, Brenda North Coast

Solis, Monica Los Angeles

Soupcoff, Bri San Bernardino - West

Spaventa, Marilynn Santa Barbara

Stachniak, Don San Bernardino - West

Stamper, Forrest North Coast

Stapleton, Denise Los Angeles

Stevens, Harriette Contra Costa

Thomas, Candy San Bernardino - West

Throop, Ryan Santa Barbara

Tomkins, Phil Fresno - West

Torre, Ken Sonoma (North Bay)

Trapp, Roland San Bernardino - West

Turner, Carole San Francisco

Usher-Staats, Lisa Los Angeles

Utter, Barbara Sonoma (North Bay)

Uy, Fred Los Angeles

Valdez, Luz Merced

Veater, Carl Fresno - West

Villeneve, Jeanne Placer–Nevada

Wagner, Bruce North Coast

Walton, Jennifer San Mateo

Ward, Tiffany Contra Costa

Westlake, Rachel Contra Costa

White, Cynthia Santa Barbara

White, Wes Los Angeles


214 A c k n o w l e d g e m e n t s / C o n t r i b u t o r s

Name: Region:

Whitley, Sheila Merced

Wigert, Scot Sonoma (North Bay)

Willis, Kelly San Bernardino - West

Witt, Robin San Bernardino - West

Wu, Ai-Chu Sonoma (North Bay)

Yokoyama, Kevin North Coast

Zimmerman, Andrea Placer–Nevada

215A c k n o w l e d g e m e n t s / C o n t r i b u t o r s

Career and Technical Education Participants

Name: Region:

Davis, Bill Fresno Area

Harvey, Elizabeth Riverside Area

Hollems, Diane Santa Barbara Area

Kellar, Raymond Siskiyou Area

Miller, Jerald Santa Rosa Area

White, Kathleen San Francisco Area

Cal-PASS Regional Coordinators

Name: Region:

Ceaser, Lisbeth Santa Barbara

Dahlstrom, Eden Siskiyou

Horowitz, Virginia Placer–Nevada

Horton, Katheryn Sonoma (North Bay)

Iwasaki, Kentaro San Francisco

Lachmayr, Lucia San Mateo

Linfor, Cali San Diego - East

Mahar, Kate San Francisco and Contra Costa

Owens, Rae Merced

Schneider, Garry Los Angeles

Schneider, Katy San Bernardino - West

Sizoo, Bob North Coast

Sutherland, Camilla Fresno - West

Tyberg, Alana San Diego - East

Wintermeyer, Lauren Santa Barbara



216 A p p e n d i x • M a p p i n g A C C E S S C o m p e t e n c i e s

A P P E N D I X

Mapping ACCESS Competencies to California Content Standards and Common Core Standards

The tables on the following pages map the relationship between ACCESS competencies and California-based and nationally based standards. The competencies are not necessarily organized in the same way as the California State Standards or Common Core Standards, nor are they necessarily in the order in which one would teach them in the course. Additionally, some of the ACCESS competencies do not have a corresponding map to California Content Standards or Common Core Standards. However, the faculty involved in this project felt that these skills were important enough that they be included as ACCESS competencies because they are necessary for the next level of coursework.

The California State Standards were designed by appointees of the State Board of Education. This document does not retain the official input of California’s public colleges and universities. The document was adopted by the State Board of Education and mandated for all public K–12 schools. These standards are available online at www.cde.ca.gov/board/.

The Common Core State Standards Initiative is a state-led effort launched in 2009 by state leaders, including governors and state commissioners of education from 48 states through membership in the National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). The process used to write the standards was informed by:

• thebestofstatestandards

• theexperienceofteachers,contentexperts,states,andleadingthinkers

• feedbackfromthegeneralpublic

To write the standards, the NGA Center and CCSSO brought together content experts, teachers, researchers, and others.

