Chapter 2 Resource Masters

Transcription

1 Chapter Resource Masters

2 Consumable Workbooks Man of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks. Stud Guide and Intervention Workbook X Skills Practice Workbook Practice Workbook ANSWERS FR WRKBKS The answers for Chapter of these workbooks can be found in the back of this Chapter Resource Masters booklet. Glencoe/McGraw-Hill Copright b The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced onl for classroom use; be provided to students, teacher, and families without charge; and be used solel in conjunction with Glencoe s Algebra. An other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: The McGraw-Hill Companies 8787 rion Place Columbus, H ISBN: Algebra Chapter Resource Masters

3 Contents Vocabular Builder vii Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson -4 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson -6 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson -7 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Chapter Assessment Chapter Test, Form Chapter Test, Form A Chapter Test, Form B Chapter Test, Form C Chapter Test, Form D Chapter Test, Form Chapter pen-ended Assessment Chapter Vocabular Test/Review Chapter Quizzes & Chapter Quizzes & Chapter Mid-Chapter Test Chapter Cumulative Review Chapter Standardized Test Practice Standardized Test Practice Student Recording Sheet A ANSWERS A A Glencoe/McGraw-Hill iii Glencoe Algebra

4 Teacher s Guide to Using the Chapter Resource Masters The Fast File Chapter Resource sstem allows ou to convenientl file the resources ou use most often. The Chapter Resource Masters includes the core materials needed for Chapter. These materials include worksheets, etensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Algebra TeacherWorks CD-RM. Vocabular Builder Pages vii viii include a student stud tool that presents up to twent of the ke vocabular terms from the chapter. Students are to record definitions and/or eamples for each term. You ma suggest that students highlight or star the terms with which the are not familiar. WHEN T USE Give these pages to students before beginning Lesson -. Encourage them to add these pages to their Algebra Stud Notebook. Remind them to add definitions and eamples as the complete each lesson. Stud Guide and Intervention Each lesson in Algebra addresses two objectives. There is one Stud Guide and Intervention master for each objective. WHEN T USE Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Skills Practice There is one master for each lesson. These provide computational practice at a basic level. WHEN T USE These masters can be used with students who have weaker mathematics backgrounds or need additional reinforcement. Practice There is one master for each lesson. These problems more closel follow the structure of the Practice and Appl section of the Student Edition eercises. These eercises are of average difficult. WHEN T USE These provide additional practice options or ma be used as homework for second da teaching of the lesson. Reading to Learn Mathematics ne master is included for each lesson. The first section of each master asks questions about the opening paragraph of the lesson in the Student Edition. Additional questions ask students to interpret the contet of and relationships among terms in the lesson. Finall, students are asked to summarize what the have learned using various representation techniques. WHEN T USE This master can be used as a stud tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner) students. Enrichment There is one etension master for each lesson. These activities ma etend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students perspectives on the mathematics the are learning. These are not written eclusivel for honors students, but are accessible for use with all levels of students. WHEN T USE These ma be used as etra credit, short-term projects, or as activities for das when class periods are shortened. Glencoe/McGraw-Hill iv Glencoe Algebra

5 Assessment ptions The assessment masters in the Chapter Resource Masters offer a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. Chapter Assessment CHAPTER TESTS Form contains multiple-choice questions and is intended for use with basic level students. Forms A and B contain multiple-choice questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Forms C and D are composed of freeresponse questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Grids with aes are provided for questions assessing graphing skills. Form is an advanced level test with free-response questions. Grids without aes are provided for questions assessing graphing skills. All of the above tests include a freeresponse Bonus question. The pen-ended Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. A Vocabular Test, suitable for all students, includes a list of the vocabular words in the chapter and ten questions assessing students knowledge of those terms. This can also be used in conjunction with one of the chapter tests or as a review worksheet. Intermediate Assessment Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. A Mid-Chapter Test provides an option to assess the first half of the chapter. It is composed of both multiple-choice and free-response questions. Continuing Assessment The Cumulative Review provides students an opportunit to reinforce and retain skills as the proceed through their stud of Algebra. It can also be used as a test. This master includes free-response questions. The Standardized Test Practice offers continuing review of algebra concepts in various formats, which ma appear on the standardized tests that the ma encounter. This practice includes multiplechoice, grid-in, and quantitativecomparison questions. Bubble-in and grid-in answer sections are provided on the master. Answers Page A is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on pages This improves students familiarit with the answer formats the ma encounter in test taking. The answers for the lesson-b-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer kes are provided for the assessment masters in this booklet. Glencoe/McGraw-Hill v Glencoe Algebra

