Linear Relations and Functions Chapter Overview and Pacing

Transcription

1 Linear Relations and Functions Chapter verview and Pacing PACING (das) Regular Block Basic/ Basic/ Average Advanced Average Advanced Relations and Functions (pp. 6 6) optional. optional Analze and graph relations. Find functional values. LESSN BJECTIVES Linear Equations (pp. 6 67) optional. optional Identif linear equations and functions. Write linear equations in standard form and graph them. Slope (pp. 68 7) optional. optional Find and use the slope of a line. Graph parallel and perpendicular lines. Writing Linear Equations (pp. 7 8) optional. optional Write an equation of a line given the slope and a point on the line. Write an equation of a line parallel or perpendicular to a given line. Modeling Real-World Data: Using Scatter Plots (pp. 8 88) optional optional Draw scatter plots. (with - Find and use prediction equations. Follow-Up) Follow-Up: Lines of Regression Special Functions (pp. 89 9) optional. optional Identif and graph step, constant, and identit functions. Identif and graph absolute value and piecewise functions. Graphing Inequalities (pp ) optional. optional Graph linear inequalities. Graph absolute value inequalities. Stud Guide and Practice Test (pp. ).. Standardized Test Practice (pp. 6 7) Chapter Assessment.. TTAL Pacing suggestions for the entire ear can be found on pages T T. A Chapter Linear Relations and Functions

2 Timesaving Tools Chapter Resource Manager All-In-ne Planner and Resource Center See pages T T. CHAPTER RESURCE MASTERS Stud Guide and Intervention Practice (Skills and Average) Reading to Learn Mathematics Enrichment Assessment Applications* -Minute Check Transparencies Interactive Chalkboard AlgePASS: Tutorial Plus (lessons) Materials spaghetti graphing calculator, spaghetti , GCS, - - SC SC - - tape measure, graph paper SM 97 (Follow-Up: graphing calculator) GCS graphing calculator, toothpicks , 6 8 *Ke to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual Chapter Linear Relations and Functions B

3 Mathematical Connections and Background Continuit of Instruction Prior Knowledge In prior ears students worked with coordinate sstems, ordered pairs, and linear equations and functions. The manipulated and solved linear equations and inequalities algebraicall. Also, the used graphs to represent two-variable data sets. This Chapter Students eplore algebraic descriptions of linear functions, graphs of lines, and how to go back and forth between linear equations and graphed lines. The find the slope of a line containing two given points, relate the slope and -intercept of a line to the values m and b in the slope-intercept form of an equation, and write equations for lines given two points or given one point and the slope. The find lines of fit for data and graph inequalities and special functions such as the greatest integer function and the absolute value function. Future Connections Students will continue to learn how algebraic epressions and the coordinate plane are related. At a simple level, the will learn how the horizontal line test identifies a one-toone function. As activities become more comple, the will use graphs to eplore quadratic and other non-linear equations and inequalities and use graphs to represent and solve sstems of equations and inequalities. Relations and Functions This lesson begins an eploration of two important themes of algebra. ne theme is how algebraic equations and the Cartesian coordinate sstem are related. The other theme is the relationships between equations that represent entire lines and numbers that represent points on or properties of that line. For this lesson the central idea is ordered pairs. First, ordered pairs are eplored as names for points in a coordinate plane. Second, several ordered pairs are used to describe how the set of the first elements (the domain) can be related to the set of the second elements (the range). Third, for equations that represent a line or a curve, ordered pairs are used to determine the graph that represents that equation in the coordinate plane. In the lesson the relation between functions and relations is eplored in two was. Mappings of domain elements to range elements are used to identif functions that are one to one, functions that are not one to one, and relations that are not functions. In the coordinate plane, the vertical line test is used to distinguish a relation from a function. Relationships between ordered pairs and functions are eplored in two was. First, students are given an equation (for a curve or for a line) and make a table of ordered pairs for the equation. Second, students are given a function and a domain value, and evaluate the function to find the range value. Linear Equations In this lesson students deal with linear functions and intercepts. Linear functions and equations can be written in slope-intercept form, f() m b or m + b, or in standard form, A B C. The graph of a linear function or equation is alwas a line. Slope Slope is a fundamental concept in algebra and higher mathematics. In this lesson, students calculate the slope of a line given two points on the line and eplore the slopes of families or pairs of lines that are parallel and the slopes of pairs of lines that are perpendicular. Students graph a line given two points or given one point and the slope. In the coordinate plane, students associate lines with slopes that are positive, negative, zero, or undefined. C Chapter Linear Relations and Functions

4 Writing Linear Equations This lesson focuses on the slope and -intercept of a linear equation. In the slope-intercept form of a linear equation, m b, m represents the slope of the line and b is the -intercept. Students use two forms of a linear equation, the slope-intercept form and the point-slope form, to write an equation given two points, given a point and the slope, or given a point and the equation of a parallel or a perpendicular line. Modeling Real-World Data: Using Scatter Plots This lesson eplores equations that approimate the relation between domain values and range values, etending the idea of using an algebraic equation to represent a set of points in a plane. Starting with a scatter plot of data, students mentall picture a line through the data. After selecting two points on that line, the calculate the slope and -intercept of that line. The equation, called a line of fit or a prediction equation, ma be used to calculate the value of one variable given a value of the other. Activities in this lesson require three steps: given a set of ordered pairs, students identif a line that represents a set of ordered pairs; then the select two ordered pairs that lie on the line; and finall the calculate the slope and -intercept for that line. Special Functions In this lesson, students eplore special functions. The identit and constant functions are special linear functions. The graph of a step function is a series of line segments. An absolute value function has a V-shaped graph made up of portions of two lines. A piecewise function is a function written using two or more algebraic epressions. Graphing Inequalities In this lesson the graph of an equation is seen as the boundar between two regions of the coordinate plane. An inequalit is a description of one of the two regions, and whether the boundar is part of that region depends on the inequalit smbol that is used. Students eplore how inequalities, including absolute value inequalities, are modeled b points in the coordinate plane, and vice versa. Additional mathematical information and teaching notes are available in Glencoe s Algebra Ke Concepts: Mathematical Background and Teaching Notes, which is available at The lessons appropriate for this chapter are as follows. Linear Relations and Functions (Lesson ) Graphing Linear Equations (Lessons 6 and ) Slope (Lesson 7) Writing Linear Equations in Point-Slope and Standard Forms (Lesson 8) Writing Linear Equations in Slope-Intercept Form (Lesson ) Integration: Geometr/Parallel and Perpendicular Lines (Lesson ) Statistics: Scatter Plots and Best-Line Fits (Lesson 9) Graphing Inequalities in Two Variables (Lesson 7) Chapter Linear Relations and Functions D

5 and Assessment Tpe Student Edition Teacher Resources Technolog/Internet ASSESSMENT INTERVENTIN ngoing Prerequisite Skills, pp., 6, 67, 7, 8, 86, 9 Practice Quiz, p. 7 Practice Quiz, p. 9 Mied Review Error Analsis Standardized Test Practice pen-ended Assessment Chapter Assessment pp. 6, 67, 7, 8, 86, 9, 99 Find the Error, pp. 6, 7 pp. 6, 67, 7, 76, 78, 8, 86, 9, 99,, 6 7 Writing in Math, pp. 6, 67, 7, 8, 86, 9, 99 pen Ended, pp. 6, 6, 7, 78, 8, 9, 98 Stud Guide, pp. Practice Test, p. -Minute Check Transparencies Quizzes, CRM pp. Mid-Chapter Test, CRM p. Stud Guide and Intervention, CRM pp. 7 8, 6 6, 69 7, 7 76, 8 8, 87 88, 9 9 Cumulative Review, CRM p. 6 Find the Error, TWE pp. 6, 7 Unlocking Misconceptions, TWE p. 8 Tips for New Teachers, TWE pp. 6, 7, 9 TWE p. 76 Standardized Test Practice, CRM pp. 7 8 Modeling: TWE pp. 67, 7, 9 Speaking: TWE pp. 6, 98 Writing: TWE pp. 8, 86 pen-ended Assessment, CRM p. Multiple-Choice Tests (Forms, A, B), CRM pp. 99 Free-Response Tests (Forms C, D, ), CRM pp. Vocabular Test/Review, CRM p. AlgePASS: Tutorial Plus Standardized Test Practice CD-RM standardized_test TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes vocabular_review Ke to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters Additional Intervention Resources The Princeton Review s Cracking the SAT & PSAT The Princeton Review s Cracking the ACT ALEKS TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment fleibilit: Worksheet Builder to make worksheet and tests Student Module to take tests on screen (optional) Management Sstem to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests. E Chapter Linear Relations and Functions

6 Intervention Technolog AlgePASS: Tutorial Plus CD-RM offers a complete, self-paced algebra curriculum. Algebra Lesson AlgePASS Lesson - Graphing Linear Equations on the Coordinate Plane -7 Graphing Linear Inequalities on the Coordinate Plane ALEKS is an online mathematics learning sstem that adapts assessment and tutoring to the student s needs. Subscribe at Intervention at Home Log on for student stud help. For each lesson in the Student Edition, there are Etra Eamples and Self-Check Quizzes. For chapter review, there is vocabular review, test practice, and standardized test practice For more information on Intervention and Assessment, see pp. T8 T. Reading and Writing in Mathematics Glencoe Algebra provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition Foldables Stud rganizer, p. Concept Check questions require students to verbalize and write about what the have learned in the lesson. (pp. 6, 6, 7, 78, 8, 9, 98, ) Writing in Math questions in ever lesson, pp. 6, 67, 7, 8, 86, 9, 99 Reading Stud Tip, pp. 6, 9, 7, 8 WebQuest, p. 8 Teacher Wraparound Edition Foldables Stud rganizer, pp., Stud Notebook suggestions, pp. 6, 6, 7, 78, 8, 9, 97 Modeling activities, pp. 67, 7, 9 Speaking activities, pp. 6, 98 Writing activities, pp. 8, 86 Differentiated Instruction, (Verbal/Linguistic), p. 9 ELL Resources, pp., 6, 66, 7, 79, 8, 9, 9, 99, Additional Resources Vocabular Builder worksheets require students to define and give eamples for ke vocabular terms as the progress through the chapter. (Chapter Resource Masters, pp. vii-viii) Reading to Learn Mathematics master for each lesson (Chapter Resource Masters, pp. 6, 67, 7, 79, 8, 9, 97) Vocabular PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabular lists that ou can customize. Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. Reading and Writing in the Mathematics Classroom WebQuest and Project Resources For more information on Reading and Writing in Mathematics, see pp. T6 T7. Chapter Linear Relations and Functions F

7 Notes Have students read over the list of objectives and make a list of an words with which the are not familiar. Point out to students that this is onl one of man reasons wh each objective is important. thers are provided in the introduction to each lesson. Linear Relations and Functions Lesson - Analze relations and functions. Lessons - and - Identif, graph, and write linear equations. Lesson - Find the slope of a line. Lesson - Draw scatter plots and find prediction equations. Lessons -6 and -7 Graph special functions, linear inequalities, and absolute value inequalities. Ke Vocabular linear equation (p. 6) linear function (p. 6) slope (p. 68) slope-intercept form (p. 7) point-slope form (p. 76) Linear equations can be used to model relationships between man real-world quantities. ne of the most common uses of a linear model is to make predictions. Most hot springs are the result of groundwater passing through or near recentl formed, hot, igneous rocks. Iceland, Yellowstone Park in the United States, and North Island of New Zealand are noted for their hot springs. You will use a linear equation to find the temperature of underground rocks in Lesson -. NCTM Local Lesson Standards bjectives -,, 7, 9, -,,, 6, 8, 9 -,,, 6, 7, 8, 9, -,, 6, 8, 9, -,,, 6, 8, 9, -,,,, 6, 9, Follow-Up -6,,, 6, 8, 9, -7,, 6, 8, 9, Ke to NCTM Standards: =Number & perations, =Algebra, =Geometr, =Measurement, =Data Analsis & Probabilit, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, =Representation Chapter Linear Relations and Functions Vocabular Builder ELL The Ke Vocabular list introduces students to some of the main vocabular terms included in this chapter. For a more thorough vocabular list with pronunciations of new words, give students the Vocabular Builder worksheets found on pages vii and viii of the Chapter Resource Masters. Encourage them to complete the definition of each term as the progress through the chapter. You ma suggest that the add these sheets to their stud notebooks for future reference when studing for the Chapter test. Chapter Linear Relations and Functions

8 Prerequisite Skills To be successful in this chapter, ou ll need to master these skills and be able to appl them in problem-solving situations. Review these skills before beginning Chapter. For Lesson - Write the ordered pair for each point.. A (, ). B (, ). C (, ). D (, ). E (, ) 6. F (, ) Identif Points on a Coordinate Plane A C E B D F This section provides a review of the basic concepts needed before beginning Chapter. Page references are included for additional student help. Prerequisite Skills in the Getting Read for the Net Lesson section at the end of each eercise set review a skill needed in the net lesson. For Lesson - Evaluate Epressions Evaluate each epression if a, b, c, and d. (For review, see Lesson -.) 7. c d 8. c b 9. a a 9. b b 7 8. a b a c. c d b c For Lesson - Simplif Epressions Simplif each epression. (For review, see Lesson -.). ( ). ( ). [ ( )] 6 6. [ ( )] 8 7. [ ( )] 8. [ ( 6)] For Prerequisite Lesson Skill - Solving Equations (p. 6) - Solving Equations (p. 7) - Finding a Median (p. 8) -6 Absolute Value (p. 86) -7 Inequalities (p. 9) For Lessons -6 and -7 Evaluate Epressions with Absolute Value Evaluate each epression if =, =, and z =.. (For review, see Lesson -.) z 9. z.. z. Make this Foldable to help ou organize information about relations and functions. Begin with two sheets of grid paper. Fold Cut and Label Fold in half along the width and staple along the fold. Cut the top three sheets and label as shown. Graphing Linear Relations Graphing Linear Functions Reading and Writing As ou read and stud the chapter, write notes, eamples, and graphs under the tabs. Chapter Linear Relations and Functions For more information about Foldables, see Teaching Mathematics with Foldables. TM rganization of Data: Annotating As students read and work their wa through the chapter, have them make annotations under the appropriate tabs of their Foldable. Eplain to them that annotations are usuall notes taken in the margins of books, which we own, to organize the tet for review or studing. Annotations often include questions that arise, reader comments and reactions, short summaries, steps or data numbered b the reader, and ke points highlighted or underlined. Chapter Linear Relations and Functions

9 Lesson Notes Relations and Functions Focus -Minute Check Transparenc - Use as a quiz or review of Chapter. Mathematical Background notes are available for this lesson on p. C. Building on Prior Knowledge In Chapter, students solved equations and inequalities. In this lesson, students relate equations to functions and relations, as well as to their graphs. do relations and functions appl to biolog? Ask students: What is the difference between average lifetime and maimum lifetime? The average lifetime is a representative number of ears for an animal of that tpe, while the maimum lifetime is the greatest age ever attained b an animal of that tpe. Wh can ou be sure that the second number in the ordered pairs for this data is alwas greater than or equal to the first? For each animal, the maimum age will alwas equal or eceed the average age. Vocabular ordered pair Cartesian coordinate plane quadrant relation domain range function mapping one-to-one function vertical line test independent variable dependent variable functional notation Stud Tip Reading Math An -coordinate is sometimes called an abscissa, and a -coordinate is sometimes called an ordinate. Analze and graph relations. Find functional values. do relations and functions appl to biolog? The table shows the average lifetime and maimum lifetime for some animals. The data can also be represented as ordered pairs. The ordered pairs for the data are (, 8), (, ), (8, ), (, ), and (, ). The first number in each ordered pair is the average lifetime, and the second number is the maimum lifetime. (, 8) average maimum lifetime lifetime GRAPH RELATINS You can graph the ordered pairs above b creating a coordinate sstem with two aes. Each point represents one of the ordered pairs above. Remember that each point in the coordinate plane can be named b eactl one ordered pair and that ever ordered pair names eactl one point in the coordinate plane. The graph of the animal lifetime data lies in onl one part of the Cartesian coordinate plane the part with all positive numbers. The Cartesian coordinate plane is composed of the -ais (horizontal) and the -ais (vertical), which meet at the origin (, ) and divide the plane into four quadrants. The points on the two aes do not lie in an quadrant. In general, an ordered pair in the coordinate plane can be written in the form (, ). A relation is a set of ordered pairs, such as the one for the longevit of animals. The domain of a relation is the set of all first coordinates (-coordinates) from the ordered pairs, and the range is the set of all second coordinates (-coordinates) from the ordered pairs. The graph of a relation is the set of points in the coordinate plane corresponding to the ordered pairs in the relation. Average Maimum Animal Lifetime Lifetime (ears) (ears) Cat 8 Cow Deer 8 Dog Horse Source: The World Almanac Maimum Lifetime Animal Lifetimes 6 Average Lifetime The vertical ais represents the maimum lifetime. The horizontal ais represents the average lifetime. Quadrant II -ais Quadrant I -coordinate (, ) origin -coordinate Quadrant III -ais Quadrant IV Assume that each square on a graph represents unit unless otherwise labeled. 6 Chapter Linear Relations and Functions Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp. 7 8 Skills Practice, p. 9 Practice, p. 6 Reading to Learn Mathematics, p. 6 Enrichment, p. 6 Transparencies -Minute Check Transparenc - Answer Ke Transparencies Technolog Interactive Chalkboard

10 A function is a special tpe of relation in which each element of the domain is paired with eactl one element of the range. A mapping shows how each member of the domain is paired with each member of the range. The first two relations shown below are functions. The third relation is not a function because the in the domain is paired with both and 6 in the range. A function like the first one below, where each element of the range is paired with eactl one element of the domain, is called a one-to-one function. Domain Range Functions {(, ), (, ), (, )} {(, ), (, ), (, )} {(, 6), (, ), (, ), (, 6)} Domain Range Domain Range Domain {(, ), (, ), (, )} Range 6 Teach GRAPH RELATINS In-Class Eample Power Point State the domain and range of the relation shown in the graph. Is the relation a function? (, ) (, ) (, ) (, ) one-to-one function function, not one-to-one not a function (, ) Eample Domain and Range State the domain and range of the relation shown in the graph. Is the relation a function? The relation is {(, ), (, ), (, ), (, ), (, )}. The domain is {,,,, }. The range is {,,, }. Each member of the domain is paired with eactl one member of the range, so this relation is a function. (, ) (, ) (, ) You can use the vertical line test to determine whether a relation is a function. Words Models If no vertical line intersects a graph in more than one point, the graph represents a function. (, ) (, ) Vertical Line Test If some vertical line intersects a graph in two or more points, the graph does not represent a function. The domain is {,,,, }. The range is {,,,, }. Each member of the domain is paired with eactl one member of the range, so this relation is a function. Teaching Tip Ask students to eplain wh the set of ordered pairs (9, ), (9, ), (, ), (, ) is not a function. Sample answer: For at least one of the values, there are two different values. In Eample, there is no vertical line that contains more than one of the points. Therefore, the relation is a function. Lesson - Relations and Functions 7 Interactive Chalkboard PowerPoint Presentations This CD-RM is a customizable Microsoft PowerPoint presentation that includes: Step-b-step, dnamic solutions of each In-Class Eample from the Teacher Wraparound Edition Additional, Your Turn eercises for each eample The -Minute Check Transparencies Hot links to Glencoe nline Stud Tools Lesson - Relations and Functions 7