The Common Core standards, which can be found online at www.corestandards.org, are divided into two categories:

• collegeandcareerreadinessstandards,whichaddresswhatstudentsareexpectedtolearn when they have graduated from high school

• K–12standards,whichaddressexpectationsforelementarythroughhighschool


217M a p p i n g A C C E S S C o m p e t e n c i e s • A p p e n d i x

The California Education RoundTable Content Standards (called the CERT Standards) were published just prior to the state’s content standards for mathematics in 1997. The standards were designed by a task sponsored by the RoundTable. The RoundTable comprised, in part, the heads of the California Department of Education, the California Community Colleges, the California State Universities, and the University of California. Serving on the task force were K–12 faculty, administrators, public participants, and academic Senate-appointed postsecondary faculty. While these standards have no official status under the Board of Education, they provide a point of contrast, noting competencies that are not addressed in the California mathematics standards. These standards can be found at www.certicc.org.

CERT Mathematics Standards for California High School Graduates (not included in the rubric on the following pages): This report identifies the kinds of activities that should be promoted in the classroom. We include them here in summary form:

• Becoming Fluent in Mathematics — Effective classes will provide experiences that lead to students’ acquisition of facility with the basic techniques of mathematics. There are certain necessary skills that students will need to be able to call upon without hesitation.

• Modeling Mathematical Thinking — Effective classes will reflect the enthusiasm that comes in learning with a teacher and others who get excited about mathematics, who work as a team, who experiment and form conjectures.

• Solving Problems — Effective classes will reflect the notion that problem solving is best conveyed by giving students appropriate experience in solving unfamiliar problems, by then engaging them in discussion of their various attempts at solutions, and by reflecting on these processes.

• Developing Analytic Ability and Logic — Effective teachers will emphasize a thorough understanding of the subject matter and the development of logical reasoning. A classroom full of discourse and interaction that focuses on reasoning is a classroom in which analytic ability and logic are being developed.

• Experiencing Mathematics in Depth — Students must delve deeply into well-chosen areas of mathematics. A shallow exploration of a broad range of topics will not contribute to the ability to understand and independently use mathematics.

• Appreciating the Beauty and Fascination of Mathematics — Effective teachers must nurture the appreciation for the inherent beauty of mathematics.

• Building Confidence — Effective teachers must create situations in which students are rewarded for being inquisitive, for experimenting, for taking risks, and for persisting in finding solutions they fully understand. This will help students generate confidence in their mathematical ability.

• Communicating — Effective classes will reflect a student’s ability to explain with confidence not only the answer to the problem but the process by which the problem was solved. Students need extensive experience in oral and written communication with extensive feedback in order to develop these skills.



Competency Statement(Algebra II/Intermediate

Algebra)

ACCESS Competency

Number

CA Content Standard

National Common Core

Standard

Use properties of absolute values to solve multistep equations and inequalities.

1 1.0 A-CED 1A-REI 3

Solve systems of linear equations in both two and three variables by substitution, elimination/addition, graphs, and matrices.

2 2.0A-REI 5–9A-CED 3N-VM 6

Factor polynomials by grouping. 3 4.0 A-SSE 2

Factor polynomials by using the sum/difference of cubes pattern. 4 4.0 A-SSE 2

Factor polynomials by extending the difference of squares pattern to polynomials of degree higher than 2.

5 4.0 A-SSE 2

Solve quadratic equations in the real and complex number systems by factoring.

6 8.0N-CN 7

A-SSE 3aF-IF 8a

Solve quadratic equations in the real and complex number systems by completing the square.

7 8.0A-SSE 3b

A-REI 4a and 4bF-IF 8a

Solve quadratic equations in the real and complex number systems by using the quadratic formula.

8 8.0 A-REI 4b

Use multiple representations (tables, graphs, and equations) to solve contextualized problems that result in quadratic equations.