6 NAME DATE PERID Reading to Learn Mathematics Vocabular Builder This is an alphabetical list of the ke vocabular terms ou will learn in Chapter. As ou stud the chapter, complete each term s definition or description. Remember to add the page number where ou found the term. Add these pages to our Algebra Stud Notebook to review vocabular at the end of the chapter. Vocabular Term absolute value function Found on Page Definition/Description/Eample Vocabular Builder boundar constant function famil of graphs function greatest integer function identit function linear equation line of fit one-to-one function (continued on the net page) Glencoe/McGraw-Hill vii Glencoe Algebra

7 NAME DATE PERID Reading to Learn Mathematics Vocabular Builder (continued) parent graph Vocabular Term Found on Page Definition/Description/Eample piecewise function PEES WYZ point-slope form prediction equation pree DIHK shuhn relation scatter plot slope slope-intercept form IHN tuhr SEHPT standard form step function Glencoe/McGraw-Hill viii Glencoe Algebra

8 NAME DATE PERID - Stud Guide and Intervention 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. 0 Lesson - Eercises Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.. {(, ), (, ),. {(, 4), (, 0),. {(0, 4), (, ), (, ), (, )} (, ), (, )} (, ), (, )} D {,,, }, D {,, }, D {, 0,, }, R {, }; es R { 4,, 0, }; no R {,,, 4}; es D all reals, D all reals, D all reals, R { }; es R all reals; es R all reals; es Glencoe/McGraw-Hill 7 Glencoe Algebra

9 - NAME DATE PERID Stud Guide and Intervention (continued) Relations and Functions Equations of Functions and Relations Equations that represent functions are often written in functional notation. For eample, 0 8 can be written as f() 0 8. This notation emphasizes the fact that the values of, the dependent variable, depend on the values of, the independent variable. To evaluate a function, or find a functional value, means to substitute a given value in the domain into the equation to find the corresponding element in the range. Eample Given the function f(), find each value. a. f() f() f() () riginal function Substitute. Simplif. b. f(a) f() f(a) (a) (a) a 0a riginal function Substitute. Simplif. Eercises Find each value if f() 4.. f() 0. f(6) 8. f(b) 4b 4 Find each value if g(). 4. g() 0. g( ) 6 6. g(7c) 4c 7c Find each value if f() and g() f(0.) 8. f( 8) 6 9. g() g(.).. f(4a) 8a. g b b. a 0. f 6 4. g(0) 8.8. f(00) Let f(). 6. Find the values of f() and f(). f () 7, f () Compare the values of f() f() and f( ). f () f () 4, f ( ) 99 Glencoe/McGraw-Hill 8 Glencoe Algebra

10 - NAME DATE PERID Skills Practice Relations and Functions Determine whether each relation is a function. Write es or no.. D R es. D R no es 4. no 4 6 Lesson - Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.. {(, ), (, 4), (, )} 6. {(, 6), (6, )} D {}, R {,, 4}; no D {, 6}, R {, 6}; es 7. {(, 4), (, 4), (, ), (, )} 8. D {,,, }, R {, 4}; es D { }, R all reals; no Find each value if f() and g(). 9. f(0) 0. f(). g(4) 4. f( ). g( ) 4. f(d) d Glencoe/McGraw-Hill 9 Glencoe Algebra

11 - NAME DATE PERID Practice (Average) Relations and Functions Determine whether each relation is a function. Write es or no.. D R no. D R es es 4. no Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.. {( 4, ), (4, 0), (0, ), (, 0)} 6. D { 4, 0,, 4}, R {, 0, }; es D all reals, R all reals; es Find each value if f() and g(). 7. f() 8. f( 4) 9. g 0. f( ) undefined. g( 6). f(m ) m. MUSIC The ordered pairs (, 6), (, 6), (, ), (4, ), and (, 48) 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 {,,, 4, }, R {6,, 48}; es 4. 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 calculations. How long would it take the computer to do billion calculations? 7. s Glencoe/McGraw-Hill 60 Glencoe Algebra