11 In-Class Eample In-Class Eample Power Point TRANSPRTATIN The table shows the average fuel efficienc in miles per gallon for light trucks for several ears. Graph this information and determine whether it represents a function. Year Fuel Efficienc (mi/gal) Source: U.S. Environmental Protection Agenc Mileage (mi/gal) Fuel Efficienc of Light Trucks Year Yes, this relation is a function. EQUATINS F FUNCTINS AND RELATINS Power Point a. Graph the relation represented b. Stud Tip Vertical Line Test You can use a pencil to represent a vertical line. Slowl move the pencil to the right across the graph to see if it intersects the graph at more than one point. Eample Vertical Line Test GEGRAPHY The table shows the population of the state of Indiana over the last several decades. Graph this information and determine whether it represents a function. Population (millions) 7 6 Population of Indiana Year Use the vertical line test. Notice that no vertical line can be drawn that contains more than one of the data points. Therefore, this relation is a function. Notice also that each ear is paired with onl one population value. EQUATINS F FUNCTINS AND RELATINS Relations and functions can also be represented b equations. The solutions of an equation in and are the set of ordered pairs (, ) that make the equation true. Consider the equation 6. Since can be an real number, the domain has an infinite number of elements. To determine whether an equation represents a function, it is often simplest to look at the graph of the relation. Eample Graph Is a Line a. Graph the relation represented b. Make a table of values to find ordered pairs that satisf the equation. Choose values for and find the corresponding values for. Then graph the ordered pairs. (, ) b. Find the domain and range. Since can be an real number, there is an infinite number of ordered pairs that can be graphed. All of them lie on the line shown. Notice that ever real number is the -coordinate of some point on the line. Also, ever real number is the -coordinate of some point on the line. So the domain and range are both all real numbers. (, ) Year (, ) (, ) Population (millions) Source: U.S. Census Bureau (, ) (, ) c. Determine whether the relation is a function. This graph passes the vertical line test. For each value, there is eactl one value, so the equation represents a function. (, ) 8 Chapter Linear Relations and Functions b. Find the domain and range. The domain and range are both all real numbers. c. Determine whether the relation is a function. Yes, the equation represents a function. Unlocking Misconceptions Relations and Functions Some students ma not realize that all functions are relations. Eplain that a function is a special tpe of relation, analogous to how a square is a special tpe of rectangle. Vertical Line Test Make sure students understand that when two points on the graph of a relation are intersected b a vertical line, this means those two points have the same value but different values. That is, one domain value is paired with more than one range value. 8 Chapter Linear Relations and Functions

12 Eample Graph Is a Curve a. Graph the relation represented b. Make a table. In this case, it is easier to choose values and then find the corresponding values for. Then sketch the graph, connecting the points with a smooth curve. (, ) (, ) (, ) (, ) (, ) In-Class Eamples Power Point a. Graph the relation represented b. (, ) (, ) (, ) (, ) (, ) Stud Tip Reading Math Suppose ou have a job that pas b the hour. Since our pa depends on the number of hours ou work, ou might sa that our pa is a function of the number of hours ou work. TEACHING TIP Your pa is the dependent variable, and the number of hours ou work is the independent variable. b. Find the domain and range. Ever real number is the -coordinate of some point on the graph, so the range is all real numbers. But, onl real numbers greater than or equal to are -coordinates of points on the graph. So the domain is { }. c. Determine whether the relation is a function. You can see from the table and the vertical line test that there are two values for each value ecept. Therefore, the equation does not represent a function. When an equation represents a function, the variable, usuall, whose values make up the domain is called the independent variable. The other variable, usuall, is called the dependent variable because its values depend on. Equations that represent functions are often written in functional notation. The equation can be written as f(). The smbol f() replaces the and is read f of. The f is just the name of the function. It is not a variable that is multiplied b. Suppose ou want to find the value in the range that corresponds to the element in the domain of the function. This is written as f() and is read f of. The value f() is found b substituting for each in the equation. Therefore, f() = () + or 9. Letters other than f can be used to represent a function. For eample, g(). Eample Evaluate a Function Given f() and g().., find each value. a. f( ) f() riginal function f( ) ( ) 9 or Substitute. Simplif. b. g(.8) g().. g(.8).(.8) (.8) c. f(z) f() f(z) (z) riginal function Substitute. riginal function Estimate: g().() (). or 7 Multipl. Compare with the estimate. b. Find the domain and range. The domain is { } and the range is all real numbers. c. Determine whether the relation is a function. No, the equation does not represent a function. Teaching Tip Some students ma recognize this curve as a parabola. It is the same shape as the more familiar graph of the equation. Given f() and h()..7, find each value. a. f( ) b. h(.6) 6.7 c. f(t) 8t 9z (ab) = a b Lesson - Relations and Functions 9 Differentiated Instruction Auditor/Musical Encourage students to relate to the mathematics in this lesson b asking those who are familiar with reading musical notation to eplain to the class how graphing points on a coordinate plane compares to writing musical notes on a staff. Lesson - Relations and Functions 9

13 Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. draw their own diagrams, similar to those in the Concept Summar on p. 7. make a sketch illustrating how to use the vertical line test, both for a function and a non-function. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Graph Relations: 7 8,, Equations of Functions and Relations: 9, 6, 6 dd/even Assignments Eercises 7 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 7 odd, 7, 7 odd, 8, 6 7 Average: 7 odd,, 7 odd, 8, 6 7 (optional: 9 6) Advanced: 8 even,, 6 even, 69 (optional: 7 7) FIND THE ERRR Suggest that students rewrite the original function b substituting the epression (a) for each variable before the begin simplifing. Concept Check. Sample answer: {(, ), (, ), (, ), (, )}. See pp. 7A 7H. Guided Practice GUIDED PRACTICE KEY Eercises Eamples 8, 9, 6 Application. D {7, 7, 88}, R {9, 97,, } 6. See margin. indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples 7 8, 9,, 6, 6 Etra Practice See page 8. 6 Chapter Linear Relations and Functions Answers. PEN ENDED Write a relation of four ordered pairs that is not a function.. Cop the graph at the right. Then draw a vertical line that shows that the graph does not represent a function.. FIND THE ERRR Teisha and Moll are finding g(a) for the function g(). Teisha Moll g(a) = (a +a ) =a +a g(a) = (a) + a = a + a Who is correct? Eplain our reasoning. See margin. Determine whether each relation is a function. Write es or no.. D R. 6. (, ) 6 es es no Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function. 7. pp. 7A 7H. 7. {(7, 8), (7, ), (7, ), (7, )} 8. {(6,.), (,.), (,.)} $ $ $ $. Moll; to find g(a), replace with a. Teisha found g(a), not g(a).. {(88, 97), (7, ), (88, 9), (7, )}. See graph at right. 6. No; the domain value 88 is paired with two range values. es es (, ) (, ). Find f() if f().. Find h( ) if h(). 7 WEATHER For Eercises 6, use the table of record high temperatures ( F) for Januar and Jul.. Identif the domain and range. Assume that the Januar temperatures are the domain.. Write a relation of ordered pairs for the data.. Graph the relation. 6. Is this relation a function? Eplain. (, ) Cit Jan. Jul Los Angeles Sacramento 7 San Diego 88 9 San Francisco 7 Source: U.S. National ceanic and Atmospheric Administration Determine whether each relation is a function. Write es or no. 7. D R 8. D R 9. es no Record High Temperatures Jul Januar no no 6 Chapter Linear Relations and Functions

14 . D {,, }, R {,, }; es. D {,, 6}, R {}; es. D {, }, R {, 7, 8}; no 6. D {,,, 6}, R {,,, 6}; es 7. D {.6,,., }, R {,.,, 8}; es Sports The major league record for runs batted in (RBIs) is 9 b Hack Wilson. Source: 8. D {.,, }, R {, }; no 9. D all reals, R all reals; es. D all reals, R all reals, es. D all reals, R all reals, es. D all reals, R all reals; es Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.. See pp. 7A 7H for. {(, ), (, ), (, )}. {(, ), (6, ), (, )} graphs.. {(, ), (, 7), (, 8)} 6. {(, ), (, ), (6, ), (, 6)} 7. {(,.), (, ), (., ), (.6, 8)} 8. {(., ), (, ), (, ), (, )} D all reals, R { }; es D { }, R all reals; no SPRTS For Eercises 7, use the table that shows the leading home run and runs batted in totals in the American League for Year HR RBI Source: The World Almanac. Make a graph of the data with home runs on the horizontal ais and runs batted in on the vertical ais. See pp. 7A 7H. 6. Identif the domain and range. 7. Does the graph represent a function? Eplain our reasoning. See margin. 6. D {7, 8,, 6}, R {, 7, 8, 7, 6} FINANCE For Eercises 8, use the table that shows a Year Price compan s stock price in recent ears. 8,. See margin. 997 $9 8. Write a relation to represent the data. 998 $ 9. Graph the relation. See pp. 7A 7H. 999 $8. Identif the domain and range. $. Is the relation a function? Eplain our reasoning. $6 Yes; each domain value is paired with onl one range value. $ GVERNMENT For Eercises, use the table below that shows the number of members of the U.S. House of Representatives with or more consecutive ears of service in Congress from 987 to 999. Source: Congressional Director. Write a relation to represent the data. See margin.. Graph the relation. See pp.7a 7H.. Identif the domain and range. See pp. 7A 7H.. Is the relation a function? If so, is it a one-to-one function? Eplain. Yes; no; see pp. 7A 7H for eplanation. Find each value if f() and g(). 6. f( ) 7. g() 6 8. g 9 9. f. f(a) a. g(n) n n. Find the value of f() + when.. What is g() if g()? Year Representatives 9 6 Lesson - Relations and Functions 6 Stud Stud Guide and and Intervention Intervention, p. 7 (shown) and p. 8 - NAME DATE PERID Relations and Functions Graph Relations A relation can be represented as a set of ordered pairs or as an equation; the relation is then the set of all ordered pairs (, ) that make the equation true. The domain of a relation is the set of all first coordinates of the ordered pairs, and the range is the set of all second coordinates. A function is a relation in which each element of the domain is paired with eactl one element of the range. You can tell if a relation is a function b graphing, then using the vertical line test. If a vertical line intersects the graph at more than one point, the relation is not a function. Eample Graph the equation and find the domain and range. Does the equation represent a function? Make a table of values to find ordered pairs that satisf the equation. Then graph the ordered pairs. The domain and range are both all real numbers. The graph passes the vertical line test, so it is function. Eercises Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.. {(, ), (, ),. {(, ), (, ),. {(, ), (, ), (, ), (, )} (, ), (, )} (, ), (, )} D {,,, }, D {,, }, D {,,, }, R {, }; es R {,,, }; no R {,,, }; es.. 6. D all reals, D all reals, D all reals, R { }; es R all reals; es R all reals; es - NAME 7 DATE PERID Practice (Average) Skills Practice, p. 9 and Practice, p. 6 (shown) Determine whether each relation is a function. Write es or no.. D R no. D R es. es. no 8 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.. {(, ), (, ), (, ), (, )} 6. (, ) Relations and Functions (, ) (, ) (, ) D {,,, }, R {,, }; es Find each value if f() and g(). D all reals, R all reals; es 7. f() 8. f( ) 9. g. f( ) undefined. g( 6). f(m ) m. MUSIC The ordered pairs (, 6), (, 6), (, ), (, ), and (, 8) represent the cost of buing various numbers of CDs through a music club. Identif the domain and range of the relation. Is the relation a function? D {,,,, }, R {6,, 8}; es. CMPUTING If a computer can do one calculation in. second, then the function T(n).n gives the time required for the computer to do n Reading to Learn Mathematics, p. 6 - NAME 6 DATE PERID Reading to Learn Mathematics Relations and Functions Pre-Activit How do relations and functions appl to biolog? Read the introduction to Lesson - at the top of page 6 in our tetbook. Refer to the table. What does the ordered pair (8, ) tell ou? For a deer, the average longevit is 8 ears and the maimum longevit is ears. Suppose that this table is etended to include more animals. Is it possible to have an ordered pair for the data in which the first number is larger than the second? Sample answer: No, the maimum longevit must alwas be greater than the average longevit. Reading the Lesson. a. Eplain the difference between a relation and a function. Sample answer: A relation is an set of ordered pairs. A function is a special kind of relation in which each element of the domain is paired with eactl one element in the range. b. Eplain the difference between domain and range. Sample answer: The domain of a relation is the set of all first coordinates of the ordered pairs. The range is the set of all second coordinates. ELL Lesson - Lesson - Answers 7. No; the domain value 6 is paired with two different range values. 8. {(997, 9), (998, ), (999, 8), (, ), (, 6), (, )}. D {997, 998, 999,,, }, R {9,, 8,,, 6}. {(987, ), (989, ), (99, ), (99, ), (99, 9), (997, 6), (999, )} - Enrichment, p. 6 Mappings NAME DATE PERID There are three special was in which one set can be mapped to another. A set can be mapped into another set, onto another set, or can have a one-to-one correspondence with another set. A mapping from set A to set B where ever element of A is mapped to one or more Into mapping elements of set B, but never to an element not in B. A mapping from set A to set B where each element of set B has at least one element of nto mapping set A mapped to it. ne-to-one A mapping from set A onto set B where each element of set A is mapped to eactl one correspondence element of set B and different elements of A are never mapped to the same element of B. State whether each set is mapped into the second set, onto the second set, or has a one-to-one correspondence with the second set. 7 a a. Write the domain and range of the relation shown in the graph. (, ) (, ) (, ) (, ) (, ) (, ) D: {,,,, }; R: {,,,,, } b. Is this relation a function? Eplain. Sample answer: No, it is not a function because one of the elements of the domain,, is paired with two elements of the range. Helping You Remember. Look up the words dependent and independent in a dictionar. How can the meaning of these words help ou distinguish between independent and dependent variables in a function? Sample answer: The variable whose values depend on, or are determined b, the values of the other variable is the dependent variable. Lesson - Relations and Functions 6

15 Assess pen-ended Assessment Speaking Ask students to discuss function notation, including what the ma find confusing and how to read it out loud. Make sure the understand and can correctl eplain the difference between f times and f of. Intervention This lesson has a number of vocabular words that ma be new or challenging for some students. Make sure that all students are comfortable with the mathematical language of this lesson before the go on. New Getting Read for Lesson - PREREQUISITE SKILL Lesson - presents identifing and graphing linear equations in two variables. The skills needed to write linear equations in standard form are similar to solving equations in one variable. Eercises 7 7 should be used to determine our students familiarit with solving single-variable equations. Answer 6. Relations and functions can be used to represent biological data. Answers should include the following. If the data are written as ordered pairs, then those ordered pairs are a relation. The maimum lifetime of an animal is not a function of its average lifetime. Standardized Test Practice Etending the Lesson Maintain Your Skills Mied Review Getting Read for the Net Lesson 6 Chapter Linear Relations and Functions. HBBIES Chaz has a collection of CDs. After he gets a part-time job, he decides to bu more CDs ever time he goes to the music store. The function C(t) t counts the number of CDs, C(t), he has after t trips to the music store. How man CDs will he have after he has been to the music store 8 times? 9. CRITICAL THINKING If f(a ) a 7, find f(). f() 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How do relations and functions appl to biolog? Include the following in our answer: an eplanation of how a relation can be used to represent data, and a sentence that includes the words average lifetime, maimum lifetime, and function. 7. If f(), then f() B A. B. C. D. 8. If g() =, then g( ) C A. B. C. D. A function whose graph consists of disconnected points is called a discrete function. A function whose graph ou can draw without lifting our pencil is called a continuous function. Determine whether each function is discrete or continuous. 9. f() discrete 6. f() continuous 6. {(, ), (, ), (, )} discrete 6. continuous Solve each inequalit. (Lessons - and -6) 6. { 8 6} 6. {m m 6} m 6.. {.} SHPPING For Eercises 66 and 67, use the following information. Javier had $. when he went to the mall. His friend Sall had $.67. Javier wanted to bu a shirt for $7.89. (Lesson -) 66. How much mone did he have to borrow from Sall to bu the shirt? $ How much mone did that leave Sall? $9.8 Simplif each epression. (Lessons - and -) 68. ( ) 69. (a 6b) 8(a b) a b PREREQUISITE SKILL Solve each equation. Check our solution. (To review solving equations, see Lesson -.) Chapter Linear Relations and Functions

16 Linear Equations Lesson Notes Vocabular linear equation linear function standard form -intercept -intercept Identif linear equations and functions. Write linear equations in standard form and graph them. do linear equations relate to time spent studing? Lolita has hours after dinner to stud and do homework. She has brought home math and chemistr. If she spends hours on math and hours on chemistr, a portion of the graph of the equation can be used to relate how much time she spends on each. Focus -Minute Check Transparenc - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on p. C. TEACHING TIP When variables other than and are used, the letter coming first in the alphabet usuall represents the domain variable or horizontal coordinate. IDENTIFY LINEAR EQUATINS AND FUNCTINS An equation such as is called a linear equation. A linear equation has no operations other than addition, subtraction, and multiplication of a variable b a constant. The variables ma not be multiplied together or appear in a denominator. A linear equation does not contain variables with eponents other than. The graph of a linear equation is alwas a line. Linear equations Not linear equations 7 7a b 8 9 6s t A linear function is a function whose ordered pairs satisf a linear equation. An linear function can be written in the form f() m b, where m and b are real numbers. Eample Identif Linear Functions State whether each function is a linear function. Eplain. a. f() This is a linear function because it can be written as f(). m, b b. g() This is not a linear function because has an eponent other than. c. h(, ) This is not a linear function because the two variables are multiplied together. Building on Prior Knowledge In Lesson -, students graphed functions and equations b using a table of points. In this lesson, the generalize these skills to write equations in standard form before graphing them. do linear equations relate to time spent studing? Ask students: What does a value of zero for mean in this situation? Lolita spends hours studing math. What does a negative value for or for mean in this situation? No meaning; she cannot spend negative hours studing. Chapter Resource Masters Stud Guide and Intervention, pp. 6 6 Skills Practice, p. 6 Practice, p. 66 Reading to Learn Mathematics, p. 67 Enrichment, p. 68 Assessment, p. Lesson - Linear Equations 6 Workbook and Reproducible Masters Resource Manager Transparencies -Minute Check Transparenc - Real-World Transparenc Answer Ke Transparencies Technolog AlgePASS: Tutorial Plus, Lesson Interactive Chalkboard Lesson - Lesson Title 6

17 Teach IDENTIFY LINEAR EQUATINS AND FUNCTINS In-Class Eamples Power Point State whether each function is a linear function. Eplain. a. g() es; m ; b b. p() No; has an eponent other than. c. t() 7 es; m 7; b METERLGY The linear function f(c).8c can be used to find the number of degrees Fahrenheit, f, that are equivalent to a given number of degrees Celsius, C. a. n the Celsius scale, normal bod temperature is 7 C. What is normal bod temperature in degrees Fahrenheit? 98.6 F b. There are Celsius degrees between the freezing and boiling points of water and 8 Fahrenheit degrees between these two points. How man Fahrenheit degrees equal Celsius degree?.8 F C Militar To avoid decompression sickness, it is recommended that divers ascend no faster than feet per minute. Source: Eample STANDARD FRM An linear equation can be written in standard form, A B C, where A, B, and C are integers whose greatest common factor is. Standard Form of a Linear Equation The standard form of a linear equation is A B C, where A, A and B are not both zero. Eample Standard Form Write each equation in standard form. Identif A, B, and C. a. riginal equation So, A, B, and C. b. Evaluate a Linear Function MILITARY In August, the Russian submarine Kursk sank to a depth of feet in the Barents Sea. The linear function P(d) 6.d 7 can be used to find the pressure (lb/ft ) at a depth of d feet below the surface of the water. a. Find the pressure at a depth of feet. P(d) 6.d 7 riginal function P() 6.() 7,99 Substitute. Simplif. The pressure at a depth of feet is about, lb/ft. b. The term 7 in the function represents the atmospheric pressure at the surface of the water. How man times as great is the pressure at a depth of feet as the pressure at the surface? Divide the pressure feet below the surface b the pressure at the surface.,99. Use a calculator. 7 The pressure at that depth is more than times as great as the pressure at the surface. Add to each side. riginal equation Subtract from each side. STANDARD FRM In-Class Eample Power Point Write each equation in standard form. Identif A, B, and C. a. 9 9; A, B, C 9 b. 6 ; A, B 6, C c. 8 6 ; A, B, C 6 Chapter Linear Relations and Functions Multipl each side b so that the coefficients are integers and A. So, A, B, and C. c Reading Tip Make sure that students understand the difference between the - and -intercepts. Some students ma use the word intersect instead of the correct term intercept. Help them see that an intercept is the nonzero coordinate of the point where the graph intersects either ais. riginal equation Add 9 to each side. Divide each side b so that the coefficients have a GCF of. So, A, B, and C. Answer (page 6). The function can be written as f(), so it is of the form f() m b, where m and b. 6 Chapter Linear Relations and Functions