9 8.0 and 9.0A-CED 1

F-IF 7a and 8aF-LE 3

Add and subtract complex numbers. 10 6.0 N-CN 2

Multiply complex numbers. 11 7.0 N-CN 2 and 5

Mapping ACCESS Algebra II/Intermediate Algebra Competencies to Standards




Algebra)

ACCESS Competency

Number

CA Content Standard


Standard

Perform division with complex numbers. 12 7.0 N-CN 3

Simplify rational expressions with higher order polynomials in the denominator by canceling common factors in the numerator and denominator.

13 3.0 A-APR 1, 6, and 7

Add and subtract rational expressions with higher order polynomials in the denominator.

14 7.0 A-APR 7

Multiply and divide rational expressions with higher order polynomials in the denominator.

15 6.0 A-APR 6 and 7

Divide polynomials by binomials using long division and synthetic division.

16 12.0 A-APR 6

Simplify a fraction where the numerator, denominator, or both contain a fraction (complex fractions).

17 12.0 A-APR 6

Use multiple representations (tables, graphs, and equations) to solve contextualized problems that result in equations involving rational expressions.

18 15.0 A-REI 2A-APR 6

Use the properties of exponents to simplify expressions with rational exponents.

19 15.0 N-RN 1 and 2

Use the properties of exponents to solve equations with rational exponents.

20 15.0 N-RN 1 and 2A-REI 2

Simplify radical expressions by removing repeated factors and performing addition and subtraction operations.

21 1.0, 9.0, 10.0, 12.0, 13.0, 16.0, and 17.0 A-REI 2




Algebra)

ACCESS Competency

Number

CA Content Standard


Standard

Simplify radical expressions by removing repeated factors and performing multiplication operations.

22 1.0, 9.0, 10.0, 12.0, 13.0, 16.0, and 17.0 A-REI 2

Rationalize the denominator of radical expressions. 23 24.0 No corresponding

standard

Use multiple representations (tables, graphs, and equations) to solve problems that result in radical equations.

24 24.0 No corresponding standard

Apply and graph transformations of parent functions (quadratic, cubic, square root, absolute value, exponential, and logarithmic).

25 12.0 F-LE 2 and 3F-IF 7

Identify the parent function (quadratic, cubic, square root, absolute value, exponential, and logarithmic) and transformations for a given transformed function in algebraic or graphical form.

26 11.0–14.0 F-LE 2F-BF 5

Compose two linear, quadratic, cubic, square root, absolute value, exponential, or logarithmic functions.

27 16.0 F-BF 1c

Find the inverse of linear, quadratic, cubic, square root, absolute value, exponential, and logarithmic functions algebraically and graphically.

28 18.0–20.0 F-BF 4 and 5

Find the equation of an exponential function given two points on the graph.

29 22.0–23.0 F-BF 2F-LE 2

Use multiple representations (tables, graphs, and equations) to solve contextualized problems that result in one- and two-step exponential and logarithmic equations.

30 Linear Algebra 1–12 F-BF 1 and 5




Algebra)

ACCESS Competency

Number

CA Content Standard


Standard

Graph simple conics, including: • acirclegivenitsequation

(centered anywhere) • anellipsegivenitsequation

(centered at the origin)• ahyperbolagivenitsequation

(centered at the origin)

31

Trigonometry 4.0, 8.0, and 19.0;

Math Analysis 6.0 and 7.0

G-GPE 1, 2, and 3

Solve statistical problems: • Computepermutationsusing

fundamental counting principle.• Computecombinations.• Computeprobabilitiesusing

combinations.• Computeprobabilitiesusing

permutations.• Expandbinomialexpressions

using the binomial theorem.

32Embedded in

Algebra I and Algebra II standards

A-APR 5S-CP 9

Compute the general term and sums of arithmetic series, finite geometric series, and infinite geometric series.

33 Algebra II 22.0 and 23.0 F-BF 2



Competency Statement(Precalculus)

ACCESS Competency

Number

CA Content Standard


Standard

Perform matrix operations, including addition and multiplication, and calculate the determinants of 2x2 and 3x3 matrices.