12 - NAME DATE PERID Reading to Learn Mathematics Relations and Functions Pre-Activit How do relations and functions appl to biolog? Reading the Lesson Read the introduction to Lesson - at the top of page 6 in our tetbook. Refer to the table. What does the ordered pair (8, 0) tell ou? For a deer, the average longevit is 8 ears and the maimum longevit is 0 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.. 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. Lesson -. a. Write the domain and range of the relation shown in the graph. (, ) (0, 4) (, ) (, 0) (, ) (, 4) D: {,,, 0, }; R: {, 4, 0,,, 4} 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. Glencoe/McGraw-Hill 6 Glencoe Algebra

13 - NAME DATE PERID Enrichment Mappings 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. Into mapping nto mapping A mapping from set A to set B where ever element of A is mapped to one or more 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 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 a g 6 9 k l 9 q into, onto into, onto into, onto, into, onto one-to-one into into, onto into, onto into, onto, one-to-one 9. Can a set be mapped onto a set with fewer elements than it has? es 0. Can a set be mapped into a set that has more elements than it has? es. If a mapping from set A into set B is a one-to-one correspondence, what can ou conclude about the number of elements in A and B? The sets have the same number of elements. Glencoe/McGraw-Hill 6 Glencoe Algebra

14 - NAME DATE PERID Stud Guide and Intervention 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 Eample Is f() 0. a linear function? Eplain. Yes; it is a linear function because it can be written in the form f() 0.. Eample Is 0 a linear function? Eplain. No; it is not a linear function because the variables and are multiplied together in the middle term. Find the -intercept and the -intercept of the graph of 4 0. Then graph the equation. The -intercept is the value of when riginal equation 4 (0) 0 Substitute 0 for. Simplif. So the -intercept is. Similarl, the -intercept is 4. Lesson - 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: 0; -int: -int: ; -int:. 8 Glencoe/McGraw-Hill 6 Glencoe Algebra

15 - NAME DATE PERID Stud Guide and Intervention (continued) Linear Equations Standard Form The standard form of a linear equation is A B C, where A, B, and C are integers whose greatest common factor is. Eample a Write each equation in standard form. Identif A, B, and C. riginal equation So A 8, B, and C. Eercises Subtract 8 from each side. Multipl each side b. b riginal equation 4 7 Add 7 to each side. Divide each side b 7. So A, B, and C. Write each equation in standard form. Identif A, B, and C ; A, ; A, ; A, B 4, C B, C B, C ; A 8, ; A 8, 4 ; A 4, B 6, C B 9, C 60 B, C ; A, 4 7; A, 6; A, B, C 0 B 4, C 7 B, C ; A 6, ; A, 6; A, B, C 0 B, C B 0, C ; A 9, 6; A, 4 0; A, B, C 6 B, C 6 B 4, C ; A 8, 0 ; A 0, 8 ; A 8, B, C B, C B, C ; A, B, C 4 ; A, B, C 00; A, B, C 00 Glencoe/McGraw-Hill 64 Glencoe Algebra

16 - NAME DATE PERID Skills Practice Linear Equations State whether each equation or function is linear. Write es or no. If no, eplain our reasoning... es es f() 4 es No; the eponent of is not No; is in a denominator. es 7. g() 8 8. h() es No; is inside a square root. Write each equation in standard form. Identif A, B, and C. 9. 0;,, 0 0. ;,, ;, 7, 4. ;,, Lesson ; 0,, ; 4,, 7 Find the -intercept and the -intercept of the graph of each equation. Then graph the equation.. 6, , 0 7., 8. 0, Glencoe/McGraw-Hill 6 Glencoe Algebra