18 Stud Tip Vertical and Horizontal Lines An equation of the form C represents a vertical line, which has onl an -intercept. C represents a horizontal line, which has onl a -intercept. TEACHING TIP Some students ma find it helpful to remember that if there is no in the equation, the graph cannot cross the -ais. Similarl, if there is no in the equation, the graph cannot cross the -ais. Concept Check In Lesson -, ou graphed an equation or function b making a table of values, graphing enough ordered pairs to see a pattern, and connecting the points with a line or smooth curve. Since two points determine a line, there are quicker was to graph a linear equation or function. ne wa is to find the points at which the graph intersects each ais and connect them with a line. The -coordinate of the point at which a graph crosses the -ais is called the -intercept. Likewise, the -coordinate of the point at which it crosses the -ais is the -intercept. Eample Use Intercepts to Graph a Line Find the -intercept and the -intercept of the graph of. Then graph the equation. The -intercept is the value of when. riginal equation () Substitute for. Subtract from each side. Divide each side b. The -intercept is. The graph crosses the -ais at (, ). Likewise, the -intercept is the value of when. riginal equation () Substitute for. Subtract from each side. Divide each side b. The -intercept is. The graph crosses the -ais at (, ). Use these ordered pairs to graph the equation.. Eplain wh f() is a linear function. See margin.. Name the - and -intercepts of the graph shown at the right.,. PEN ENDED Write an equation of a line with an -intercept of. Sample answer: (, ) (, ) In-Class Eample Practice/Appl Stud Notebook Power Point Find the -intercept and the -intercept of the graph of. Then graph the equation. -intercept: ; -intercept: (, ) (, ) Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. include an other items(s) that students find helpful in mastering the skills in the lesson. Guided Practice GUIDED PRACTICE KEY Eercises Eamples, 6 8 9, State whether each equation or function is linear. Write es or no. If no, eplain our reasoning.. No, the variables have an. h(). es eponent other than. Write each equation in standard form. Identif A, B, and C ;,, ;,, ;,, Find the -intercept and the -intercept of the graph of each equation. Then graph the equation. 9. See pp. 7A 7H for graphs. 9.,.,. 6,. 8, Differentiated Instruction Lesson - Linear Equations 6 Visual/Spatial Encourage students to relate the intercepts and the graph of the equation to the standard form of the equation, A B C. Point A out that the ratio is equivalent to the ratio of the graph s intercepts: -intercept B. These ratios can be visualized on the graph b counting grid -intercept squares verticall and horizontall from one intercept to the other. Recognition of the ratios leads to the discussion of slope in Lesson -. About the Eercises rganization b bjective Identif Linear Equations and Functions: 6 Standard Form: 7 dd/even Assignments Eercises and 7 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: odd,, 6, 7 odd, 9 7 odd,, 6 78 Average: odd,, 6, 7 9 odd,, 6 78 Advanced: 6 even, 8 even, 7 (optional: 7 78) Lesson - Linear Equations 6

19 Stud Stud Guide and and Intervention Intervention, p. 6 (shown) and p. 6 - NAME DATE PERID Linear Equations Identif Linear Equations and Functions A linear equation has no operations other than addition, subtraction, and multiplication of a variable b a constant. The variables ma not be multiplied together or appear in a denominator. A linear equation does not contain variables with eponents other than. The graph of a linear equation is a line. A linear function is a function whose ordered pairs satisf a linear equation. An linear function can be written in the form f() m b, where m and b are real numbers. If an equation is linear, ou need onl two points that satisf the equation in order to graph the equation. ne wa is to find the -intercept and the -intercept and connect these two points with a line. Eample Is f(). a Eample Find the -intercept and the linear function? Eplain. -intercept of the graph of. Yes; it is a linear function because it can Then graph the equation. be written in the form The -intercept is the value of when. f().. riginal equation () Substitute for. Eample Is a Simplif. linear function? Eplain. So the -intercept is. No; it is not a linear function because Similarl, the the variables and are multiplied -intercept is. together in the middle term. Eercises State whether each equation or function is linear. Write es or no. If no, eplain es. 9 No; the. f() es variable appears in the denominator. Find the -intercept and the -intercept of the graph of each equation. Then graph the equation int: 7; -int: -int: ; -int: -int: ; -int:. - NAME 6 DATE PERID Practice (Average) Skills Practice, p. 6 and Practice, p. 66 (shown) State whether each equation or function is linear. Write es or no. If no, eplain our reasoning.. h() es. es. No; is a denominator.. 9 No; and are multiplied. Write each equation in standard form. Identif A, B, and C ; 7,, 6. 8 ;, 8, 7 7. ;,, ; 8, 8, Find the -intercept and the -intercept of the graph of each equation. Then graph the equation. 9.,. 7 7, (, ).,. 6 6, (, ) Linear Equations (, ) (, ). MEASURE The equation. gives the length in centimeters corresponding to a length in inches. What is the length in centimeters of a -foot ruler?.8 cm LNG DISTANCE For Eercises and, use the following information. For Meg s long-distance calling plan, the monthl cost C in dollars is given b the linear function C(t) 6.t, where t is the number of minutes talked.. What is the total cost of talking 8 hours? of talking hours? $; $66. What is the effective cost per minute (the total cost divided b the number of minutes talked) of talking 8 hours? of talking hours? $.6; $. - (, ) Reading to Learn Mathematics, p. 67 NAME 66 DATE PERID (, ) (, ) (7, ) Reading to Learn Mathematics Linear Equations Pre-Activit How do linear equations relate to time spent studing? Read the introduction to Lesson - at the top of page 6 in our tetbook. If Lolita spends hours studing math, how man hours will she have to stud chemistr? hours Suppose that Lolita decides to sta up one hour later so that she now has hours to stud and do homework. Write a linear equation that describes this situation. Reading the Lesson. Write es or no to tell whether each linear equation is in standard form. If it is not, eplain wh it is not. a. No; A is negative. b. 9 es ELL Lesson - Application indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples, , Etra Practice See page ;,, 8. ;,, 9. ;,,. 7 ;, 7,. ;,,. ;,,. ;,,. 6;,, 6.,. See margin for graph. The lines are parallel but have different -intercepts. 66 Chapter Linear Relations and Functions ECNMICS For Eercises and, use the following information. n Januar, 999, the euro became legal tender in participating countries in Europe. Based on the echange rate on March,, the linear function d().888 could be used to convert euros to U.S. dollars.. n that date, what was the value in U.S. dollars of euros? $77.6. n that date, what was the value in euros of U.S. dollars? 6. euros nline Research Data Update How do the dollar and the euro compare toda? Visit to convert among currencies. State whether each equation or function is linear. Write es or no. If no, eplain our reasoning. 6 9,. See margin for eplanations.. es 6. no 7. no 8. h() no 9. g() no. f() 6 9 es. f() 7 no. no. Which of the equations 9 7,, and is not linear?. Which of the functions f(), g() 7, and h() is not linear? h() PHYSICS For Eercises and 6, use the following information. When a sound travels through water, the distance in meters that the sound travels in seconds is given b the equation.. How far does a sound travel underwater in seconds? 7 m 6. In air, the equation is. Does sound travel faster in air or water? Eplain. Sound travels onl 7 m in seconds in air, so it travels faster underwater. Write each equation in standard form. Identif A, B, and C ;,, ;,, 9;,, 9 ;,, Find the -intercept and the -intercept of the graph of each equation. Then graph the equation. 9. See pp. 7A 7H for graphs. 9.,. 6 6,.. +,.,.,. none, 6. none, , none 8., none 9. f(),. g(). 6, CRITICAL THINKING For Eercises and, use,, and.. Graph the equations on a coordinate plane. Compare and contrast the graphs.. Write a linear equation whose graph is between the graphs of and. Sample answer: c. 7 es d. 7 No; B is not an integer. e. No; A and B are both. f. 8 No; The greatest common factor of,, and 8 is, not.. How can ou use the standard form of a linear equation to tell whether the graph is a horizontal line or a vertical line? If A, then the graph is a horizontal line. If B, then the graph is a vertical line. Helping You Remember. ne wa to remember something is to eplain it to another person. Suppose that ou are studing this lesson with a friend who thinks that she should let to find the -intercept and let to find the -intercept. How would ou eplain to her how to remember the correct wa to find intercepts of a line? Sample answer: The -intercept is the -coordinate of a point on the -ais. Ever point on the -ais has -coordinate, so let to find an -intercept. The -intercept is the -coordinate of a point on the -ais. Ever point on the -ais has -coordinate, so let to find a -intercept. 66 Chapter Linear Relations and Functions NAME DATE PERID - Enrichment, p. 68 Greatest Common Factor Suppose we are given a linear equation a b c where a, b, and c are nonzero integers, and we want to know if there eist integers and that satisf the equation. We could tr guessing a few times, but this process would be time consuming for an equation such as B using the Euclidean Algorithm, we can determine not onl if such integers and eist, but also find them. The following eample shows how this algorithm works. Eample Find integers and that satisf Divide the greater of the two coefficients b the lesser to get a quotient and remainder. Then, repeat the process b dividing the divisor b the remainder until ou get a remainder of. The process can be written as follows. 88 () 6 () 6() () 6 () 6 () 6() () () Answers 6. No; appears in a denominator. 7. No; is inside a square root. 8. No; has eponents other than. 9. No; appears in a denominator.. Yes; this is a linear function because it is written as f() 6 9, m 6, b 9.. No; has an eponent other than.. No; is inside a square root.

20 Geolog Geothermal energ from hot springs is being used for electricit in California, Ital, and Iceland. 8. Yes; the graph passes the vertical line test. Standardized Test Practice GELGY For Eercises, use the following information. Suppose the temperature T ( C) below Earth s surface is given b T(d) d, where d is the depth (km).. Find the temperature at a depth of kilometers. 9 C. Find the depth if the temperature is 6 C. km. Graph the linear function. See margin. FUND-RAISING For Eercises 6 9, use the following information. The Jackson Band Boosters sell beverages for $.7 and cand for $. at home games. Their goal is to have total sales of $ for each game. 6. Write an equation that is a model for the different numbers of beverages and cand that can be sold to meet the goal..7b.c 7. Graph the equation. See margin. 8. Does this equation represent a function? Eplain. 9. If the sell beverages and pieces of cand, will the Band Boosters meet their goal? no 6. GEMETRY Find the area of the shaded region in the graph. (Hint: The area of a trapezoid is given b A h(b b ).) units 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How do linear equations relate to time spent studing? Include the following in our answer: See margin. wh onl the part of the graph in the first quadrant is shown, and an interpretation of the graph s intercepts in terms of the amount of time Lolita spends on each subject. 6. Which function is linear? B A f() B g().7 C g() D f() 9 6. What is the -intercept of the graph of? B A B C 6 D Assess pen-ended Assessment Modeling Have students place a piece of spaghetti or a pencil on a large coordinate plane to model the graphs of these equations:,,,,, and. Getting Read for Lesson - BASIC SKILL Lesson - presents the fact that perpendicular lines have slopes that are negative reciprocals. Eercises 7 78 should be used to determine our students familiarit with finding reciprocals. Assessment ptions Quiz (Lessons - and -) is available on p. of the Chapter Resource Masters. Maintain Your Skills Mied Review 6. D {,,, }, R {,, }; es 6. D {,, }, R {,,, }; no Getting Read for the Net Lesson State the domain and range of each relation. Then graph the relation and determine whether it is a function. (Lesson -) 6 6. See pp. 7A 7H for 6. {(, ), (, ), (, ), (, )} 6. {(, ), (, ), (, ), (, )} graphs. Solve each inequalit. (Lesson -6) { } 67. { 6 or } 68. TAX Including a 6% sales ta, a paperback book costs $8.. What is the price before ta? (Lesson -) $7.9 Simplif each epression. (Lesson -) 69. (9s ) (s 6) s 7. [9 (8 )] BASIC SKILL Find the reciprocal of each number Lesson - Linear Equations T(d ) 7. c 8 d 8 T(d ) d 6.7b.c b Answers 6. A linear equation can be used to relate the amounts of time that a student spends on each of two subjects if the total amount of time is fied. Answers should include the following. and must be nonnegative because Lolita cannot spend a negative amount of time studing a subject. The intercepts represent Lolita spending all of her time on one subject. The -intercept represents her spending all of her time on math, and the -intercept represents her spending all of her time on chemistr. Lesson - Linear Equations 67

21 Lesson Notes Slope Focus -Minute Check Transparenc - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on p. C. Building on Prior Knowledge In Lesson -, students wrote linear equations in standard form and graphed them using intercepts. In this lesson, the appl this skill to finding the slope of a line and to graphing parallel and perpendicular lines. does slope appl to the steepness of roads? Ask students: An engineer designed a road with a rise of feet for each horizontal distance of feet. What is the grade of this road? % If a road has a grade of %, what is its rise for each horizontal distance of feet?. ft Vocabular slope rate of change famil of graphs parent graph oblique Stud Tip Slope The formula for slope is often remembered as rise over run, where the rise is the difference in -coordinates and the run is the difference in -coordinates. TEACHING TIP Make sure students realize that must come from the same ordered pair as. Find and use the slope of a line. Graph parallel and perpendicular lines. SLPE The slope of a line is the ratio of the change in -coordinates to the corresponding change in -coordinates. The slope measures how steep a line is. Suppose a line passes through points (, ) and (, ). (, ) The change in -coordinates is. The change in ( -coordinates is., ) change in -coordinates slope change in -coordinates The slope of a line is the same, no matter what two points on the line are used. Words Smbols Eample does slope appl to the steepness of roads? The grade of a road is a percent that measures the steepness of the road. It is found b dividing the amount the road rises b the corresponding horizontal distance. Slope of a Line The slope of a line is the ratio of the change in -coordinates to the change in -coordinates. The slope m of the line passing through (, ) and (, ) is given b m, where. Find Slope horizontal distance Find the slope of the line that passes through (, ) and (, ). Then graph the line. m Slope formula (, ) ( ( ), ) (, ), (, ) (, ) 6 or Simplif. (, ) The slope of the line is. Graph the two ordered pairs and draw the line. Use the slope to check our graph b selecting an point on the line. Then go down units and right unit or go up units and left unit. This point should also be on the line. rise 68 Chapter Linear Relations and Functions Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 7 Practice, p. 7 Reading to Learn Mathematics, p. 7 Enrichment, p. 7 Transparencies -Minute Check Transparenc - Answer Ke Transparencies Technolog Interactive Chalkboard

22 Eample Use Slope to Graph a Line Graph the line passing through (, ) with a slope of. Graph the ordered pair (, ). Then, according to the slope, go up units and right units. Plot the new point at (, ). You can also go right units and then up units to plot the new point. Draw the line containing the points. The slope of a line tells the direction in which it rises or falls. (, ) (, ) Teach SLPE In-Class Eamples Power Point Find the slope of the line that passes through (, ) and (, ). Then graph the line. (, ) If the line rises to the right, then the slope is positive. (, ) If the line is horizontal, then the slope is zero. (, ) (, ) If the line falls to the right, then the slope is negative. (, ) If the line is vertical, then the slope is undefined. (, ) (, ) (, ) m ( ) ( ) m ( ) (, ) m (, ), so m is undefined. Graph the line passing through (, ) with a slope of. Log on for: Updated data More activities on rate of change com/usa_toda Eample Slope is often referred to as rate of change. It measures how much a quantit changes, on average, relative to the change in another quantit, often time. Rate of Change TRAVEL Refer to the graph at the right. Find the rate of change of the number of people taking cruises from 98 to. m Slope formula 6.9. Substitute. 98. Simplif. Between 98 and, the number of people taking cruises increased at an average rate of about.(,) or, people per ear. Teacher to Teacher Jud Buchholtz USA TDAY Snapshots Cruises grow in popularit The number of North Americans taking cruises, b ear: Millions Source: Cruise Lines International Association 6.9 B Hilar Wasson and Quin Tian, USA TDAY Lesson - Slope 69 Dublin Scioto H.S., Dublin, H A match it, graph it CBL-motion detector lab is a fun wa for students to eperience slope." (, ) CMMUNICATIN Refer to the graph below. Find the rate of change of the number of radio stations on the air in the United States from 99 to 998. Number (thousands) (, ) U.S. Radio Stations on the Air Year Source: The New York Times Almanac Between 99 and 998, the number of radio stations on the air in the United States increased at an average rate of.() or stations per ear. Lesson - Slope 69

23 PARALLEL AND PERPENDICULAR LINES In-Class Eample Power Point Graph the line through (, ) that is parallel to the line with equation. (, ) (, ) PARALLEL AND PERPENDICULAR LINES A famil of graphs is a group of graphs that displas one or more similar characteristics. The parent graph is the simplest of the graphs in a famil. A graphing calculator can be used to graph several graphs in a famil on the same screen. Lines with the Same Slope The calculator screen shows the graphs of,,, and. Think and Discuss. See margin.. Identif the parent function and describe the famil of graphs. What is similar about the graphs? What is different about the graphs?. Find the slope of each line.. Write another function that has the same characteristics as this famil of graphs. Check b graphing. Sample answer: [, ] scl: b [, ] scl: Answer. ; The graphs are parallel lines, but the have different -intercepts. Stud Tip Horizontal Lines All horizontal lines are parallel because the all have a slope of. In the Investigation, ou saw that lines that have the same slope are parallel. These and other similar eamples suggest the following rule. Words In a plane, nonvertical lines with the same slope are parallel. All vertical lines are parallel. Model same slope Parallel Lines Eample Parallel Lines Graph the line through (, ) that is parallel to the line with equation. The -intercept is, and the -intercept is. Use the intercepts to graph. The line falls unit for ever units it moves to the right, so the slope is. Now use the slope and the point at (, ) to graph the line parallel to the graph of. (, ) (, ) The figure at the right shows the graphs of two lines that are perpendicular. You know that parallel lines have the same slope. What is the relationship between the slopes of two perpendicular lines? slope of line AB slope of line CD or 6 6 or ( ) The slopes are opposite reciprocals of each other. This relationship is true in general. When ou multipl the slopes of two perpendicular lines, the product is alwas. C (, ) A(, ) B(, ) D(, ) 7 Chapter Linear Relations and Functions Lines with the Same Slope Point out that the simplest of the graphs in a famil is often the one that passes through the origin, where the values of and are both zero. Suggest that students substitute for in each equation to find a point that will help them identif which graph goes with each equation. 7 Chapter Linear Relations and Functions

24 Stud Tip Reading Math An oblique line is a line that is neither horizontal nor vertical. Words Smbols In a plane, two oblique lines are perpendicular if and onl if the product of their slopes is. Suppose m and m are the slopes of two oblique lines. Then the lines are perpendicular if and onl if m m, or m. m Model Perpendicular Lines slope m slope m In-Class Eample Power Point Graph the line through (, ) that is perpendicular to the line with equation. Concept Check. Sometimes; the slope of a vertical line is undefined. Guided Practice GUIDED PRACTICE KEY Eercises Eamples 6 7, 8 9, Eample An vertical line is perpendicular to an horizontal line. Perpendicular Line Graph the line through (, ) that is perpendicular to the line with equation. The -intercept is, and the -intercept is. Use the intercepts to graph. (, 6) The line falls units for ever units it moves to the right, so the slope is. The slope of the perpendicular line is the opposite reciprocal of, or. (, ) Start at (, ) and go up units and right units. Use this point and (, ) to graph the line.. PEN ENDED Write an equation of a line with slope. Sample answer:. Decide whether the statement below is sometimes, alwas, or never true. Eplain. The slope of a line is a real number.. FIND THE ERRR Mark and Luisa are finding the slope of the line through (, ) and (, ). Mark m= or ( ) Luisa m= or ( ) Who is correct? Eplain our reasoning. See margin. Find the slope of the line that passes through each pair of points.. (, ), (, ). (, ), (, ) 6. (, ), (, ) Graph the line passing through the given point with the given slope. 7. (, ), 8. (, ), 7 8. See pp. 7A 7H. Graph the line that satisfies each set of conditions. 9. See pp. 7A 7H. 9. passes through (, ), parallel to graph of 6. passes through (, ), perpendicular to graph of 6. passes through (, ), perpendicular to graph of Differentiated Instruction Lesson - Slope 7 Naturalist Ask students to appl what the have learned about slope, grade, or steepness as the describe or sketch what the see around them in nature, such as the shapes of hills, branches of trees, and roof lines of houses. Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. draw sample graphs to compare and contrast lines with positive slope, negative slope, zero slope, and a slope that is undefined. include an other item(s) that the find helpful in mastering the skills in this lesson. FIND THE ERRR Point out that when finding a slope, if ou use as the numerator, ou must use as the denominator. To find the slope, ou can move from point A to point B, or from point B to point A, but ou must move in a consistent direction for both the rise and the run. Answer (, ). Luisa; Mark did not subtract in a consistent manner when using the slope formula. If and, then must be and must be, not vice versa. Lesson - Slope 7