1 Linear Algebra 1.0–12.0 N-VM 6 and 8

Describe different types of functions (polynomial, exponential, logarithmic, and trigonometric) using a table, a graph, an equation, and a verbal description..

2

Trigonometry 4.0, 8.0, and 19.0;

Math Analysis 6.0 and 7.0

F-IF 1, 2, 4, 5, 7, 8a, and 9

Graph piecewise-defined functions. 3 No corresponding

standard F-IF 7a, 7b, and 7c

Find all real zeros (roots) of polynomial functions exactly using the rational root theorem.

4 No corresponding standard A-APR 3

Find all the zeros (roots) of a polynomial function. 5 No corresponding

standard N-CN 7 and 8

Solve polynomial inequalities. 6 No corresponding standard

F-IF 7c A-APR 3

Solve rational inequalities. 7 No corresponding standard

No corresponding standard

Find the composition of two or more functions, each containing two or more operations (such as linear, quadratic, rational, cubic, square root, cube root, and trigonometric functions).

8Higher level

complexity than Algebra II 24.0

F-BF 1, 3, and 4

Given a function, determine if an inverse function exists. If so:• Findtheinverseofthefunction

algebraically and graphically.• Identifythedomainandrangeof

the original and inverse functions.

9

Higher level complexity than

Algebra II11.0 and 24.0

F-BF 4F-IF 5

Mapping ACCESS Precalculus Competencies to Standards




ACCESS Competency

Number

CA Content Standard


Standard

Simplify expressions involving exponents. 10 No corresponding

standardNo corresponding

standard

Expand logarithmic expressions. 11 No corresponding standard


Condense logarithmic expressions. 12 No corresponding standard


Solve multistep exponential equations. 13

Higher level complexity than Algebra II 12.0


Solve multistep logarithmic equations. 14 No corresponding

standardNo corresponding

standard

Sketch the graph of a conic, identifying center, vertices, foci, and asymptotes (if present).

15 Math Analysis 5.0, 5.1, and 5.2 G-GPE 1, 2, and 3

Graph trigonometric functions of the form

( ) .y A f Bx C k= + +

• Determinetheamplitudeofasine or cosine function from its equation.

• Determinetheperiodofasine,cosine, or tangent function from its equation.

• Determinetheverticalshiftofasine, cosine, or tangent function from its equation.

• Determinethephaseshiftofasine, cosine, or tangent function from its equation.

• Sketchbyhandthegraphofasine, cosine, or tangent function.

16 Trigonometry 2.0, 4.0, 5.0, and 9.0 F-TF 5

Given the graph of a trigonometric function, write the equation (sine, cosine, tangent).

17 Trigonometry 4.0 F-IF 7F-TF 5

Graph inverse trigonometric functions. 18 No corresponding

standard F-TF 6

Prove trigonometric identities. 19 Trigonometry 3.0, 3.1, 3.2, 10.0, and 11.0 F-TF 8, 9, and 10




ACCESS Competency

Number

CA Content Standard


Standard

Solve trigonometry equations. 20 Trigonometry 19.0 F-TF 7

Solve for all unknown parts of a given triangle:• byusingrighttriangle

trigonometry• byusingtheLawofSines• byusingtheLawofCosines

21 Trigonometry 12.0 and 13.0 G-SRT 8–11

Convert between polar and rectangular coordinates. 22 Math Analysis 1.0

G-PCC 1 and 2(CA add-on standard)

Graph equations written in polar form. 23 Trigonometry

15.0 and 16.0

G-PCC 1 and 2(CA add-on standard)

Graph equations written in parametric form. 24 Math Analysis 7.0

G-PCC 3(CA add-on standard)

Given the components of a two-dimensional vector, determine the magnitude and direction of the vector.

25 Math Analysis 7.0 N-VM 1, 2, 3, and 4

Perform the vector operations of addition, subtraction, and scalar multiplication.