17 - State whether each equation or function is linear. Write es or no. If no, eplain our reasoning.. h() es. es NAME DATE PERID Practice (Average) Linear Equations. No; is a denominator No; and are multiplied. Write each equation in standard form. Identif A, B, and C ; 7,, ;, 8, ; 0,, ; 8, 8, Find the -intercept and the -intercept of the graph of each equation. Then graph the equation. 9. 4, ,. 4, ,. MEASURE The equation.4 gives the length in centimeters corresponding to a length in inches. What is the length in centimeters of a -foot ruler? 0.48 cm LNG DISTANCE For Eercises 4 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 0.0t, where t is the number of minutes talked. 4. What is the total cost of talking 8 hours? of talking 0 hours? $0; $66. What is the effective cost per minute (the total cost divided b the number of minutes talked) of talking 8 hours? of talking 0 hours? $0.06; $0.0 Glencoe/McGraw-Hill 66 Glencoe Algebra

18 - NAME DATE PERID Reading to Learn Mathematics Linear Equations Pre-Activit How do linear equations relate to time spent studing? Reading the Lesson 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.. 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 c. 7 es Lesson d. No; B is not an integer. e No; A and B are both 0. f. 4 8 No; The greatest common factor of, 4, 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 0, then the graph is a horizontal line. If B 0, 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 0 to find the -intercept and let 0 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 0, so let 0 to find an -intercept. The -intercept is the -coordinate of a point on the -ais. Ever point on the -ais has -coordinate 0, so let 0 to find a -intercept. Glencoe/McGraw-Hill 67 Glencoe Algebra

19 NAME DATE PERID - Enrichment 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 0. The process can be written as follows. 88 4() 6 () 4 6() 0 () 6 0() 6 () 0 6() (4) 6 () The last nonzero remainder is the GCF of the two coefficients. If the constant term 7 is divisible b the GCF, then integers and do eist that satisf the equation. To find and, work backward in the following manner [0 6()] Substitute for using (4) 6(0) 8(6) 6(0) 8[6 0()] Substitute for 6 using () 8(6) 4(0) 8(6) 4[4 6()] Substitute for 0 using () 4(4) 66(6) 4(4) 66[88 4()] Substitute for 6 using () 88( 66) 4(90) Thus, 66 and 90. Find integers and, if the eist, that satisf each equation Glencoe/McGraw-Hill 68 Glencoe Algebra

20 - Slope NAME DATE PERID Stud Guide and Intervention Slope change in Slope m of a Line For points (, ) and (, ), where, m change in Eample Eample Determine the slope of the line that passes through (, ) and ( 4, ). m Slope formula ( ) (, ) (, ), (, ) ( 4, ) 4 6 Simplif. 6 The slope of the line is. Eercises Graph the line passing 4 through (, ) with a slope of. Graph the ordered pair (, ). Then, according to the slope, go up 4 units and right units. Plot the new point (4,). Connect the points and draw the line. Find the slope of the line that passes through each pair of points.. (4, 7) and (6, ). (6, 4) and (, 4) 0. (, ) and (7, ) 4. (, ) and ( 4, ). (, 0) and (, ) 6. (, 4) and (, ) 7. (7, ) and (, ) 8. (, 9) and (, ) 9. (4, ) and ( 4, 8) 4 4 Graph the line passing through the given point with the given slope. 0. slope. slope. slope 0 passes through (0, ) passes through (, 4) passes through (, ) Lesson -. slope 4. slope. slope 4 passes through ( 4, 6) passes through (, 0) passes through (0, 0) Glencoe/McGraw-Hill 69 Glencoe Algebra

21 - NAME DATE PERID Stud Guide and Intervention (continued) Slope Parallel and Perpendicular Lines In a plane, nonvertical lines with the same slope are parallel. All vertical lines are parallel. slope m In a plane, two oblique lines are perpendicular if and onl if the product of their slopes is. An vertical line is perpendicular to an horizontal line. slope m slope m slope m Eample Are the line passing through (, 6) and (, ) and the line passing through (, 0) and (0, 4) parallel, perpendicular, or neither? Find the slopes of the two lines. 6 The slope of the first line is. ( ) The slope of the second line is. 0 The slopes are not equal and the product of the slopes is not, so the lines are neither parallel nor perpendicular. Eercises Are the lines parallel, perpendicular, or neither?. the line passing through (4, ) and (. ) and the line passing through (, ) and (, ) perpendicular. the line passing through (, 8) and (, ) and the line passing through (0, 9) and (6, 0) neither. the line passing through (, 9) and (, ) and the graph of parallel 4. the line with -intercept and -intercept and the line with -intercept and -intercept parallel. the line with -intercept and -intercept and the line with -intercept and -intercept neither 6. the line passing through (, ) and (, ) and the graph of 0 perpendicular 7. the line passing through ( 4, 8) and (6, 4) and the graph of parallel Glencoe/McGraw-Hill 70 Glencoe Algebra