25 About the Eercises rganization b bjective Slope: Parallel and Perpendicular Lines: dd/even Assignments Eercises 6 and are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercises 6 and 7 require a graphing calculator. Assignment Guide Basic: odd, odd, 7 9, 7 odd,, 8 7 Average: odd, 7 9, odd,, 8 7 (optional: 6, 7) Advanced: 6 6 even,, even, 69 (optional: 7 7) All: Practice Quiz ( ) Answers. Application indicates increased difficult Practice and Appl Homework Help For See Eercises Eamples 6 7, Etra Practice See page 8. WEATHER For Eercises, use the table that shows the temperatures at different times on March,. Time Temp ( F) 8: A.M. : A.M. : P.M. : P.M. : P.M What was the average rate of change of the temperature from 8: A.M. to : A.M.?. /h. What was the average rate of change of the temperature from : P.M. to : P.M.?. /h. During what -hour period was the average rate of change of the temperature the least? : P.M. : P.M. Find the slope of the line that passes through each pair of points... (6, ), (8, ) 6. (6, 8), (, ) 7. ( 6, ), (, ) 8. (, 7), (, ) 9. (7, 8), (, 8). (, ), (, ). (., ), (, 9) 8. (,.), (,.) undefined. (a, ), (a, ) undefined 6. (, b), (, b).,,,.,, 6, 7. Determine the value of r so that the line through (6, r) and (9, ) has slope. 8. Determine the value of r so that the line through (, r) and (, ) has slope. 9 ANCIENT CULTURES Maan Indians of Meico and Central America built pramids that were used as their temples. Ancient Egptians built pramids to use as tombs for the pharohs. Estimate the slope that a face of each pramid makes with its base The Pramid of the Sun in Teotihuacán, Meico, measures about 7 feet on each side of its square base and is about feet high. about.6 The Great Pramid in Egpt measures 76 feet on each side of its square base and was originall 8 feet high. about. 6. See margin. Graph the line passing through the given point with the given slope.. (, 6), m. (, ), m. (, ), m. (, ), m. (6, ), m 6. (, ), undefined 7 Chapter Linear Relations and Functions Chapter Linear Relations and Functions

26 7. about 68 million per ear 8. about million per ear ENTERTAINMENT For Eercises CD and Tape Shipments 7 9, refer to the graph that shows the number of CDs and cassette tapes shipped b manufacturers to 8 retailers in recent ears. CDs 6 7. Find the average rate of change of Cassettes the number of CDs shipped from 99 to. 8. Find the average rate of change of the number of cassette tapes shipped Year from 99 to. Source: Recording Industr Association of America 9. Interpret the sign of our answer to Eercise 8. The number of cassette tapes shipped has been decreasing. TRAVEL For Eercises, use the following information. Mr. and Mrs. Wellman are taking their daughter to college. The table shows their distance from home after various amounts of time.. Find the average rate of change of their distance from home between and hours after leaving home. mph. Find the average rate of change of their distance from home between and hours after leaving home. mph. What is another word for rate of change in this situation? speed or velocit Graph the line that satisfies each set of conditions.. See pp. 7A 7H.. passes through (, ), parallel to a line whose slope is. passes through (, ), perpendicular to a line whose slope is. passes through (, ), perpendicular to graph of 6. passes through (, ), parallel to graph of 7. passes through (, ), parallel to graph of 6 8. passes through origin, parallel to graph of 9. perpendicular to graph of, intersects that graph at its -intercept. perpendicular to graph of, intersects that graph at its -intercept GEMETRY Determine whether quadrilateral ABCD with vertices A(, ), B(, ), C(, ), and D(, ) is a rectangle. Eplain. Yes; slopes show that adjacent sides are perpendicular.. CRITICAL THINKING If the graph of the equation a 9 is perpendicular to the graph of the equation, find the value of a.. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 7A 7H. How does slope appl to the steepness of roads? Include the following in our answer: a few sentences eplaining the relationship between the grade of a road and the slope of a line, and a graph of.8, which corresponds to a grade of 8%. (A road with a grade of 6% to 8% is considered to be fairl steep. The scales on our - and -aes should be the same.) Number (millions) Time Distance (h) (mi) 6 6 Lesson - Slope 7 Stud Stud Guide and and Intervention Intervention, p. 69 (shown) and p. 7 - Slope NAME DATE PERID Slope change in Slope m of a Line For points (, ) and (, ), where, m change in Eample Determine the slope of Eample Graph the line passing the line that passes through (, ) and through (, ) with a slope of. (, ). Graph the ordered m Slope formula pair (, ). Then, according to the ( ) slope, go up units (, ) (, ), (, ) (, ) and right units. 6 Plot the new point Simplif. 6 (,). Connect the points and draw The slope of the line is. the line. Eercises Find the slope of the line that passes through each pair of points.. (, 7) and (6, ). (6, ) and (, ). (, ) and (7, ). (, ) and (, ). (, ) and (, ) 6. (, ) and (, ) 7. (7, ) and (, ) 8. (, 9) and (, ) 9. (, ) and (, 8) Graph the line passing through the given point with the given slope.. slope. slope. slope passes through (, ) passes through (, ) passes through (, ). slope. slope. slope passes through (, 6) passes through (, ) passes through (, ) - NAME 69 DATE PERID Practice (Average) Skills Practice, p. 7 and Practice, p. 7 (shown) Find the slope of the line that passes through each pair of points.. (, 8), (, ). (, ), (7, ) 7. ( 7, 6), (, 6) 7. (8, ), (8, ) undefined. (, ), (7, ) 6. ( 6, ), ( 8, ) Graph the line passing through the given point with the given slope. 7. (, ), m 8. (, ), m (, ) 9. (, ), m. (, ), m Graph the line that satisfies each set of conditions.. passes through (, ), perpendicular. passes through (, ), parallel to a line whose slope is to a line whose slope is Slope (, ) (, ) DEPRECIATIN For Eercises, use the following information. A machine that originall cost $,6 has a value of $7 at the end of ears. The same machine has a value of $8 at the end of 8 ears.. Find the average rate of change in value (depreciation) of the machine between its purchase and the end of ears. $7 per ear. Find the average rate of change in value of the machine between the end of ears and the end of 8 ears. $9 per ear. Interpret the sign of our answers. It is negative because the value is decreasing. Reading to Learn Mathematics, p. 7 - NAME 7 DATE PERID (, ) (, ) (, ) Reading to Learn Mathematics Slope Pre-Activit How does slope appl to the steepness of roads? Read the introduction to Lesson - at the top of page 68 in our tetbook. What is the grade of a road that rises feet over a horizontal distance of feet? % What is the grade of a road that rises meters over a horizontal distance of kilometers? ( kilometer meters).% Reading the Lesson. Describe each tpe of slope and include a sketch. Tpe of Slope Description of Graph Sketch Positive The line rises to the right. ELL Lesson - Zero The line is horizontal. NAME DATE PERID - Enrichment, p. 7 Aerial Surveors and Area Man land regions have irregular shapes. Aerial surveors suppl aerial mappers with lists of coordinates and elevations for the areas that need to be photographed from the air. These maps provide information about the horizontal and vertical features of the land. Step List the ordered pairs for the vertices in counterclockwise order, repeating the first ordered pair at the bottom of the list. (, ) (, ) (, 7) (6, ) Negative The line falls to the right. Undefined The line is vertical.. a. How are the slopes of two nonvertical parallel lines related? The are equal. b. How are the slopes of two oblique perpendicular lines related? Their product is. Lesson - Step Find D, the sum of the downward diagonal products (from left to right). D ( ) ( ) ( ) (6 7) 6 or 7 Step Find U, the sum of the upward diagonal products (from left to right). ) ( ) ) (, 7) (, ) (, ) (6, ) Helping You Remember. Look up the terms grade, pitch, slant, and slope. How can everda meanings of these words help ou remember the definition of slope? Sample answer: All these words can be used when ou describe how much a thing slants upward or downward. You can describe this numericall b comparing rise to run. Lesson - Slope 7

27 Assess pen-ended Assessment Modeling Have students use a piece of spaghetti or a pencil on a coordinate plane to model lines with slopes of,, and, as well as a line with undefined slope. Then have them use a second piece of spaghetti or a second pencil to model a pair of parallel lines and then a pair of perpendicular lines. Intervention If there is an doubt whether our students thoroughl understand the concept of slope, consider spending an etra da on this lesson. Use the Etra Practice on p. 8, the Stud Guide and Intervention Masters, or the Practice Masters to reinforce this concept. New Getting Read for Lesson - PREREQUISITE SKILL Lesson - presents writing the equation of a line given the slope and a point on the line. Eercises 7 7 should be used to determine our students familiarit with solving equations for a given variable. Assessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons - through -. Lesson numbers are given to the right of eercises or instruction lines so students can review concepts not et mastered. Answers 6. The graphs have the same -intercept. As the slopes increase, the lines get steeper. 7. The graphs have the same -intercept. As the slopes become more negative, the lines get steeper. P Standardized Test Practice Graphing Calculator Maintain Your Skills Mied Review Getting Read for the Net Lesson ractice Quiz. State the domain and range of the relation {(, ), (, ), (, ), ( 7, ), (, )}. (Lesson -) D { 7,,, }, R {,,,, }. Find the value of f() if f(). (Lesson -) 7. Write 6 in standard form. (Lesson -) 6. Find the -intercept and the -intercept of the graph of. Then graph the equation. (Lesson -), 6; See margin for graph.. Graph the line that goes through (, ) and is parallel to the line whose equation is. (Lesson -) See margin. 7 Chapter Linear Relations and Functions What is the slope of the line shown in the graph at the right? D A B C D. What is the slope of a line perpendicular to a line with slope? D A B C D FAMILY F GRAPHS Use a graphing calculator to investigate each famil of graphs. Eplain how changing the slope affects the graph of the line. 6.,, 8, 6 7. See margin. 7.,,, 7 Find the -intercept and the -intercept of the graph of each equation. Then graph the equation. (Lesson -) 8 6. See pp. 7A 7H for graphs. 8., 9. 8, , Find each value if f(). (Lesson -) 6. f( ) 7 6. f() 6. f 6. f(a) a Solve each inequalit. (Lessons - and -6) 6. 7 { } 66. z 7 {z z 7} 67. SCHL A test has multiple-choice questions worth points each and truefalse questions worth points each. Marco answers multiple-choice questions correctl. How man true-false questions must he answer correctl to get at least 8 points total? (Lesson -) at least 8 Simplif. (Lessons - and -) 68. (a 9b) (8b 8a) 7a b 69. ( 7) PREREQUISITE SKILL Solve each equation for. (To review solving equations, see Lesson -.) 7 7. See margin Answers (Practice Quiz ).. Lessons through 7 Chapter Linear Relations and Functions

28 Writing Linear Equations Lesson Notes Vocabular slope-intercept form point-slope form Stud Tip Slope-intercept Form The equation of a vertical line cannot be written in slope-intercept form because its slope is undefined. Write an equation of a line given the slope and a point on the line. Write an equation of a line parallel or perpendicular to a given line. When a compan manufactures a product, the must consider two tpes of cost. There is the fied cost, which the must pa no matter how man of the product the produce, and there is variable cost, which depends on how man of the product the produce. In some cases, the total cost can be found using a linear equation such as.7. Words do linear equations appl to business? FRMS F EQUATINS Consider the graph at the right. The line passes through A(, b) and C(, ). Notice that b is the -intercept of AC. You can use these two points to find the slope of AC. Substitute the coordinates of points A and C into the slope formula. m Slope formula b m (, ) (, b), (, ) (, ) m b Simplif. Now solve the equation for. m b Multipl each side b. m b Add b to each side. m b Smbols m b slope Smmetric Propert of Equalit When an equation is written in this form, it is in slope-intercept form. The slope-intercept form of the equation of a line is m b, where m is the slope and b is the -intercept. Slope-Intercept Form of a Linear Equation -intercept Model (, b) C(, ) m b A(, b) Focus -Minute Check Transparenc - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on p. D. Building on Prior Knowledge In Lesson -, students studied slope and the graphs of parallel and perpendicular lines. In this lesson, students appl these ideas to writing the equations of lines. do linear equations appl to business? Ask students: What are some eamples of costs that might be included in the fied cost? utilities, salaries, rent What are some eamples of costs that might be included in the variable cost? raw materials, packaging, shipping If ou are given the slope and -intercept of a line, ou can find an equation of the line b substituting the values of m and b into the slope-intercept form. For eample, if ou know that the slope of a line is and the -intercept is, the equation of the line is, or, in standard form,. You can also use the slope-intercept form to find an equation of a line if ou know the slope and the coordinates of an point on the line. Lesson - Writing Linear Equations 7 Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 77 Practice, p. 78 Reading to Learn Mathematics, p. 79 Enrichment, p. 8 Assessment, pp., Graphing Calculator and Spreadsheet Masters, p. School-to-Career Masters, p. Transparencies -Minute Check Transparenc - Answer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 7

29 Teach FRMS F EQUATINS In-Class Eamples Power Point Write an equation in slopeintercept form for the line that has a slope of and passes through (, ). Teaching Tip Make sure that students understand that the letter m is alwas used for slope, and b for the -intercept, in m b. What is an equation of the line through (, ) and (, 7)? D A B C D Standardized Test Practice Test-Taking Tip To check our answer, substitute each ordered pair into our answer. Each should satisf the equation. Eample Write an Equation Given Slope and a Point Write an equation in slope-intercept form for the line that has a slope of and passes through (, ). Substitute for m,, and in the slope-intercept form. (, ) m b Slope-intercept form ( ) b (, ) (, ), m 6 b Simplif. b Subtract 6 from each side. The -intercept is. So, the equation in slope-intercept form is. If ou are given the coordinates of two points on a line, ou can use the point-slope form to find an equation of the line that passes through them. Words Eample The point-slope form of the equation of a line is m( ), where (, ) are the coordinates of a point on the line and m is the slope of the line. Multiple-Choice Test Item Point-Slope Form of a Linear Equation Smbols slope m( ) Write an Equation Given Two Points coordinates of point on line What is an equation of the line through (, ) and (, )? A B C D Read the Test Item You are given the coordinates of two points on the line. Notice that the answer choices are in slope-intercept form. Solve the Test Item First, find the slope of the line. m Slope formula ( ( ), ) (, ), (, ) (, ) or Simplif. The slope is. That eliminates choices B and D. Then use the point-slope form to find an equation. m( ) Point-slope form [ ( )] m ; ou can use either point for (, ). Distributive Propert The answer is A. 76 Chapter Linear Relations and Functions Standardized Test Practice Eample Point out that finding the slope eliminated two of the choices. Emphasize that eliminating some of the answer choices helps ou to use our time efficientl when taking a timed test. 76 Chapter Linear Relations and Functions

30 Stud Tip Alternative Method You could also find Mr. Fu s salar in part c b etending the graph. Then find the value when is. When changes in real-world situations occur at a linear rate, a linear equation can be used as a model for describing the situation. Eample Write an Equation for a Real-World Situation SALES As a salesperson, Eric Fu is paid a dail salar plus commission. When his sales are $, he makes $. When his sales are $, he makes $. a. Write a linear equation to model this situation. Let be his sales and let be the amount of mone 6 he makes. Use the points (, ) and (, ) to make a graph to represent the situation. (, ) m Slope formula 8 (, ) (, ) (, ), (, ) (, ). Simplif. Now use the slope and either of the given points with the point-slope form to write the equation. m( ) Point-slope form.( ) m., (, ) (, ).. Distributive Propert Add to each side. The slope-intercept form of the equation is.. b. What are Mr. Fu s dail salar and commission rate? The -intercept of the line is. The -intercept represents the mone Eric would make if he had no sales. In other words, $ is his dail salar. The slope of the line is.. Since the slope is the coefficient of, which is his sales, he makes % commission. c. How much would he make in a da if Mr. Fu s sales were $? Find the value of when.. Use the equation ou found in part a..() Replace with. or Simplif. Mr. Fu would make $ if his sales were $. 8 6 PARALLEL AND PERPENDICULAR LINES The slope-intercept and pointslope forms can be used to find equations of lines that are parallel or perpendicular to given lines. In-Class Eample In-Class Eample Power Point SALES As a part-time salesperson, Jean Stock is paid a dail salar plus commission. When her sales are $, she makes $8. When her sales are $, she make $78. a. Write a linear equation to model this situation.. 8 b. What are Ms. Stock s dail salar and commission rate? $8; % c. How much would Jean make in a da if her sales were $? $98 Teaching Tip Point out that when the units differ on the two aes, ou cannot estimate the slope of the graphed line b comparing it to the slope of as the -degree line with a slope of. PARALLEL AND PERPENDICULAR LINES Power Point Write an equation for the line that passes through (, ) and is perpendicular to the line whose equation is. Eample Write an Equation of a Perpendicular Line Write an equation for the line that passes through (, ) and is perpendicular to the line whose equation is. The slope of the given line is. Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular line is. (continued on the net page) Lesson - Writing Linear Equations 77 Lesson - Writing Linear Equations 77

31 Practice/Appl TEACHING TIP You could also use slopeintercept form. Use the point-slope form and the ordered pair (, ) to write the equation. m( ) Point-slope form Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. add the slope-intercept and pointslope forms of a linear equation to their notebooks, with eamples of both tpes of equations. add items from Eample to their list of test-taking tips, which the can review as the prepare for standardized tests. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Forms of Equations: Parallel and Perpendicular Lines: 8 dd/even Assignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic:,, 9 odd, 9,,, 9, 7 67 Average: 9 odd,,, 7 67 (optional:, 6) Advanced: even,, 6 6 (optional: 6 67) Answer. Solve the equation for to get. The slope of this line is. The slope of a parallel line is the same. Concept Check. Sample answer: Guided Practice GUIDED PRACTICE KEY Eercises Eamples 7 8, 9,,, 9. 6 Standardized Test Practice indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples 8, 8 9,,, 9, 9, 8 Etra Practice See page 8. 6., 6 78 Chapter Linear Relations and Functions [ ( )] (, ) (, ), m Distributive Propert Add to each side. An equation of the line is.. PEN ENDED Write an equation of a line in slope-intercept form.. Identif the slope and -intercept of the line with equation 6. 6,. Eplain how to find the slope of a line parallel to the graph of. See margin. State the slope and -intercept of the graph of each equation..,., Write an equation in slope-intercept form for the line that satisfies each set of conditions slope., passes through (6, ) 7. slope 6, passes through, 8. passes through (6, ) and (8, ) 9. passes through (, ) and (, ). passes through (, ), perpendicular to the graph of. Write an equation in slope-intercept form for the graph at the right. 7. What is an equation of the line through (, ) and (, )? B 6 A B 6 C D State the slope and -intercept of the graph of each equation..,.,., c d c, d undefined, none Write an equation in slope-intercept form for each graph (., ) Differentiated Instruction (, ) (7, ) Intrapersonal Have students write a paragraph summarizing an problems the have with relating graphs to their equations. Ask them to include specific techniques the find useful to help them with this task. (, 7) (, ) 78 Chapter Linear Relations and Functions