26 Linear Algebra 4.0, 5.0, and 7.0 N-VM 4 and 5

Use factorial notation. 27 No corresponding standard


Expand binomial expressions. 28Higher level

compexity than Algebra II 20.0

A-APR 5



Mapping ACCESS (Non-STEM) Statistics Competencies to Standards

Competency Statement(Non-STEM Statistics)

ACCESS Competency

Number

CA Content Standard


Standard

Determine whether a variable is categorical or quantitative, given a situation.

1 Not explicitly stated, but implied in 8.0

Not explicitly stated, but

implied in S-ID

Identify the sampling method used (random, systematic, convenience, stratified, cluster, or voluntary response), given a scenario in which data were collected from a population.

2 No corresponding standard


implied in S-IC 3

Identify the basic terminology of experimental design used in a given scenario.



implied in S-IC

Identify biases in experimental designs or in the creation of a sample.

4 No corresponding standard S-IC 3

Determine an appropriate experimental design for a given scenario.


Construct and describe an appropriate visual display given a data set (histogram, bar graph, stem plot, scatter plot, box and whisker plot, and pie chart).

6 8.0 S-ID 1 and 6

Identify which measure of the center (mean, median, or mode) is most appropriate for a given situation and what the relationship between the mean, median, and mode indicates about the distribution of the data.

7 6.0 S-ID 2

Determine standard deviation from multiple representations of data (table, graph, and word problems) and explain standard deviation in the context of a given situation.

8 5.0 and 7.0 S-ID 2




ACCESS Competency

Number

CA Content Standard


Standard

Use the normal distribution to solve for the probability of an event or solve for the boundaries for a particular event.

9 4.0 S-ID 4

Determine the size of a sample space using counting principles, permutations, and combinations.

10 1.0 and 2.0 S-CP 9

Compute basic probability of independent events and solve problems involving the binomial distribution:• Determinesituationsthatare

Bernoulli trials.• Calculatetheprobabilityof

an event (success) and its complementary probability (failure).

• Discernthedifferencebetweengeometric probability and binomial probability of Bernoulli trials.

• Applytheformulaforbinomialprobability for x successes in n trials ( ) .x n x

n rP x C p q −=

• Usethenormalmodel ( , ),N µ σ

where npµ = and npqσ =

to approximate a binomial

probability when the number of

trials and desired successes is

inordinately large.

11 1.0 and 4.0Not explicitly

stated, but implied in S-MD

Apply the Central Limit Theorem to sampling distributions. 12 No corresponding

standard


implied in S-IC

Estimate population parameters using confidence intervals for means and proportions.





ACCESS Competency

Number

CA Content Standard


Standard

Perform appropriate hypothesis test (state the hypotheses, determine the significance level, check conditions/criteria, calculate the test statistic, determine the p-value, make a decision and interpret it in the context of the problem) for a given situation.


Not explicitly stated, but implied in

S-IC 4 and 5

Explain the relationship between parameters and statistics. 15 No corresponding

standard S-IC 1

Given a p-value, interpret its meaning in the context of the variables of a problem: • Definethep-value. • Identifytheappropriatenulland

alternative hypotheses, and use the proper notation of H0 and Ha.

• Interpretagivenp-value in the context of a hypothesis testing situation as it relates to the rejection of or failure to reject H0.


Not explicitly stated, but implied in

S-IC 4 and 5

Find and apply the equation of the regression line when linear regression is appropriate:• Determinewhenlinear

regression is appropriate for a set of bivariate data.

• Findthelinearcorrelationcoefficient r and the coefficient of determination r 2.

• Usetheregressionlinetomakeappropriate predictions.

• Interpretr, r 2, and slope in the context of a given situation.

17 No corresponding standard S-ID 6, 7, and 8

www.iebcnow.org

2011 For more information, contact Shelly Valdez, EdD, at [email protected]

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