22 - NAME DATE PERID Skills Practice Slope Find the slope of the line that passes through each pair of points.. (, ), (, ) 4. (0, ), (, 0). (, 9), (0, 6) 4. (8, ), (4, ). (, ), (, ) undefined 6. (, ), (0, ) (4, ), (, 7) 8. (, 4), (, ) 9. (, ), (, ) 0 Graph the line passing through the given point with the given slope. 0. (0, 4), m. (, 4), m. (, ), m. (, ), m Lesson - Graph the line that satisfies each set of conditions. 4. passes through (0, ), perpendicular to. passes through (0, ), parallel to the a line whose slope is graph of 6. HIKING Naomi left from an elevation of 7400 feet at 7:00 A.M. and hiked to an elevation of 9800 feet b :00 A.M. What was her rate of change in altitude? 600 ft/h Glencoe/McGraw-Hill 7 Glencoe Algebra

23 - NAME DATE PERID Practice (Average) Slope Find the slope of the line that passes through each pair of points.. (, 8), (, ). ( 0, ), (7, ). ( 7, 6), (, 6) (8, ), (8, ) undefined. (4, ), (7, ) 6. ( 6, ), ( 8, 4) Graph the line passing through the given point with the given slope. 7. (0, ), m 8. (, ), m 4 9. (0, ), m 0 0. (, ), m 4 Graph the line that satisfies each set of conditions.. passes through (, 0), perpendicular. passes through (, ), parallel to a line to a line whose slope is whose slope is DEPRECIATIN For Eercises, use the following information. A machine that originall cost $,600 has a value of $700 at the end of ears. The same machine has a value of $800 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. $700 per ear 4. Find the average rate of change in value of the machine between the end of ears and the end of 8 ears. $940 per ear. Interpret the sign of our answers. It is negative because the value is decreasing. Glencoe/McGraw-Hill 7 Glencoe Algebra

24 - NAME 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 40 feet over a horizontal distance of 000 feet? 4% What is the grade of a road that rises meters over a horizontal distance of 0 kilometers? ( kilometer 000 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. Zero The line is horizontal. Negative The line falls to the right. Lesson - 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. 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. Glencoe/McGraw-Hill 7 Glencoe Algebra

25 - NAME DATE PERID Enrichment 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, ) Step Step Step 4 Find D, the sum of the downward diagonal products (from left to right). D ( ) ( ) ( ) (6 7) 6 4 or 7 Find U, the sum of the upward diagonal products (from left to right). U ( 7) ( ) (6 ) ( ) or 4 Use the formula A (D U) to find the area. A (7 4) (0) or (, 7) (, ) (, ) (6, ) (, 7) The area is square units. Count the number of square units enclosed b the polgon. Does this result seem reasonable? Use the coordinate method to find the area of each region in square units.... Glencoe/McGraw-Hill 74 Glencoe Algebra

26 -4 NAME DATE PERID Stud Guide and Intervention 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 Eample Eample Write an equation in slope-intercept form for the line that has slope and passes through the point (, 7). Substitute for m,, and in the slope-intercept form. m b Slope-intercept form 7 ( )() b (, ) (, 7), m 7 6 b Simplif. b Add 6 to both sides. The -intercept is. The equation in slope-intercept form is. Eercises Write an equation in slope-intercept form for the line that has slope and -intercept. m b Slope-intercept form 0 () b (, ) (, 0), m 0 b Simplif. b Subtract from both sides. The -intercept is. The slope-intercept form is. Write an equation in slope-intercept form for the line that satisfies each set of conditions.. slope, passes through ( 4, 6). slope, -intercept 4 4. slope, passes through (, ) 4. slope, passes through (, 7) 6 Write an equation in slope-intercept form for each graph (, 6) (4, ) (, ) ( 4, ) Lesson -4 (, 0) (0, 0) Glencoe/McGraw-Hill 7 Glencoe Algebra