32 Write an equation in slope-intercept form for each graph... (, ) Stud Stud Guide and and Intervention Intervention, p. 7 (shown) and p NAME DATE PERID Writing Linear Equations Forms of Equations Slope-Intercept Form of a Linear Equation Point-Slope Form of a Linear Equation m b, where m is the slope and b is the -intercept m( ), where (, ) are the coordinates of a point on the line and m is the slope of the line no slope-intercept form for (, ) Write an equation in slope-intercept form for the line that satisfies each set of conditions.. slope, passes through (, 6). slope., passes through (, ). slope, passes through (, ) 6. slope, passes through, ) 7. slope., passes through (, ) 8. slope, passes through the origin 9. passes through (, ) and (, ). passes through (7, ) and (7, 8). passes through (, ) and (, ). passes through (, ) and (, ). -intercept, -intercept. -intercept, -intercept. passes through (, 6), parallel to the graph of 6. passes through (, ), perpendicular to the graph of 7 7. passes through (6, ), perpendicular to the line whose equation is 8. passes through (, ), parallel to the line that passes through (, ) and (, 6) 9. Write an equation in slope-intercept form of the line that passes through the points indicated in the table.. Write an equation in slope-intercept form of the line that passes through (, ), (, ), and (, ). 6 7 Eample Write an equation in Eample Write an equation in slope-intercept form for the line that slope-intercept form for the line that has slope and passes through the has slope and -intercept. point (, 7). m b Slope-intercept form Substitute for m,, and in the slope-intercept form. () b (, ) (, ), m m b Slope-intercept form 7 ( )() b b Simplif. (, ) (, 7), m 7 6 b Simplif. b Subtract from both sides. b Add 6 to both sides. The -intercept is. The equation in The -intercept is. The slope-intercept slope-intercept form is. form is. Eercises Write an equation in slope-intercept form for the line that satisfies each set of conditions.. slope, passes through (, 6). slope, -intercept. slope, passes through (, ). slope, passes through (, 7) 6 Write an equation in slope-intercept form for each graph (, 6) (, ) (, ) (, ) (, ) NAME 7 DATE PERID Practice (Average) Skills Practice, p. 77 and Practice, p. 78 (shown) Writing Linear Equations State the slope and -intercept of the graph of each equation.. 8 8,...,., 7. 7,., 6., Write an equation in slope-intercept form for each graph (, ) (, ) (, ) (, ) (, ) (, ) Lesson - GEMETRY For Eercises, use the equation d 8(c ) that gives the total number of degrees d in an conve polgon with c sides.. Write this equation in slope-intercept form. d 8c 6. Identif the slope and d-intercept. 8, 6. Find the number of degrees in a pentagon. ECLGY A park ranger at Blendon Woods estimates there are 6 deer in the park. She also estimates that the population will increase b 7 deer each ear thereafter. Write an equation that represents how man deer will be in the park in ears BUSINESS Refer to the signs below. At what distance do the two stores charge the same amount for a balloon arrangement? mi Conrad s Balloon Bouquets $ balloon arrangements Deliver: $ per mile The Balloon House $ Balloon Arrangements $ per mile deliver Lesson - Writing Linear Equations 79 Write an equation in slope-intercept form for the line that satisfies each set of conditions.. slope, passes through (, 8). slope, passes through (, ). slope, passes through (, ). slope, passes through (6, 8). passes through (, ) and ( 6, ). passes through (7, ) and (, ) intercept, -intercept 7. -intercept, -intercept passes through ( 8, 7), perpendicular to the graph of 9 9. RESERVIRS The surface of Grand Lake is at an elevation of 68 feet. During the current drought, the water level is dropping at a rate of inches per da. If this trend continues, write an equation that gives the elevation in feet of the surface of Grand Lake after das BUSINESS Ton Marconi s compan manufactures CD-RM drives. The compan will make $, profit if it manufactures, drives, and $,7, profit if it manufactures, drives. The relationship between the number of drives manufactured and the profit is linear. Write an equation that gives the profit P when n drives are manufactured. P n, Reading to Learn Mathematics, p NAME 78 DATE PERID Reading to Learn Mathematics Writing Linear Equations Pre-Activit How do linear equations appl to business? Read the introduction to Lesson - at the top of page 7 in our tetbook. If the total cost of producing a product is given b the equation.7, what is the fied cost? What is the variable cost (for each item produced)? $; $.7 Write a linear equation that describes the following situation: A compan that manufactures computers has a fied cost of $8,7 and a variable cost of $8 to produce each computer. 8,7 8 Reading the Lesson ELL. a. Write the slope-intercept form of the equation of a line. Then eplain the meaning of each of the variables in the equation. m b; m is the slope and b is the -intercept. The variables and are the coordinates of an point on the line. - Enrichment, p. 8 Two-Intercept Form of a Linear Equation You are alread familiar with the slope-intercept form of a linear equation, m b. Linear equations can also be written in the form b with a -intercept a and -intercept b. This is called two-intercept form. Eample Draw the graph of. 6 The graph crosses the -ais at and the -ais at 6. Graph (, ) and (, 6), then draw a straight line through them. Eample NAME DATE PERID Write in two-intercept form. Divide b to obtain on the right side. b. Write the point-slope form of the equation of a line. Then eplain the meaning of each of the variables in the equation. m( ); m is the slope. and are the coordinates of an point on the line. and are the coordinates of one specific point on the line.. Suppose that our algebra teacher asks ou to write the point-slope form of the equation of the line through the points ( 6, 7) and (, ). You write ( ) and our classmate writes 7 ( 6). Which of ou is correct? Eplain. You are both correct. Either point ma be used as (, ) in the point-slope form. You used (, ), and our classmate used ( 6, 7).. You are asked to write an equation of two lines that pass through (, ), one of them parallel to and one of them perpendicular to the line whose equation is. The first step in finding these equations is to find their slopes. What is the slope of the parallel line? What is the slope of the perpendicular line? ; Helping You Remember. Man students have trouble remembering the point-slope form for a linear equation. How can ou use the definition of slope to remember this form? Sample answer: Write the definition of slope: m. Multipl both sides of this equation b. Drop the subscripts in and. This gives the point-slope form of the equation of a line. Lesson - Lesson - Writing Linear Equations 79

33 Assess pen-ended Assessment Writing Have students write their own summar of how to relate graphs to their equations, including slope, intercepts, and parallel and perpendicular lines. Getting Read for Lesson - PREREQUISITE SKILL Lesson - presents modeling real-world data using scatter plots. Eercises 6 67 should be used to determine our students familiarit with finding the median of a set of numbers. Assessment ptions Quiz (Lessons - and -) is available on p. of the Chapter Resource Masters. Mid-Chapter Test (Lessons - through -) is available on p. of the Chapter Resource Masters. Answers A linear equation can sometimes be used to relate a compan s cost to the number the produce of a product. Answers should include the following. The -intercept,, is the cost the compan must pa if the produce units, so it is the fied cost. The slope,.7, means that it costs $.7 to produce each unit. The variable cost is.7. $677 Science Ice forms at a temperature of C, which corresponds to a temperature of F. A temperature of C corresponds to a temperature of F. Standardized Test Practice Etending the Lesson Maintain Your Skills Mied Review Getting Read for the Net Lesson 8 Chapter Linear Relations and Functions SCIENCE For Eercises 6 8, use the information on temperatures at the left. 6. Write and graph the linear equation that gives the number of degrees Fahrenheit in terms of the number of degrees Celsius. 9 ; See margin for graph. 7. What temperature corresponds to C? 68 F 8. What temperature is the same on both scales? TELEPHNES For Eercises 9 and, use the following information. Namid is eamining the calling card portion of his phone bill. A -minute call at the night rate cost $.6. A -minute call at the night rate cost $ Write a linear equation to model this situation.... How much would it cost to talk for half an hour at the night rate? $.7. CRITICAL THINKING Given ABC with vertices A( 6, 8), B(6, ), and C( 6, ), write an equation of the line containing the altitude from A. (Hint: The altitude from A is a segment that is perpendicular to B C.). WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How do linear equations appl to business? Include the following in our answer: See margin. the fied cost and the variable cost in the equation.7, where is the cost for a compan to produce units of its product, and the cost for the compan to produce units of its product.. Find an equation of the line through (, ) and (, ). C A B C D. Choose the equation of the line through, and,. A A B C D For Eercises and 6, use the following information. The form a is known as the b a is the -intercept and b is the -intercept. intercept form of the equation of a line because. Write the equation in intercept form. 6. Identif the - and -intercepts of the graph of., Find the slope of the line that passes through each pair of points. (Lesson -) 7. (7, ), (, 6) 8. (, ), (, ) 9. (, ), (, ) 6. INTERNET A Webmaster estimates that the time (seconds) required to connect to the server when n people are connecting is given b t(n).n.. Estimate the time required to connect when people are connecting. (Lesson -). s Solve each inequalit. (Lessons - and -6) (r ) 9 { 6} {r r 6} PREREQUISITE SKILL Find the median of each set of numbers. (To review finding a median, see pages 8 and 8.) 6. {,,,,, 8, } 6. {9,, 7,, 6,, 7, 9} {8,, 976,,, 66}. 67. {., 7.8,.,., 6., 7.8}.8 8 Chapter Linear Relations and Functions

34 Modeling Real-World Data: Using Scatter Plots Lesson Notes Vocabular scatter plot line of fit prediction equation Draw scatter plots. Find and use prediction equations. can a linear equation model the number of Calories ou burn eercising? The table shows the number of Calories burned per hour b a -pound person running at various speeds. A linear function can be used to model these data. Speed (mph) Calories Focus -Minute Check Transparenc - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on p. D. Stud Tip Choosing the Independent Variable Letting be the number of ears since the first ear in the data set sometimes simplifies the calculations involved in finding a function to model the data. SCATTER PLTS To model data with a function, it is helpful to graph the data. A set of data graphed as ordered pairs in a coordinate plane is called a scatter plot. A scatter plot can show whether there is a relationship between the data. Eample Draw a Scatter Plot HUSING The table below shows the median selling price of new, privatelowned, one-famil houses for some recent ears. Make a scatter plot of the data. Year Price ($) Source: U.S. Census Bureau and U.S. Department of Housing and Urban Development Graph the data as ordered pairs, with the number of ears since 99 on the horizontal ais and the price on the vertical ais. Price ($) Median House Prices 6 8 Years Since 99 PREDICTIN EQUATINS Ecept for (,.9), the data in Eample appear to lie nearl on a straight line. When ou find a line that closel approimates a set of data, ou are finding a line of fit for the data. An equation of such a line is often called a prediction equation because it can be used to predict one of the variables given the other variable. Lesson - Modeling Real-World Data: Using Scatter Plots 8 Building on Prior Knowledge In Lesson -, students wrote linear equations based on information provided about their graphs. In this lesson, students appl those skills to modeling real-world data with scatter plots and writing their prediction equations. can a linear equation model the number of Calories ou burn eercising? Ask students: Will a person that weighs less than pounds burn more or less Calories than shown in the table at the given speeds? less What is a reasonable estimate of the number of Calories a -pound person burns running at a speed of miles per hour? about Calories Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp. 8 8 Skills Practice, p. 8 Practice, p. 8 Reading to Learn Mathematics, p. 8 Enrichment, p. 86 School-to-Career Masters, p. Science and Mathematics Lab Manual, pp. 97 Teaching Algebra With Manipulatives Masters, p. 8 Transparencies -Minute Check Transparenc - Answer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 8

35 Teach SCATTER PLTS In-Class Eample In-Class Eample Power Point EDUCATIN The table below shows the approimate percent of students who sent applications to two colleges in various ears since 98. Make a scatter plot of the data. Years 6 9 Since 98 Percentage 8 Source: U.S. News & World Report Percentage PREDICTIN EQUATINS Power Point EDUCATIN Refer to the data in In-Class Eample. a. Draw a line of fit for the data. How well does the line fit the data? Percentage Percentage of Students Appling to Two Colleges Years Since 98 Percentage of Students Appling to Two Colleges Years Since 98 Ecept for (6, ), the line fits the data fairl well. (continued on the net page) Stud Tip Reading Math A data point that does not appear to belong with the rest of the set is called an outlier. Stud Tip Reading Math When ou are predicting for an value greater than an in the data set, the process is known as etrapolation. When ou are predicting for an value between the least and greatest in the data set, the process is known as interpolation. TEACHING TIP n pages 87 and 88, students will learn about a number that measures how well a line fits a set of data. 8 Chapter Linear Relations and Functions To find a line of fit and a prediction equation for a set of data, select two points that appear to represent the data well. This is a matter of personal judgment, so our line and prediction equation ma be different from someone else s. Eample Find and Use a Prediction Equation HUSING Refer to the data in Eample. a. Draw a line of fit for the data. How well does the line fit the data? Ignore the point (,.9) since it would not be close to a line that represents the rest of the data points. The points (,.) and (8,.) appear to represent the data well. Draw a line through these two points. Ecept for (,.9), this line fits the data ver well. b. Find a prediction equation. What do the slope and -intercept indicate? Find an equation of the line through (,.) and (8,.). Begin b finding the slope. m Slope formula.. Substitute. 8 Simplif..6 m( ) Point-slope form..6( ) m.6, (, ) (,.) Distributive Propert Add. to each side. ne prediction equation is The slope indicates that the median price is increasing at a rate of about $6 per ear. The -intercept indicates that, according to the trend of the rest of the data, the median price in 99 should have been about $7,8. c. Predict the median price in. The ear is ears after 99, so use the prediction equation to find the value of when Prediction equation.6() Price ($) Simplif. Median House Prices 6 8 Years Since 99 The model predicts that the median price in will be about $,. d. How accurate is the prediction? Ecept for the outlier, the line fits the data ver well, so the predicted value should be fairl accurate. Differentiated Instruction Naturalist Ask students how scatter plots and prediction equations might be used to relate local insect and animal populations to food supplies, temperature, and rainfall. Have interested students devise a plan for conducting the research necessar for developing such a prediction equation. 8 Chapter Linear Relations and Functions

36 Stud Tip utliers If our scatter plot includes points that are far from the others on the graph, check our data before deciding it is an outlier. You ma have made a graphing or recording mistake. Concept Check. D {,,, }, R {,, }; Sample answer using (, ) and (, ): Guided Practice GUIDED PRACTICE KEY Eercises Eamples,, Head versus Height Collect the Data Collect data from several of our classmates. Use a tape measure to measure the circumference of each person s head and his or her height. Record the data as ordered pairs of the form (height, circumference). Analze the Data. See students work.. Graph the data in a scatter plot.. Choose two ordered pairs and write a prediction equation.. Eplain the meaning of the slope in the prediction equation. Make a Conjecture. Predict the head circumference of a person who is 66 inches tall.. Predict the height of an individual whose head circumference is 8 inches.. Choose the scatter plot with data that could best be modeled b a linear function. d a. b. c. d.. Identif the domain and range of the relation in the graph at the right. Predict the value of when.. PEN ENDED Write a different prediction equation for the data in Eamples and on pages 8 and 8. Sample answer using (,.) and (6,.): Complete parts a c for each set of data in Eercises and. a. Draw a scatter plot. b. Use two ordered pairs to write a prediction equation. c. Use our prediction equation to predict the missing value.. SCIENCE Whether ou are climbing a mountain or fling in an airplane, the higher ou go, the colder the air gets. The table shows the temperature in the atmosphere at various altitudes.. See pp. 7A 7H. b. Find a prediction equation. What do the slope and -intercept indicate? Using (, 8) and (, ):. 9.; The slope indicates that the percent is falling at about.% each ear. The -intercept indicates the percent in 98 should have been about 9%. c. Predict the percent in. about 9% d. How accurate is the prediction? The fit is onl approimate, so the prediction ma not be ver accurate. Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. write a summar of the procedure for placing a line of fit on a scatter plot and how to find a prediction equation using this line. include an other item(s) that the find helpful in mastering the skills in this lesson. Altitude (ft) Temp ( C) ? Source: NASA. TELEVISIN As more channels have been added, cable television has become attractive to more viewers. The table shows the number of U.S. households with cable service in some recent ears. Year Households (millions) Source: Nielsen Media Research ? Lesson - Modeling Real-World Data: Using Scatter Plots 8 Algebra Activit Materials: tape measure, grid paper You can use string and a ruler as an alternative wa to measure the circumference of a person s head as well as their height. Measurements can be made in inches or in centimeters. You ma want to have half the class use one sstem and the other half the alternate sstem. Lesson - Modeling Real-World Data: Using Scatter Plots 8

37 About the Eercises rganization b bjective Scatter Plots: 6a, 7a, 8a, 9a, Prediction Equations: 6b, 6c, 7b, 7c, 8b, 8c, 9b, 9c,, 6 dd/even Assignments Eercises 6 9 are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercise involves research on the Internet or other reference materials. Assignment Guide Basic: 7,, 9, Average: 7, 9,,, (optional: ) Advanced: 6, 8, 7 (optional: 8 ) Answers 6a. Lives Saved b Minimum Drinking Age Year Lives (thousands) 7a. Detroit Red Wings 6 Assists 8a. Bottled Water Consumption Gallons 8 6 Goals Year indicates increased difficult Practice and Appl Homework Help For See Eercises Eamples 6, Etra Practice See page 8. 6a. See margin. 6b. Sample answer using (996, 6.) and (998, 8.): c. Sample answer: 8, 7a. See margin. 7b. Sample answer using (,) and (,7):.. 7c. Sample answer: about A scatter plot of loan paments can help ou analze home loans. Visit com/webquest to continue work on our WebQuest project. 8a. See margin. 8b. Sample answer using (99, 9.) and (996,.):..9 8c. Sample answer: about 6.9 gal 9a. See pp. 7A 7H. 9b. Sample answer using (, 99) and (, 88):.., where is the number of seasons since c. Sample answer: about $78 million or $. billion 8 Chapter Linear Relations and Functions Complete parts a c for each set of data in Eercises 6 9. a. Draw a scatter plot. b. Use two ordered pairs to write a prediction equation. c. Use our prediction equation to predict the missing value. 6. SAFETY All states and the District of Columbia have enacted laws setting as the minimum drinking age. The table shows the estimated cumulative number of lives these laws have saved b reducing traffic fatalities. Year Lives (s) Source: National Highwa Traffic Safet Administration 7. HCKEY Each time a hocke plaer scores a goal, up to two teammates ma be credited with assists. The table shows the number of goals and assists for some of the members of the Detroit Red Wings in the NHL season. Goals Assists Source: 8. HEALTH Bottled water has become ver popular. The table shows the number of gallons of bottled water consumed per person in some recent ears. Year Gallons Source: U.S. Department of Agriculture 9. THEATER Broadwa, in New York Cit, is the center of American theater. The table shows the total revenue of all Broadwa plas for some recent seasons. Season Revenue ($ millions) Source: The League of American Theatres and Producers, Inc. MEDICINE For Eercises, use the graph that shows how much Americans spent on doctors visits in some recent ears.. Write a prediction equation from the data for 99, 99, and.. Use our equation to predict the amount for.. Compare our prediction to the one given in the graph. The value predicted b the equation is somewhat lower than the one given in the graph ? 7 6 8? ?. Sample answer using (99, 8) and (99, 79):. 6,. Sample answer: $ nline Lesson Plans ? Cost of seeing the doctor How much Americans spend a ear on doctor visits: USA TDAY Education s nline site offers resources and interactive features connected to each da s newspaper. Eperience TDAY, USA TDAY s dail lesson plan, is available on the site and delivered dail to subscribers. This plan provides instruction for integrating USA TDAY graphics and ke editorial features into our mathematics classroom. Log on to USA TDAY Snapshots $6 $79 Source: U.S. Health Care Financing Administration $96 $,7 projected B Mark Pearson and Jerr Mosemak, USA TDAY 8 Chapter Linear Relations and Functions

38 Finance A financial analst can advise people about how to invest their mone and plan for retirement. nline Research For information about a career as a financial analst, visit: careers 6. Sample answer using (, 6) and (98, ):.. 9. Sample answer using (97, 6.) and (99, 8.7): 96.% FINANCE For Eercises and, use the following information. Della has $ that she wants to invest in the stock market. She is considering buing stock in either Compan or Compan. The values of the stocks at the ends of the last four months are shown in the tables below.. Based onl on these data, which stock should Della bu? Eplain.. Do ou think investment decisions should be based on this tpe of reasoning? If not, what other factors should be considered?. See pp. 7A 7H. GEGRAPHY For Eercises 8, use the table below that shows the elevation and average precipitation for selected cities. Source: World Meteorological Association Compan Month Share Price ($) Aug.. Sept..9 ct..9 Nov..6. Draw a scatter plot with elevation as the independent variable. See pp. 7A 6. Write a prediction equation. 7H. 7. Predict the average annual precipitation for Dublin, Ireland, which has an elevation of 79 feet. Sample answer: about in. 8. Compare our prediction to the actual value of 9 inches. See margin. CRITICAL THINKING For Eercises 9 and, use the table that shows the percent of people ages and over with a high school diploma over the last few decades. 9. Use a prediction equation to predict the percent in.. Do ou think our prediction is accurate? Eplain. See margin. Compan Month Share Price ($) Aug.. Sept..8 ct..6 Nov.. Cit Average Average Elevation Elevation Precip. Cit Precip. (feet) (feet) (inches) (inches) Rome, Ital 79 London, England Algiers, Algeria 8 7 Paris, France 6 Istanbul, Turke 8 7 Bucharest, Romania 98 Montreal, Canada 8 7 Budapest, Hungar 6 Stockholm, Sweden 7 Toronto, Canada 67 Berlin, German 9. RESEARCH Use the Internet or other resource to look up the population of our communit or state in several past ears. Use a prediction equation to predict the population in some future ear. See students work. High School Graduates Year Percent Source: U.S. Census Bureau Lesson - Modeling Real-World Data: Using Scatter Plots 8 Stud Stud Guide and and Intervention Intervention, p. 8 (shown) and p. 8 - NAME DATE PERID Modeling Real-World Data: Using Scatter Plots Scatter Plots When a set of data points is graphed as ordered pairs in a coordinate plane, the graph is called a scatter plot. A scatter plot can be used to determine if there is a relationship among the data. Eample BASEBALL The table below shows the number of home runs and runs batted in for various baseball plaers who won the Most Valuable Plaer Award during the 99s. Make a scatter plot of the data. Home Runs Runs Batted In Source: New York Times Almanac Eercises Runs Batted In 7 MVP HRs and RBIs Home Runs Make a scatter plot for the data in each table below.. FUEL EFFICIENCY The table below shows the average Average Fuel Efficienc fuel efficienc in miles per gallon of new cars 6 manufactured during the ears listed. Year Fuel Efficienc (mpg) Year Source: New York Times Almanac. CNGRESS The table below shows the number of Women in Congress women serving in the United States Congress during 7 the ears Congressional Session Number of Women 8 Session of Congress 8 6 Source: Wall Street Journal Almanac NAME 8 DATE PERID - Skills Practice Practice, p. 8 and (Average) Practice, Modeling Real-World p. 8 (shown) Data: Using Scatter Plots For Eercises, complete parts a c for each set of data. a. Draw a scatter plot. b. Use two ordered pairs to write a prediction equation. c. Use our prediction equation to predict the missing value.. FUEL ECNMY The table gives the Weight (tons) approimate weights in tons and estimates for overall fuel econom in miles per gallon Miles per Gallon 9? 7 for several cars. Fuel Econom Versus Weight b. Sample answer using (., ) and (., ): c. Sample answer: 8.6 mi/gal..... Weight (tons). ALTITUDE In most cases, temperature decreases with increasing altitude. As Anchara drives into the mountains, her car thermometer registers the temperatures ( F) shown in the table at the given altitudes (feet). Altitude (ft) ,, Temperature Versus Altitude Temperature ( F) ? 6 6 b. Sample answer using (7, 6) and (97, ):. 98. c. Sample answer: 8. F. HEALTH Alton has a treadmill that uses the time on the treadmill and the speed of walking or running to estimate the number of Calories he burns during a workout. The table gives workout times and Calories burned for several workouts. Calories Burned - Time (min) Calories Burned ? b. Sample answer using (, 8) and Burning Calories (8, ): c. Sample answer: about calories Time (min) Reading to Learn Mathematics, p. 8 NAME 8 DATE PERID Reading to Learn Mathematics Fuel Econom (mi/gal) Temperature ( F) Miles per Gallon Number of Women 7, 8, 9,, Altitude (ft) ELL Modeling Real-World Data: Using Scatter Plots Pre-Activit How can a linear equation model the number of Calories ou burn eercising? Read the introduction to Lesson - at the top of page 8 in our tetbook. If a woman runs. miles per hour, about how man Calories will she burn in an hour? Sample answer: 7 Calories If a man runs 7. miles per hour, about how man Calories will he burn in half an hour? Sample answer: 97 Calories Reading the Lesson. Suppose that a set of data can be modeled b a linear equation. Eplain the difference between a scatter plot of the data and a graph of the linear equation that models that data. Sample answer: The scatter plot is a discrete graph. It is made up just of the individual points that represent the data points. The linear equation has a continuous graph that is the line that best fits the data points. Lesson - Answers 8. Sample answer: The predicted value differs from the actual value b more than %, possibl because no line fits the data ver well.. Sample answer: The predicted percent is almost certainl too high. Since the percent cannot eceed %, it cannot continue to increase indefinitel at a linear rate. NAME DATE PERID - Enrichment, p. 86 Median-Fit Lines A median-fit line is a particular tpe of line of fit. Follow the steps below to find the equation of the median-fit line for the data. Approimate Percentage of Violent Crimes Committed b Juveniles That Victims Reported to Law Enforcement Year 996 ffenders Source: U.S. Bureau of Justice Statistics. Divide the data into three approimatel equal groups. There should alwas be the same number of points in the first and third groups. In this case, there will be three data points in each group. Group Group Group Year ffenders Year ffenders Year ffenders. Suppose that tuition at a state college was $ per ear in 99 and has been increasing at a rate of $ per ear. a. Write a prediction equation that epresses this information. b. Eplain the meaning of each variable in our prediction equation. represents the number of ear since 99 and represents the tuition in that ear.. Use this model to predict the tuition at this college in 7. $6 Helping You Remember. Look up the word scatter in a dictionar. How can its definition help ou to remember the meaning of the difference between a scatter plot and the graph of a linear equation? Sample answer: To scatter means to break up and go in man directions. The points on a scatter plot are broken up. In a scatter plot, the points are scattered or broken up. In the graph of a linear equation, the points are connected to form a continuous line. Lesson - Lesson - Modeling Real-World Data: Using Scatter Plots 8

39 Assess pen-ended Assessment Writing Ask students to write instructions that could be used to teach a friend how to read a graph such as the one shown in Eample on p. 8. The importance of reading titles, captions, and tet, as well as the scales should be included as part of the instructions. Getting Read for Lesson -6 PREREQUISITE SKILL Lesson -6 presents the graphing of special functions, including absolute value functions. Eercises 8 should be used to determine our students familiarit with finding absolute values. Answer. Data can be used to write a linear equation that approimates the number of Calories burned per hour in terms of the speed that a person runs. Answers should include the following. Calories Calories Burned While Running Speed (mph) Sample answer using (, 8) and (8, 88): about 97 calories; Sample answer: The predicted value differs from the actual value b onl about %. Standardized Test Practice Etending the Lesson. 988, 99, 998; 7, 6., 6 8. about (99, 6.7) Maintain Your Skills Mied Review Getting Read for the Net Lesson. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can a linear equation model the number of Calories ou burn eercising? Include the following in our answer: See margin. a scatter plot and a prediction equation for the data, and a prediction of the number of Calories burned in an hour b a -pound person running at 9 miles per hour, with a comparison of our predicted value with the actual value of 9.. Which line best fits the data in the graph at the right? D A B. C. D... A prediction equation for a set of data is.6.. For which value is the predicted value 6.? A A C B. 6 D 8. For Eercises, use the following information. A median-fit line is a particular tpe of line of fit. Follow the steps below to find the equation of the median-fit line for the data. Year Prisoners Source: U.S. Bureau of Justice Statistics 86 Chapter Linear Relations and Functions Federal and State Prisoners (per, U.S. citizens) Divide the data into three approimatel equal groups. There should alwas be the same number of points in the first and third groups. Find,, and, the medians of the values in Groups,, and, respectivel. Find,, and, the medians of the values in Groups,, and, respectivel. 6. Find an equation of the line through (, ) and (, ).., Find Y, the -coordinate of the point on the line in Eercise 6 with an -coordinate of. 8. The median-fit line is parallel to the line in Eercise 6, but is one-third closer to (, ). This means it passes through, Y. Find this ordered pair. 9. Write an equation of the median-fit line..,9.. Predict the number of prisoners per, citizens in and. about 6, about 7 Write an equation in slope-intercept form that satisfies each set of conditions. (Lesson -). 6. slope, passes through (, 6). passes through (, ) and (, ) Find each value if g() 7. (Lesson -). g(). g() 7. g( ) 9 6. g( ) 7 7. Solve. (Lesson -6) { 7 or } PREREQUISITE SKILL Find each absolute value. (To review absolute value, see Lesson -.) Teacher to Teacher Susan Nelson Spring H.S., Spring, TX "I have m students take measurements of their height and arm span and record them. We enter the entire class data into a graphing calculator and find the linear regression. Then we use the regression equation to make predictions." 86 Chapter Linear Relations and Functions

40 Lines of Regression You can use a TI-8 Plus graphing calculator to find a line that best fits a set of data. This line is called a regression line or line of best fit. You can also use the calculator to draw scatter plots and make predictions. A Follow-Up of Lesson - INCME The table shows the median income of U.S. families for the period Source: U.S. Census Bureau Find and graph a regression equation. Then predict the median income in. Find a regression equation. Enter the ears in L and the incomes in L. KEYSTRKES: STAT ENTER 97 ENTER Find the regression equation b selecting LinReg(a b) on the STAT CALC menu. KEYSTRKES: STAT ENTER The regression equation is about.9,6.7. Year Income ($) The slope indicates that famil incomes were increasing at a rate of about $ per ear. The number r is called the linear correlation coefficient. The closer the value of r is to or, the closer the data points are to the line. If the values of r and r are not displaed, use Diagnosticn from the CATALG menu. Predict using the regression equation. Find when. Use value on the CALC menu. KEYSTRKES: nd CALC ENTER According to the regression equation, the median famil income in will be about $6, , 7,7,,6 6,77 Graph the regression equation. Use STAT PLT to graph a scatter plot. KEYSTRKES: nd ENTER [STAT PLT] Select the scatter plot, L as the Xlist, and L as the Ylist. Cop the equation to the Y list and graph. KEYSTRKES: VARS GRAPH ENTER [96, ] scl: b [,,] scl:, Notice that the regression line does not pass through an of the data points, but comes close to all of them. The line fits the data ver well. Graphing Calculator Investigation A Follow-Up of Lesson - Getting Started Know Your Calculator The TI-8 Plus graphing calculator has a linear regression function, LinReg(a b), that uses a leastsquares fit method to determine the values for a and b. This involves calculus and finding the distance from each point to the line of best fit. Correlation Coefficient With Diagnosticn, the calculator also displas values for r and r. The closer the value of r is to, the better the equation fits the data. Teach Make sure students have cleared the L and L lists before the enter the new data. After students have entered data, have them work in pairs to verif that the data displa shows the correct numbers before proceeding. Have students complete Eercises. Investigating Slope-Intercept Form 87 Graphing Calculator Investigation Lines of Regression 87 Graphing Calculator Investigation Lines of Regression 87

41 Graphing Calculator Investigation Assess Refer students to the calculator displa shown in Step on p. 87, and ask the following questions. What is the regression equation for the income data?.9,6,.7 What does the value of r tell ou about the regression equation? The value of r is ver close to, so the data points are ver close to the graph of that equation. Answers.. 7. [, ] scl: b [, 6] scl: [98, ] scl: b [, ] scl: [99, ] scl: b [,, 8,] scl:. The prediction ma not be accurate because different parts of the data could be represented b lines with different slopes. The sales could drop, as the did in 99, or the could level out, as the did in 996 and The correlation coefficient, , is closer to. The new regression line fits the data better. Eercises GVERNMENT For Eercises, use the table below that shows the population and the number of representatives in Congress for selected states. State Population (millions) Representatives Source: The World Almanac. Make a scatter plot of the data. See margin.. Find a regression equation for the data Predict the number of representatives for regon, which has a population of about.8 million. BASEBALL For Eercises 6, use the table at the right that shows the total attendance for minor league baseball in some recent ears.. Make a scatter plot of the data. See margin.. Find a regression equation for the data Predict the attendance in.,, TRANSPRTATIN For Eercises 7, use the table below that shows the retail sales of motor vehicles in the United States for the period Make a scatter plot of the data. See margin. 8. Find a regression equation for the data ,7. 9. According to the regression equation, what was the average rate of change of vehicle sales during the period?. Predict the sales in. about,88,. How accurate do ou think our prediction is? Eplain. See margin. RECREATIN For Eercises, use the table at the right that shows the amount of mone spent on skin diving and scuba equipment in some recent ears.. about $,,. Find a regression equation for the data. 6.9,9.. Delete the outlier (997, ) from the data set. Then find a new regression equation for the data. 7.6,.. Use the new regression equation to predict the sales in.. Compare the new correlation coefficient to the old value and state whether the regression line fits the data better. See margin. 88 Investigating Slope-Intercept Form 88 Chapter Linear Relations and Functions CA NY TX FL NC IN AL Motor Vehicle Sales Year Vehicles (thousands),8,99,,8,6,98,96 7, Source: American Automobile Manufacturers Association 9. about 7, vehicles more per ear Year Attendance (millions) Source: National Association of Professional Baseball Leagues Skin Diving and Scuba Equipment Year Sales ($ millions) Source: National Sporting Goods Association 88 Chapter Linear Relations and Functions

42 Special Functions Lesson Notes Vocabular step function greatest integer function constant function identit function absolute value function piecewise function Identif and graph step, constant, and identit functions. Identif and graph absolute value and piecewise functions. do step functions appl to postage rates? The cost of the postage to mail a letter is a function of the weight of the letter. But the function is not linear. It is a special function called a step function. Weight not over (ounces)... Price ($) Focus -Minute Check Transparenc -6 Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on p. D. Stud Tip Greatest Integer Function Notice that the domain of this step function is all real numbers and the range is all integers. STEP FUNCTINS, CNSTANT FUNCTINS, AND THE IDENTITY FUNCTIN The graph of a step function is not linear. It consists of line segments or ras. The greatest integer function, written f() [[]], is an eample of a step function. The smbol [[]] means the greatest integer less than or equal to. For eample, [[7.]] 7 and [[.]] because.. Stud the table and graph below. Eample Step Function BUSINESS Labor costs at the Fi-It Auto Repair Shop are $6 per hour or an fraction thereof. Draw a graph that represents this situation. Eplore The total labor charge must be a multiple of $6, so the graph will be the graph of a step function. Plan Solve f() [[]] f() A dot means that the point is included in the graph. f() f() If the time spent on labor is greater than hours, but less than or equal to hour, then the labor cost is $6. If the time is greater than hour but less than or equal to hours, then the labor cost is $, and so on. Use the pattern of times and costs to make a table, where is the number of hours of labor and C() is the total labor cost. Then draw the graph. (continued on the net page) A circle means that the point is not included in the graph. Lesson -6 Special Functions 89 Building on Prior Knowledge In Lesson -, students drew scatter plots. In this lesson the draw graphs of several special functions: step, constant, identit, absolute value, and piecewise. do step functions appl to postage rates? Ask students: What is the cost of mailing a letter when the weight is.9 ounces? $. when it is. ounces? $. What is the ratio of the change in price over the change in weight from.9 ounce to. ounces?. from.8 ounces to. ounces? How can ou tell that this is not a linear function? Sample answer: The slope between some pairs of points is not the same as the slope between other pairs of points. Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 89 Practice, p. 9 Reading to Learn Mathematics, p. 9 Enrichment, p. 9 Assessment, p. Graphing Calculator and Spreadsheet Masters, p. 9 Teaching Algebra With Manipulatives Masters, p. 9 Transparencies -Minute Check Transparenc -6 Answer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 89

43 Teach STEP FUNCTINS, CNSTANT FUNCTINS, AND THE IDENTITY FUNCTIN In-Class Eamples Power Point PSYCHLGY ne pschologist charges for counseling sessions at the rate of $8 per hour or an fraction thereof. Draw a graph that represents this situation. 7 8 C() 6 Teaching Tip Help clear up confusions about step functions b asking students to choose several sample times, in hours and minutes, and find the associated costs. Lead them to see that two different times ( values) can have the same cost (C value). Graph g(). g() Intervention Students who are having trouble with the absolute value graphs ma see more clearl what is happening if the draw their own graph of the parent function. New Stud Tip Absolute Value Function Notice that the domain is all real numbers and the range is all nonnegative real numbers. Eamine 9 Chapter Linear Relations and Functions C() $6 $ $8 $ $ 6 7 Since the shop rounds an fraction of an hour up to the net whole number, each segment on the graph has a circle at the left endpoint and a dot at the right endpoint. You learned in Lesson - that the slope-intercept form of a linear function is m b, or in functional notation, f() m b. When m, the value of the function is f() b for ever value. So, f() b is called a constant function. The function f() is called the zero function. Eample Constant Function Graph f(). For ever value of, f(). The graph is a horizontal line. Another special case of slopeintercept form is m, b. This is the function f(). The graph is the line through the origin with slope. Since the function does not change the input value, f() is called the identit function. ABSLUTE VALUE AND PIECEWISE FUNCTINS Another special function is the absolute value function, f() C() f() f(). f() f().. f() f() f() f() f() f() f() f() 9 Chapter Linear Relations and Functions

44 Stud Tip Look Back To review families of graphs, see Lesson -. TEACHING TIP abs( is located on the MATH NUM menu.. All of the graphs have a corner point at the origin.. The graph becomes narrower.. The graphs are reflections of each other about the -ais.. The graph opens downward. if The absolute value function can be written as f() if. A function that is written using two or more epressions is called a piecewise function. Recall that a famil of graphs is a group of graphs that displas one or more similar characteristics. The parent graph of most absolute value functions is. Eample Absolute Value Functions Graph f() and g() on the same coordinate plane. Determine the similarities and differences in the two graphs. Find several ordered pairs for each function. Graph the points and connect them. The domain of each function is all real numbers. The range of f() is { }. The range of g() is { }. The graphs have the same shape, but different -intercepts. The graph of g() is the graph of f() translated down units. f() g() You can also use a graphing calculator to investigate families of absolute value graphs. Families of Absolute Value Graphs The calculator screen shows the graphs of,,, and. f() Think and Discuss. What do these graphs have in common?. Describe how the graph of a changes as a increases. Assume a.. Write an absolute value function whose graph [ 8, 8] scl: b [, ] scl: is between the graphs of and. Sample answer:.. Graph and on the same screen. Then graph and on the same screen. What is true in each case?. In general, what is true about the graph of a when a? ABSLUTE VALUE AND PIECEWISE FUNCTINS In-Class Eample Power Point Teaching Tip Lead students to see that the -intercept, b, of the graph of an absolute value function of the form b indicates how the parent graph of is translated. Graph f() and g() on the same coordinate plane. Determine the similarities and differences in the two graphs. The domain of both graphs is all real numbers. The range of f() is { }. The range of g() is { }. The graphs have the same shape, but different -intercepts. The graph of g() is the graph of f() translated units to the left. Lesson -6 Special Functions 9 Families of Absolute Value Graphs It is important that students arrive at the conclusion in Eercise that the graph of the function narrows as the coefficient a increases. As an etension, ask students to compare the graph of with the graph of and with. Make sure the see the difference between adding and having as a coefficient. Lesson -6 Special Functions 9