27 -4 NAME DATE PERID Stud Guide and Intervention (continued) Writing Linear Equations Parallel and Perpendicular Lines Use the slope-intercept or point-slope form to find equations of lines that are parallel or perpendicular to a given line. Remember that parallel lines have equal slope. The slopes of two perpendicular lines are negative reciprocals, that is, their product is. Eample Eample Write an equation of the line that passes through (8, ) and is perpendicular to the line whose equation is. The slope of the given line is. Since the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line is. Use the slope and the given point to write the equation. m( ) Point-slope form ( 8) (, ) (8, ), m 6 Distributive Prop. 4 Add to each side. An equation of the line is 4. Eercises Write an equation of the line that passes through (, ) and is parallel to the graph of. The slope of the given line is. Since the slopes of parallel lines are equal, the slope of the parallel line is also. Use the slope and the given point to write the equation. m( ) Point-slope form ( ( )) (, ) (, ), m Distributive Prop. 8 Add to each side. An equation of the line is 8. Write an equation in slope-intercept form for the line that satisfies each set of conditions.. passes through ( 4, ), parallel to the line whose equation is 4. passes through (, ), perpendicular to the graph of. passes through (, ), parallel to the line that passes through (4, ) and (, ) 4. passes through (4, 7), perpendicular to the line that passes through (, 6) and (, ) 7. passes through (8, 6), perpendicular to the graph of 4 6. passes through (, ), perpendicular to the graph of 6 7. passes through (6, ), parallel to the line with -intercept and -intercept 9 8. passes through (, ), perpendicular to the line 4 4 Glencoe/McGraw-Hill 76 Glencoe Algebra

28 -4 NAME DATE PERID Skills Practice Writing Linear Equations State the slope and -intercept of the graph of each equation.. 7 7,.,., , , , 7 7., 8. 6, Write an equation in slope-intercept form for each graph (, ) (0, ) (, ) (4, ) (, 4) (, ) Write an equation in slope-intercept form for the line that satisfies each set of conditions.. slope, passes through (, ). slope, passes through (0, 0) 6 4. slope, passes through (0, ). slope, passes through (, 0) 6 6. passes through (, ) and (, ) 7. passes through (, 4) and (, 8) Lesson intercept, -intercept intercept, -intercept 6 0. passes through (, ), perpendicular to the graph of 4. 0 Glencoe/McGraw-Hill 77 Glencoe Algebra

29 -4 NAME DATE PERID Practice (Average) Writing Linear Equations State the slope and -intercept of the graph of each equation.. 8 8, ,., ,., 6. 0, Write an equation in slope-intercept form for each graph (4, 4) (0, ) (, ) (0, ) (, ) Write an equation in slope-intercept form for the line that satisfies each set of conditions. 0. slope, passes through (, 8) 4. slope, passes through (0, ) 4. slope 0, passes through (0, 0). 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 RESERVIRS The surface of Grand Lake is at an elevation of 648 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 $0,000 profit if it manufactures 00,000 drives, and $,70,000 profit if it manufactures 00,000 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 4n 0,000 Glencoe/McGraw-Hill 78 Glencoe Algebra

30 -4 NAME DATE PERID Reading to Learn Mathematics Writing Linear Equations Pre-Activit How do linear equations appl to business? Reading the Lesson Read the introduction to Lesson -4 at the top of page 7 in our tetbook. If the total cost of producing a product is given b the equation 400.7, what is the fied cost? What is the variable cost (for each item produced)? $400; $.7 Write a linear equation that describes the following situation: A compan that manufactures computers has a fied cost of $8,70 and a variable cost of $8 to produce each computer. 8,70 8. 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. 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 4. 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 4. 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 Lesson -4 equation b.drop the subscripts in and.this gives the point-slope form of the equation of a line. Glencoe/McGraw-Hill 79 Glencoe Algebra

31 -4 NAME DATE PERID Enrichment 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 a b with -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 (, 0) and (0, 6), then draw a straight line through them. Eample 4 Write 4 in two-intercept form. Divide b to obtain on the right side. 4 Simplif. The -intercept is 4; the -intercept is. Use the given intercepts a and b, to write an equation in two-intercept form. Then draw the graph. See students graphs.. a, b 4. a, b 8. a, b 4. a 6, b 9 Write each equation in two-intercept form. Then draw the graph Glencoe/McGraw-Hill 80 Glencoe Algebra