45 In-Class Eamples Power Point if Graph f() if. Identif the domain and range. Stud Tip Graphs of Piecewise Functions The graphs of each part of a piecewise function ma or ma not connect. A graph ma stop at a given value and then begin again at a different value for the same value. To graph other piecewise functions, eamine the inequalities in the definition of the function to determine how much of each piece to include. Eample Piecewise Function Graph f() if. Identif the domain and range. if Step Graph the linear function f() for. Since does not satisf this inequalit, stop f() with an open circle at (, ). Step Graph the constant function f() for. Since does satisf this inequalit, begin with a closed circle at (, ) and draw a horizontal ra to the right. The function is defined for all values of, so the domain is all real numbers. The values that are -coordinates of points on the graph are and all real numbers less than, so the range is { or }. The domain is all real numbers. The range is { }. Determine whether each graph represents a step function, a constant function, an absolute value function, or a piecewise function. a. Step Function f() Constant Function f() Absolute Value Function f() Special Functions Piecewise Function f() horizontal segments and ras horizontal line V-shape different ras, segments, and curves Eample Identif Functions Determine whether each graph represents a step function, a constant function, an absolute value function, or a piecewise function. a. b. f() f() piecewise function b. Since this graph consists of multiple horizontal segments, it represents a step function. Since this graph is a horizontal line, it represents a constant function. absolute value function Concept Check. Sample answer: f () 9 Chapter Linear Relations and Functions. Find a countereample to the statement To find the greatest integer function of when is not an integer, round to the nearest integer. Sample answer: [[.9]]. Evaluate g(.) if g() [[ ]].. PEN ENDED Write a function involving absolute value for which f( ). Differentiated Instruction ELL Verbal/Linguistic Have students eplain wh step functions, constant functions, and piecewise functions are so named. 9 Chapter Linear Relations and Functions

46 Guided Practice GUIDED PRACTICE KEY Eercises Eamples, 6 6. D all reals, R all integers 7. D all reals, R all integers 8. D all reals, R all nonnegative reals 9. D all reals, R all nonnegative reals Application indicates increased difficult Practice and Appl Homework Help For See Eercises Eamples 9 7, 7, 9 8,,, 8,, Etra Practice See page 8. Identif each function as S for step, C for constant, A for absolute value, or P for piecewise.. f() A. f() S Graph each function. Identif the domain and range. 6. See pp. 7A 7H for 6. f() [[]] 7. g() [[]] graphs. 8. h() 9. f() if if. g(). h() if D all reals, R { }. TRANSPRTATIN Bluffton High School chartered buses so the student bod could attend the girls basketball state tournament games. Each bus held a maimum of 6 students. Draw a graph of a step function that shows the relationship between the number of students who went to the game and the number of buses that were needed. See margin. if D all reals, R all reals PARKING For Eercises, use the following information. A downtown parking lot charges $ for the first hour and $ for each additional hour or part of an hour.. What tpe of special function models this situation? step function. Draw a graph of a function that represents this situation. See margin.. Use the graph to find the cost of parking there for hours. $6 Identif each function as S for step, C for constant, A for absolute value, or P for piecewise.. f() C 6. f() A 7. f() S 8. f() S 9. f() A. P f() Lesson -6 Special Functions 9 Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. draw their own graphs to compare and contrast step, constant, absolute value, and piecewise functions. Include an other items(s) that the find helpful in mastering the skills in the lesson. About the Eercises rganization b bjective Step Functions, Constant Functions, and the Identit Function: 9 Absolute Value and Piecewise Functions: dd/even Assignments Eercises and are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: odd,,, odd, 9 6 Average: odd,,, odd, 9 6 Advanced: 6 even,,, even, 9 (optional: 6 6) All: Practice Quiz ( ) Answers.. Cost ($) 6 8 Time (hr) Lesson -6 Special Functions 9

47 Stud Stud Guide and and Intervention Intervention, p. 87 (shown) and p NAME DATE PERID Special Functions Step Functions, Constant Functions, and the Identit Function The chart below lists some special functions ou should be familiar with. Function Written as Graph Constant f() c horizontal line Identit f() line through the origin with slope Greatest Integer Function f() one-unit horizontal segments, with right endpoints missing, arranged like steps The greatest integer function is an eample of a step function, a function with a graph that consists of horizontal segments. Eample Identif each function as a constant function, the identit function, or a step function. a. f() b. f() a constant function Eercises a step function Identif each function as a constant function, the identit function, a greatest integer function, or a step function.. f(). f(). f() a constant function a step function the identit function -6 NAME 87 DATE PERID Practice (Average) Special Functions Skills Practice, p. 89 and Practice, p. 9 (shown) Graph each function. Identif the domain and range.. f().. f() D all reals, R all integers. g(). f() D all reals, R nonpositive reals NAME 9 DATE PERID D all reals, R all integers D all reals, R nonnegative reals if if. f() 6. h() if if f() h() D all reals, R all reals D all nonzero reals, R all reals 7. BUSINESS A Stitch in Time charges 8. BUSINESS A wholesaler charges a store $. $ per hour or an fraction thereof per pound for less than pounds of cand and for labor. Draw a graph of the step $. per pound for or more pounds. Draw a function that represents this situation. graph of the function that represents this Labor Costs situation. Cand Costs Total Cost ($) -6 f() g() 6 7 Hours Reading to Learn Mathematics, p. 9 Reading to Learn Mathematics Special Functions f() f() Cost ($) Pounds Pre-Activit How do step functions appl to postage rates? Read the introduction to Lesson -6 at the top of page 89 in our tetbook. What is the cost of mailing a letter that weighs. ounce? $. or cents Give three different weights of letters that would each cost cents to mail. Answers will var. Sample answer:. ounces,.9 ounces,. ounces Reading the Lesson. Find the value of each epression. a. b c.... ELL Lesson -6 Nutrition Good sources of vitamin C include citrus fruits and juices, cantaloupe, broccoli, brussels sprouts, potatoes, sweet potatoes, tomatoes, and cabbage. Source: The World Almanac. f() 9 Chapter Linear Relations and Functions TELEPHNE RATES For Eercises and, use the following information. Sarah has a long-distance telephone plan where she pas for each minute or part of a minute that she talks, regardless of the time of da.. Graph a step function that represents this situation. See pp. 7A 7H.. Sarah made a call to her brother that lasted 9 minutes and seconds. How much did the call cost? $. Graph each function. Identif the domain and range.. See pp. 7A 7H.. f() [[ ]]. g() [[ ]] 6. f() [[]] 7. h() [[]] 8. g() [[]] 9. f() [[ ]]. f(). h(). g(). g(). h(). f() 6. f() 7. f() 8. f() if 9. h() if if if. f() if if if. g() if if if. f() [[ ]]. g() [[]] f(). Write the function shown in the graph. if f() if if NUTRITIN For Eercises 7, use the following information. The recommended dietar allowance for vitamin C is micrograms per da.. Write an absolute value function for the difference between the number of micrograms of vitamin C ou ate toda and the recommended amount. 6. What is an appropriate domain for the function? { } 7. Use the domain to graph the function. See pp. 7A 7H. 8. INSURANCE According to the terms of Lavon s insurance plan, he must pa the first $ of his annual medical epenses. The insurance compan pas 8% of the rest of his medical epenses. Write a function for how much the insurance compan pas if represents Lavon s annual medical epenses. See margin. 9. CRITICAL THINKING Graph. See pp. 7A 7H.. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How do step functions appl to postage rates? Include the following in our answer: See pp. 7A 7H. an eplanation of wh a step function is the best model for this situation, while our gas mileage as a function of time as ou drive to the post office cannot be modeled with a step function, and a graph of a function that represents the cost of a first-class letter.. Tell how the name of each kind of function can help ou remember what the graph looks like. a. constant function Sample answer: Something is constant if it does not change. The -values of a constant function do not change, so the graph is a horizontal line. b. absolute value function Sample answer: The absolute value of a number tells ou how far it is from on the number line. It makes no difference whether ou go to the left or right so long as ou go the same distance each time. c. step function Sample answer: A step function s graph looks like steps that go up or down. d. identit function Sample answer: The - and -values are alwas identicall the same for an point on the graph. So the graph is a line through the origin that has slope. Helping You Remember. Man students find the greatest integer function confusing. Eplain how ou can use a number line to find the value of this function for an real number. Answers will var. Sample answer: Draw a number line that shows the integers. To find the value of the greatest integer function for an real number, place that number on the number line. If it is an integer, the value of the function is the number itself. If not, move to the integer directl to the left of the number ou chose. This integer will give the value ou need. NAME DATE PERID -6 Enrichment, p. 9 Greatest Integer Functions Use the greatest integer function to eplore some unusual graphs. It will be helpful to make a chart of values for each functions and to use a colored pen or pencil. Graph each function... Answer if 8. f().8( ) if 9 Chapter Linear Relations and Functions

48 Standardized Test Practice Maintain Your Skills Mied Review Getting Read for the Net Lesson P ractice Quiz. For which function does f? B A f() B f() C f() [[ ]] D f() [[ ]]. For which function is the range { }? D A f() B f() [[ ]] C f() D f(). Write an equation in slope-intercept form of the line with slope that passes through (, ). (Lesson -) BASKETBALL For Eercises, use the following information. n August 6,, the Houston Comets beat the New York Libert to win their fourth straight WNBA championship. The table shows the heights and weights of the Comets who plaed in that final game. (Lesson -) Source: WNBA. Draw a scatter plot. See margin. Sample answer using (66, 8) and. Use two ordered pairs to write a prediction equation. (7, 78): 9. Use our prediction equation to predict the missing value. Sample answer: 68 lb. Graph f(). Identif the domain and range. (Lesson -6) See margin. Answers Height (in.) Weight (lb). Life Epectanc Epectanc (r) HEALTH For Eercises, use the table that shows the life epectanc for people born in various ears. (Lesson -) Source: National Center for Health Statistics. Draw a scatter plot in which is the number of ears since 9. See margin.. Find a prediction equation.. Predict the life epectanc of a person born in. Sample answer: 78.7 r. Sample answer using (, 69.7) and (7, 76.): Write an equation in slope-intercept form that satisfies each set of conditions. (Lesson -) 6. slope, passes through (, ) 7. passes through (, ) and (, ) Solve each inequalit. Graph the solution set. (Lesson -) 8. { } See margin for graphs. PREREQUISITE SKILL Determine whether (, ) satisfies each inequalit. Write es or no. (To review inequalities, see Lesson -.) 6. es 6. no 6. no 6. 6 es 6. no 6. es Years Since 9 Year Epectanc ? Lessons - through -6 Lesson -6 Special Functions 9 Assess pen-ended Assessment Modeling Have students draw a large coordinate plane on a sheet of paper. Then have them use toothpicks (or other similar objects) to model the general shapes of step, constant, and absolute value functions. Students should identif each tpe of graph as the model it. Getting Read for Lesson -7 PREREQUISITE SKILL Lesson -7 presents graphing inequalities. It is sometimes necessar to solve an inequalit for in order to determine whether to shade above or below the boundar. Eercises 6 6 should be used to determine our students familiarit with inequalities. Assessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons - through -6. Lesson numbers are given to the right of eercises or instruction lines so students can review concepts not et mastered. Quiz (Lessons - and -6) is available on p. of the Chapter Resource Masters. Answers (Practice Quiz ).. D = all reals R = nonnegative reals Houston Comets Weight (lb) f() Height (in.) f() Lesson -6 Special Functions 9

49 Lesson Notes Graphing Inequalities Focus -Minute Check Transparenc -7 Use as a quiz or review of Lesson -6. Mathematical Background notes are available for this lesson on p. D. do inequalities appl to fantas football? Ask students: What is the meaning of receiving ards? the number of ards that a team advances down the field when the receiver of a pass catches the ball Teach GRAPH LINEAR INEQUALITIES In-Class Eample Graph. Power Point Vocabular boundar TEACHING TIP Students ma wish to test a point in the shaded region as a check of their work. Graph linear inequalities. Graph absolute value inequalities. Dana has Vikings receiver Rand Moss as a plaer on his online fantas football team. Dana gets points per receiving ard that Moss gets and points per touchdown that Moss scores. He considers points or more to be a good game. Dana can use a linear inequalit to check whether certain combinations of ardage and touchdowns, such as those in the table, result in points or more. GRAPH LINEAR INEQUALITIES A linear inequalit resembles a linear equation, but with an inequalit smbol instead of an equals smbol. For eample, is a linear inequalit and is the related linear equation. The graph of separates the coordinate plane into two regions. The line is the boundar of each region. The graph of the inequalit is the shaded region. Ever point in the shaded region satisfies the inequalit. The graph of is drawn as a solid line to show that points on the line satisf the inequalit. If the inequalit smbol were or, then points on the boundar would not satisf the inequalit, so the boundar would be drawn as a dashed line. You can graph an inequalit b following these steps. Step Determine whether the boundar should be solid or dashed. Graph the boundar. Step Choose a point not on the boundar and test it in the inequalit. Step If a true inequalit results, shade the region containing our test point. If a false inequalit results, shade the other region. Eample do inequalities appl to fantas football? Dashed Boundar Graph 6. The boundar is the graph of 6. Since the inequalit smbol is, the boundar will be dashed. Use the slope-intercept form,. 6 Now test the point (, ). The point (, ) is usuall a good point to test because it results in eas calculations. 6 riginal inequalit () () 6 (, ) (, ) 6 false Shade the region that does not contain (, ). 96 Chapter Linear Relations and Functions Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp. 9 9 Skills Practice, p. 9 Practice, p. 96 Reading to Learn Mathematics, p. 97 Enrichment, p. 98 Assessment, p. Transparencies -Minute Check Transparenc -7 Answer Ke Transparencies Technolog AlgePASS: Tutorial Plus, Lesson Interactive Chalkboard

50 Stud Tip Look Back To review translating verbal epressions to inequalities, see Lesson -. Inequalities can sometimes be used to model real-world situations. Eample Solid Boundar BUSINESS A mail-order compan is hiring temporar emploees to help in their packing and shipping departments during their peak season. a. Write an inequalit to describe the number of emploees that can be assigned to each department if the compan has temporar emploees available. Let p be the number of emploees assigned to packing and let s be the number assigned to shipping. Since the compan can assign at most emploees total to the two departments, use a smbol. The number the number of emploees of emploees is at for packing and for shipping most twent. } } p s b. Graph the inequalit. Since the inequalit smbol is, the graph of the related linear equation p + s = is solid. This is the boundar of the inequalit. Test (, ). p s riginal inequalit (p, s) (, ) true } } } Shade the region that contains (, ). Since the variables cannot be negative, shade onl the part in the first quadrant. s = p In-Class Eample Power Point EDUCATIN The SAT has two parts. ne tutoring compan advertises that it specializes in helping students who have a combined score on the SAT that is 9 or less. a. Write an inequalit to describe the combined scores of students who are prospective tutoring clients. 9 b. Graph the inequalit c. Does a student with a verbal score of 8 and a math score of fit the tutoring compan s guidelines? es c. Can the compan assign 8 emploees to packing and emploees to shipping? The point (8, ) is in the shaded region, so it satisfies the inequalit. The compan can assign 8 emploees to packing and to shipping. GRAPH ABSLUTE VALUE INEQUALITIES Graphing absolute value inequalities is similar to graphing linear inequalities. The inequalit smbol determines whether the boundar is solid or dashed, and ou can test a point to determine which region to shade. Eample Absolute Value Inequalit Graph. Since the inequalit smbol is, the graph of the related equation is dashed. Graph the equation. Test (, ). riginal inequalit (, ) = (, ) = true Shade the region that includes (, ). Differentiated Instruction Lesson -7 Graphing Inequalities 97 Interpersonal Have students work in pairs as the graph the inequalities, so the can check and correct each other s procedures and results. When graphing, one partner can work to determine the location of the boundar while the other partner determines whether the shaded region is above or below the boundar. GRAPH ABSLUTE VALUE INEQUALITIES In-Class Eample Graph. Power Point Practice/Appl Stud Notebook Have students complete the definitions/eamples for the remaining terms on their Vocabular Builder worksheets for Chapter. include an other items(s) that the find helpful in mastering the skills in this lesson. Lesson -7 Graphing Inequalities 97

51 About the Eercises rganization b bjective Graph Linear Inequalities:, 9 Graph Absolute Value Inequalities: dd/even Assignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercises 7 require a graphing calculator. Assignment Guide Basic: 7 odd,,,,, 8 6 Average: odd, 7,, 8 6 (optional: 7) Advanced: even, 8 6 Assess pen-ended Assessment Speaking Have students eplain how to tell just from looking at an inequalit with alone on the left whether the shaded area will be below or above the boundar, as well as whether the boundar is solid or dashed. Assessment ptions Quiz (Lesson -7) is available on p. of the Chapter Resource Masters. Answers. Substitute the coordinates of a point not on the boundar into the inequalit. If the inequalit is satisfied, shade the region containing the point. If the inequalit is not satisfied, shade the region that does not contain the point.. Concept Check. Guided Practice Application GUIDED PRACTICE KEY Eercises Eamples Practice and Appl 98 Chapter Linear Relations and Functions. Write an inequalit for the graph at the right.. Eplain how to determine which region to shade when graphing an inequalit. See margin.. PEN ENDED Write an absolute value inequalit for which the boundar is solid and the solution is the region above the graph of the related equation. Sample answer: Graph each inequalit. 9. See pp. 7A 7H SHPPING For Eercises, use the following information. Gwen wants to bu some cassettes that cost $ each and some CDs that cost $ each. She has $ to spend.. Write an inequalit to represent the situation, where c is the number of cassettes she bus and d is the number of CDs. c d. Graph the inequalit. See pp. 7A 7H. 7, 8, 9. Can she bu cassettes and CDs? Eplain. No; (, ) is not in the shaded region. indicates increased difficult Homework Help For Eercises See Eamples,,,, Etra Practice See page 8. Graph each inequalit.. See pp. 7A 7H Graph all the points on the coordinate plane to the left of the graph of. Write an inequalit to describe these points.. Graph all the points on the coordinate plane below the graph of. Write an inequalit to describe these points.. See pp. 7A 7H for graphs. SCHL For Eercises and, use the following information. Rosa s professor sas that the midterm eam will count for % of each student s grade and the final eam will count for 6%. A score of at least 9 is required for an A.. The inequalit..6 9 represents this situation, where is the midterm score and is the final eam score. Graph this inequalit. See pp. 7A 7H.. If Rosa scores 8 on the midterm and 9 on the final, will she get an A? es DRAMA For Eercises 7, use the following information. Tickets for the Prestonville High School Drama Club s spring pla cost $ for adults and $ for students. In order to cover epenses, at least $ worth of tickets must be sold.. Write an inequalit that describes this situation. a s 6. Graph the inequalit. See pp. 7A 7H. 7. If 8 adult and 6 student tickets are sold, will the club cover its epenses? es [, ] scl: b [, ] scl: [, ] scl: b [, ] scl: [, ] scl: b [, ] scl: [, ] scl: b [, ] scl: 98 Chapter Linear Relations and Functions

52 Finance A dividend is a pament from a compan to an investor. It is a wa to make mone on a stock without selling it. Standardized Test Practice Graphing Calculator FINANCE For Eercises 8, use the following Dividend information. Compan per Share Carl Talbert estimates that he will need to earn at least $9 per ear combined in dividend income from the Able Rentals $. two stocks he owns to supplement his retirement plan. Best Bikes $.8 8. Write and graph an inequalit for this situation. See pp. 7A 7H. 9. Will he make enough from shares of each compan? es. CRITICAL THINKING Graph. See pp. 7A 7H.. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How do inequalities appl to fantas football? Include the following in our answer: See pp. 7A 7H. an inequalit, and an eplanation of how ou obtained it, to represent a good game for Rand Moss in Dana s fantas football league, a graph of our inequalit (remember that the number of touchdowns cannot be negative, but receiving ardage can be), and which of the games with statistics in the table qualif as good games.. Which could be the inequalit for the graph? A A B C D. Which point satisfies? B A C (, ) B (, ) (, 7) D (, ) SHADE( CMMAND You can graph inequalities with a graphing calculator b using the Shade( command located in the DRAW menu. You must enter two functions. The first function defines the lower boundar of the region to be shaded. The second function defines the upper boundar of the region. If the inequalit is, use the Ymin window value as the lower boundar. If the inequalit is, use the Yma window value as the upper boundar. Graph each inequalit. 7. See margin Stud Stud Guide and and Intervention Intervention, p. 9 (shown) and p. 9-7 NAME DATE PERID Graphing Inequalities Graph Linear Inequalities. A linear inequalit, like, resembles a linear equation, but with an inequalit sign instead of an equals sign. The graph of the related linear equation separates the coordinate plane into two half-planes. The line is the boundar of each half-plane. To graph a linear inequalit, follow these steps.. Graph the boundar, that is, the related linear equation. If the inequalit smbol is or, the boundar is solid. If the inequalit smbol is or, the boundar is dashed.. Choose a point not on the boundar and test it in the inequalit. (, ) is a good point to choose if the boundar does not pass through the origin.. If a true inequalit results, shade the half-plane containing our test point. If a false inequalit results, shade the other half-plane. Eample Graph. The boundar is the graph of. Use the slope-intercept form,, to graph the boundar line. The boundar line should be solid. Now test the point (, ). ()? (, ) (, ) false Shade the region that does not contain (, ). Eercises Graph each inequalit NAME 9 DATE PERID Practice (Average) Skills Practice, p. 9 and Practice, p. 96 (shown) Graph each inequalit Graphing Inequalities Lesson -7 Maintain Your Skills Mied Review 8. See pp. 7A 7H for graphs. 8. D all reals, R all integers 9. D all reals, R { }. D all reals, R all nonnegative reals. Sample answer using (, 6) and (6, 8): Graph each function. Identif the domain and range. (Lesson -6) 8. f() [[]] 9. g(). h() SALES For Eercises, use the table that shows the ears of eperience for eight sales representatives and their sales during a given period of time. (Lesson -) Years Sales ($). Draw a scatter plot. See margin.. Find a prediction equation.. Predict the sales for a representative with 8 ears of eperience. Sample answer: $, Solve each equation. Check our solution. (Lesson -) z 6z. Sales vs. Eperience Sales ($), Years NAME DATE PERID -7 Enrichment, p. 98 Algebraic Proof Lesson -7 Graphing Inequalities 99 The following paragraph states a result ou might be asked to prove in a mathematics course. Parts of the paragraph are numbered. Let n be a positive integer. Also, let n s(n ) be the sum of the squares of the digits in n. Then n s(n ) is the sum of the squares of the digits of n, and n s(n ) is the sum of the squares of the digits of n. In general, n k s(n k ) is the sum of the squares of the digits of n k. Consider the sequence: n, n, n, n,, n k,. 6 In this sequence either all the terms from some k on have the value, 7 or some term, sa n j, has the value, so that the eight terms, 6, 7, 8, 89,,, and keep repeating from that point on. Use the paragraph to answer these questions. e in line CMPUTERS For Eercises, use the following information. A school sstem is buing new computers. The will Computers Purchased bu desktop computers costing $ per unit, and 8 notebook computers costing $ per unit. The total 7 cost of the computers cannot eceed $8,. 6. Write an inequalit that describes this situation. d n 8,. Graph the inequalit.. If the school wants to bu of the desktop computers and of the notebook computers, Desktops will the have enough mone? es Reading to Learn Mathematics, p NAME 96 DATE PERID Reading to Learn Mathematics Graphing Inequalities Pre-Activit How do inequalities appl to fantas football? Read the introduction to Lesson -7 at the top of page 96 in our tetbook. Which of the combinations of ards and touchdowns listed would Dana consider a good game? The first one: 68 ards and touchdowns Suppose that in one of the games Dana plas, Moss gets 7 receiving ards. What is the smallest number of touchdowns he must get in order for Dana to consider this a good game? Reading the Lesson. When graphing a linear inequalit in two variables, how do ou know whether to make the boundar a solid line or a dashed line? If the smbol is or, the line is solid. If the smbol is or, the line is dashed.. How do ou know which side of the boundar to shade? Sample answer: If the test point gives a true inequalit, shade the region containing the test point. If the test point gives a false inequalit, shade the region not containing the test point.. Match each inequalit with its graph. a. iii b. iv c. ii d. i i. ii. iii. iv. Helping You Remember. Describe some was in which graphing an inequalit in one variable on a number line is similar to graphing an inequalit in two variables in a coordinate plane. How can what ou know about graphing inequalities on a number line help ou to graph inequalities in a coordinate plane? Sample answer: A boundar on a coordinate graph is similar to an endpoint on a number line graph. A dashed line is similar to a circle on a number line: both are open and mean not included; the represent the smbols and. A solid line is similar to a dot on a number line: both are closed and mean included; the represent the smbols and. Lesson -7 Graphing Inequalities 99 Notebooks ELL Lesson -7

53 Stud Guide and Review Vocabular and Concept Check This alphabetical list of vocabular terms in Chapter includes a page reference where each term was introduced. Assessment A vocabular test/review for Chapter is available on p. of the Chapter Resource Masters. Lesson-b-Lesson Review For each lesson, the main ideas are summarized, additional eamples review concepts, and practice eercises are provided. Vocabular and Concept Check absolute value function (p. 9) boundar (p. 96) Cartesian coordinate plane (p. 6) constant function (p. 9) dependent variable (p. 9) domain (p. 6) famil of graphs (p. 7) function (p. 7) functional notation (p. 9) greatest integer function (p. 89) identit function (p. 9) independent variable (p. 9) linear equation (p. 6) linear function (p. 6) line of fit (p. 8) mapping (p. 7) one-to-one function (p. 7) ordered pair (p. 6) parent graph (p. 7) piecewise function (p. 9) point-slope form (p. 76) prediction equation (p. 8) quadrant (p. 6) range (p. 6) rate of change (p. 69) relation (p. 6) scatter plot (p. 8) slope (p. 68) slope-intercept form (p. 7) standard form (p. 6) step function (p. 89) vertical line test (p. 7) -intercept (p. 6) -intercept (p. 6) Choose the correct term to complete each sentence.. The (constant, identit) function is a linear function described b f().. The graph of the (absolute value, greatest integer) function forms a V-shape and is described b f().. The (slope-intercept, standard) form of the equation of a line is A B C, where A and B are not both zero.. Two lines in the same plane having the same slope are (parallel, perpendicular).. The (domain, range) is the set of all -coordinates of the ordered pairs of a relation. 6. The set of all -coordinates of the ordered pairs of a relation is the (domain, range). 7. The ratio of the change in -coordinates to the corresponding change in -coordinates is called the (slope, -intercept) of a line. 8. The (line of fit, vertical line test) can be used to determine if a relation is a function. Vocabular PuzzleMaker ELL The Vocabular PuzzleMaker software improves students mathematics vocabular using four puzzle formats crossword, scramble, word search using a word list, and word search using clues. Students can work on a computer screen or from a printed handout. MindJogger Videoquizzes ELL MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams in a game show format to gain points for correct answers. The questions are presented in three rounds. Round Concepts ( questions) Round Skills ( questions) Round Problem Solving ( questions) - See pages 6 6. Eample Chapter Linear Relations and Functions For more information about Foldables, see Teaching Mathematics with Foldables. TM Relations and Functions Concept Summar A relation is a set of ordered pairs. The domain is the set of all -coordinates, and the range is the set of all -coordinates. A function is a relation where each member of the domain is paired with eactl one member of the range. Graph the relation {(, ), (, ), (, )} and find the domain and range. Then determine whether the relation is a function. The domain is {,, }, and the range is {,, }. Graph the ordered pairs. Since each value is paired with eactl one value, the relation is a function. (, ) (, ) (, ) Some students ma need help in deciding which tabs to use. Ask student volunteers to share the thinking process the used to decide where various material should go. Remind students that their notations should be complete sentences that will make sense when the are reviewed weeks later. Encourage students to refer to their Foldables while completing the Stud Guide and Review and to use them in preparing for the Chapter Test. Chapter Linear Relations and Functions

54 Chapter Stud Guide and Review Stud Guide and Review - See pages Eample Eercises Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function. See Eamples and on pages 7 and See margin for graphs. 9. {(6, ), (, ), (, )}. {(, ), (, ), (, ), (, )}.. D all reals, R all reals; es. D all reals, R all reals; es Find each value if f() 9. See Eample on page 9.. f(6). f( ) 9. f() 9 6. f( v) v 9 9. D {,, 6}, R {, }; es. D {, }, R {,,, }; no Linear Equations Concept Summar A linear equation is an equation whose graph is a line. A linear function can be written in the form f() m b. The standard form of a linear equation is A B C. Write 6 8 in standard form. Identif A, B, and C. 6 = riginal equation Subtract from each side. Add 6 to each side. The standard form is. So, A, B, and C. Eercises State whether each equation or function is linear. Write es or no. If no, eplain our reasoning. See Eample on page No; has 8. es 9. h() No; is an eponent other than. inside a square root. Write each equation in standard form. Identif A, B, and C. See Eample on page ; 7,, ;,, 8 9 7; 8, 9, 7 Find the -intercept and the -intercept of the graph of each equation. Then graph the equation. See Eample on page 6.. See margin for graphs.., ,. 9 9, 9 - Slope See pages Concept Summar The slope of a line is the ratio of the change in -coordinates to the corresponding change in -coordinates. m Lines with the same slope are parallel. Lines with slopes that are opposite reciprocals are perpendicular. 9.. (, ) (, ) (6, ) (, ) (, ) (, ) (, ) (, ) (, ) Chapter Stud Guide and Review Answers Chapter Stud Guide and Review

55 Stud Guide and Review Chapter Stud Guide and Review Answers 9. Eample Find the slope of the line that passes through (, ) and (7, 9). m Slope formula 9 ( 7 ( ), ) (, ), (, ) (7, 9) 6 or Simplif. Eercises Find the slope of the line that passes through each pair of points. See Eample on page ( 6, ), (6, 7) 6 7. (.,.), (, 7) 8. (, ), (, ). Graph the line passing through the given point with the given slope. See Eample on page See margin. 9. (, ), m. (, ), m. (, ), m Graph the line that satisfies each set of conditions. See Eamples and on pages 7 and 7.. See margin.. passes through (, ), parallel to a line whose slope is. passes through (, ), perpendicular to a line whose slope is. passes through (, ), perpendicular to graph of. passes through (, ), parallel to graph of.. - See pages 7 8. Eample Writing Linear Equations Concept Summar Slope-Intercept Form: m b Point-Slope Form: m( ) Write an equation in slope-intercept form for the line through (, ) that is parallel to the line through (, ) and (, ). First, find the slope of the given line. The parallel line will also have slope m. Slope formula m( ) Point-slope form ( ) (, ( ) ) = (, ), (, ) (, ), m ( ) (, ) (, ) 7 Slope-intercept form Simplif. Eercises Write an equation in slope-intercept form for the line that satisfies each set of conditions. See Eamples,, and on pages slope, passes through ( 6, 9) passes through (, 8) and (, ) 8. passes through (, ), parallel to the graph of 7 9. passes through (, ), perpendicular to the graph of 7 Chapter Linear Relations and Functions.. Chapter Linear Relations and Functions

56 Chapter Stud Guide and Review Stud Guide and Review - See pages Eample Modeling Real-World Data: Using Scatter Plots Concept Summar A scatter plot is a graph of ordered pairs of data. A prediction equation can be used to predict one of the variables given the other variable. WEEKLY PAY The table below shows the median weekl earnings for American workers for the period Predict the median weekl earnings for. Year Earnings ($) 79 9? Source: U.S. Bureau of Labor Statistics A scatter plot suggests that an two points could be used to find a prediction equation. Use (98, ) and (99, ). Median Weekl Earnings 7 m 6 79 Slope formula 9 (, ) (98, ), (, ) (99, ) 6 9 or.8 Simplif m( ) Point-slope form Year.8( 98).8 7, Substitute. Add to each side. Earnings ($) Source: U.S. Bureau of Labor Statistics Answer. People (millions) Year To predict the earnings for, substitute for..8() 7, 688 Simplif. The model predicts median weekl earnings of $688 in. Eercises For Eercises, use the table that shows the number of people below the povert level for the period See Eamples and on pages 8 and 8.. Draw a scatter plot. See margin.. Use two ordered pairs to write a prediction equation.. Use our prediction equation to predict the number for. Sample answer:. million. Sample answer using (98, 9.) and (99,.6):. 8. Year People (millions) ? Source: U.S. Census Bureau Chapter Stud Guide and Review Chapter Stud Guide and Review

57 Stud Guide and Review Etra Practice, see pages 8 8. Mied Problem Solving, see page 86. Answers See pages Special Functions Concept Summar Greatest Integer Constant Absolute Value Piecewise f() f() f() f() f() f(). Eample Graph the function f(). Identif the domain and range. The domain is all real numbers. The range is all real numbers greater than or equal to. f(). Eercises Graph each function. Identif the domain and range. See Eamples on pages See pp. 7A 7H.. f() [[]]. h() [[ ]]. g() 6. h() 7 7. f() if 8. g() if if if.. -7 See pages Eample Graphing Inequalities Concept Summar You can graph an inequalit b following these steps. Step Determine whether the boundar is solid or dashed. Graph the boundar. Step Choose a point not on the boundar and test it in the inequalit. Step If a true inequalit results, shade the region containing our test point. If a false inequalit results, shade the other region. Graph. Since the inequalit smbol is, the graph of the boundar should be solid. Graph the equation. Test (, ). + riginal inequalit () (, ) = (, ) Shade the region that contains (, ).. Eercises Graph each inequalit. See Eamples on pages 96 and See margin. Chapter Linear Relations and Functions. Chapter Linear Relations and Functions

58 Practice Test Practice Test Vocabular and Concepts Choose the correct term to complete each sentence.. The variable whose values make up the domain of a function is called the (independent, dependent) variable.. To find the (-intercept, -intercept) of the graph of a linear equation, let.. An equation of the form (A B C, m b) is in slope-intercept form. Skills and Applications Graph each relation and find the domain and range. Then determine whether the relation is a function.. See pp. 7A 7H for graphs.. {(, -8), (, ), (, ), (, ), (, 9)}. D all reals, R all reals; es D {,,,, ), R { 9, 8,,, }; es Find each value. 6. f() if f() 7 7. f() if f() = Graph each equation or inequalit See pp. 7A 7H f(). f() [[]]. g(). h() if if Find the slope of the line that passes through each pair of points.. (8, ), (6, ). (, ), (, ). (, 7), (, 6) Graph the line passing through the given point with the given slope.. See pp. 7A 7H.. (, ),. (, ),. (, ), undefined Write an equation in slope-intercept form for the line that satisfies each set of conditions. 6. slope, -intercept 7. -intercept 9, -intercept 9 8. passes through ( 6, ), parallel to the graph of 9. passes through (, ), perpendicular to the graph of 7 RECREATIN For Eercises, use the Year table that shows the amount Americans Amount ($ billions) spent on recreation in recent ears. Source: U.S. Bureau of Economic Analsis. Draw a scatter plot, where represents the number of ears since 99. See margin.. Write a prediction equation. Sample answer using (,.6) and (, 9.6): 8.6. Predict the amount that will be spent on recreation in. Sample answer: $8.6 billion. STANDARDIZED TEST PRACTICE What is the slope of a line parallel to ( )? D A B C D Portfolio Suggestion Chapter Practice Test Introduction In this chapter, ou have graphed man different kinds of functions. The appearances of these graphs were also ver different from one another. Ask Students Select one kind of graph that ou found difficult to master and eplain wh ou felt this to be the case. Suggest was that this topic might be presented in a different wa to help other students who have the same difficult. Assessment ptions Vocabular Test A vocabular test/review for Chapter can be found on p. of the Chapter Resource Masters. Chapter Tests There are si Chapter Tests and an pen- Ended Assessment task available in the Chapter Resource Masters. Chapter Tests Form Tpe Level Pages MC basic 99 A MC average B MC average C FR average 6 D FR average 7 8 FR advanced 9 pen-ended Assessment Performance tasks for Chapter can be found on p. of the Chapter Resource Masters. A sample scoring rubric for these tasks appears on p. A8. TestCheck and Worksheet Builder This networkable software has three modules for assessment. Worksheet Builder to make worksheets and tests. Student Module to take tests on-screen. Management Sstem to keep student records. Answer. Mone Spent on Recreation Amount ($ billions) MC = multiple-choice questions FR = free-response questions Years Since 99 Chapter Practice Test

59 Standardized Test Practice These two pages contain practice questions in the various formats that can be found on the most frequentl given standardized tests. A practice answer sheet for these two pages can be found on p. A of the Chapter Resource Masters. Standardized Test Practice NAME DATE PERID Standardized Test Practice Student Recording Sheet (Use with pages Sheet, 6 7 ofp. the Student AEdition.) Part Multiple Choice Select the best answer from the choices given and fill in the corresponding oval. A B C D A B C D 7 A B C D A B C D A B C D 8 A B C D A B C D 6 A B C D 9 A B C D Part Short Response/Grid In Solve the problem and write our answer in the blank. For Questions 7, also enter our answer b writing each number or smbol in a bo. Then fill in the corresponding oval for that number or smbol. 6 Part Multiple Choice Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. In the figure, B and BCD are right angles. B C is 9 units, A B is units, and C D is 8 units. What is the area, in square units, of ACD? A A 6 A B B 6 C 7 9 D D C 8. If is an even integer, then could be which of the following? B A B C D 6. If one side of a triangle is three times as long as a second side and the second side is s units long, then the length of the third side of the triangle can be A A s. B s. C s. D 6s. 7. Which of the following sets of numbers has the propert that the product of an two numbers is also a number in the set? D I the set of positive numbers II the set of prime numbers III the set of even integers A I onl B II onl C III onl D I and III onl 7 / / / / / / / / / / / / / / Answers. What is the slope of the line that contains the points (, 7) and (6, )? B A B C 8 D 8. If 7 7, then. D 7 A 7 B C 7 D Part Quantitative Comparison Select the best answer from the choices given and fill in the corresponding oval. 8 A B C D A B C D A B C D 9 A B C D A B C D Additional Practice See pp. 7 8 in the Chapter Resource Masters for additional standardized test practice.. In, Matt had a collection of music CDs. Since then he has given awa CDs, purchased 6 new CDs, and traded of his CDs to Kashan for of Kashan s CDs. Since, what has been the percent of increase in the number of CDs in Matt s collection? D A % B % C 7 % 6 %. If the product of ( ), ( ), and ( ) is equal to three times the sum of and, then. B A B 6 C D 9 D 9. The average (arithmetic mean) of r, s,, and is 8, and the average of and is. What is the average of r and s? D A B 6 C 8 D Test-Taking Tip Questions 9 n multiple-choice questions, tr to compute the answer first. Then compare our answer to the given answer choices. If ou don t find our answer among the choices, check our calculations. 6 Chapter Linear Relations and Functions Log n for Test Practice The Princeton Review offers additional test-taking tips and practice problems at their web site. Visit or TestCheck and Worksheet Builder Special banks of standardized test questions similar to those on the SAT, ACT, TIMSS 8, NAEP 8, and Algebra End-of-Course tests can be found on this CD-RM. 6 Chapter Linear Relations and Functions

60 Aligned and verified b Part Short Response/Grid In Part Quantitative Comparison Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. If n is a prime integer such that n n what is one possible value of n?,, 7, or 9. If A C is units, what is the value of t? t A B C. If.8 8., what is the value of? / or.. In ABC, what is the value of w z? B t / or. Compare the quantit in Column A and the quantit in Column B. Then determine whether: A the quantit in Column A is greater, B the quantit in Column B is greater, C the two quantities are equal, or D the relationship cannot be determined from the information given. Column A 8. m is an integer greater than. A 9. C P Q Column B m w A 6 z. In an election, a total of votes were cast for three candidates, A, B, and C. Candidate C received 8 votes. If candidate B received more votes than candidate C and candidate A received more votes than candidate B, what is the least number of votes that candidate A could have received? 6. If the points P(, ), Q(, ), and R(, ) are vertices of a triangle, what is the area of the triangle? 6. How man of the first one hundred positive integers contain the digit 7? 9 C the -coordinate of point Q. The cost of bananas and apples is $.. D cost of one apple. The average (arithmetic mean) of three integers,,, and z is. B. m A (, k) the -coordinate of point P cost of one banana the average (arithmetic mean) of,, z, and 9 7. A triangle has a base of length 7, and the other two sides are equal in length. If the lengths of the sides of the triangle are integers, what is the shortest possible length of a side? 9 ( k, ) the slope of line m Chapter Standardized Test Practice 7 Chapter Standardized Test Practice 7