32 - NAME DATE PERID Stud Guide and Intervention 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 990s. Make a scatter plot of the data. Home Runs Source: New York Times Almanac Make a scatter plot for the data in each table below.. FUEL EFFICIENCY The table below shows the average fuel efficienc in miles per gallon of new cars manufactured during the ears listed. Year Runs Batted In Eercises Fuel Efficienc (mpg) Year 990 Source: New York Times Almanac Runs Batted In MVP HRs and RBIs Home Runs Miles per Gallon Average Fuel Efficienc CNGRESS The table below shows the number of women serving in the United States Congress during the ears Congressional Session Source: Wall Street Journal Almanac Number of Women Number of Women Women in Congress Session of Congress Lesson - Glencoe/McGraw-Hill 8 Glencoe Algebra

33 - Prediction Equations A line of fit is a line that closel approimates a set of data graphed in a scatter plot. The equation of a line of fit is called a prediction equation because it can be used to predict values not given in the data set. To find a prediction equation for a set of data, select two points that seem to represent the data well. Then to write the prediction equation, use what ou know about writing a linear equation when given two points on the line. Eample STRAGE CSTS According to a certain prediction equation, the cost of 00 square feet of storage space is $60. The cost of square feet of storage space is $60. a. Find the slope of the prediction equation. What does it represent? Since the cost depends upon the square footage, let represent the amount of storage space in square feet and represent the cost in dollars. The slope can be found using the formula m So,m The slope of the prediction equation is 0.8. This means that the price of storage increases 80 for each one-square-foot increase in storage space. b. Find a prediction equation. Using the slope and one of the points on the line, ou can use the point-slope form to find a prediction equation. m( ) Point-slope form ( 00) (, ) (00, 60), m Distributive Propert Add 60 to both sides. A prediction equation is Eercises NAME DATE PERID Stud Guide and Intervention (continued) Modeling Real-World Data: Using Scatter Plots SALARIES The table below shows the ears of eperience for eight technicians at Lewis Techomatic and the hourl rate of pa each technician earns. Use the data for Eercises and. Eperience (ears) Hourl Rate of Pa $7 $0 $0 $7 $9 $ $0 $. Draw a scatter plot to show how ears of eperience are related to hourl rate of pa. Draw a line of fit. See graph.. Write a prediction equation to show how ears of eperience () are related to hourl rate of pa (). Sample answer using (, 7) and (9, 7):..7 Hourl Pa ($) Technician Salaries Eperience (ears) Glencoe/McGraw-Hill 8 Glencoe Algebra

34 - NAME DATE PERID Skills Practice Modeling Real-World 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.. a ? b. Sample answer using (, ) and (8, ): c. Sample answer: 9. a ? b. Sample answer using (, 9) and (40, 44): 4 c. Sample answer: 4. a. 6 6? b. Sample answer using (, 6) and (7, 4): c. Sample answer: 9.6 Lesson - Glencoe/McGraw-Hill 8 Glencoe Algebra

35 - NAME DATE PERID Practice (Average) Modeling Real-World 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 approimate weights in tons and estimates for overall fuel econom in miles per gallon for several cars. b. Sample answer using (.4, 4) and (.4, ): c. Sample answer: 8.6 mi/gal Weight (tons) Miles per Gallon 9 4? 7 Fuel Econom (mi/gal) Fuel Econom Versus Weight 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) ,400,000 Temperature ( F) ? b. Sample answer using (700, 6) and (9700, 0): 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. Time (min) Calories Burned ? Temperature ( F) Temperature Versus Altitude 0 7,000 8,000 9,000 0,000 Altitude (ft) b. Sample answer using (4, 80) and (48, 440): c. Sample answer: about 0 calories Glencoe/McGraw-Hill 84 Glencoe Algebra

36 - NAME DATE PERID Reading to Learn Mathematics 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.. Suppose that tuition at a state college was $00 per ear in 99 and has been increasing at a rate of $ per ear. a. Write a prediction equation that epresses this information. 00 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 007. $600 Helping You Remember 4. 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 - Glencoe/McGraw-Hill 8 Glencoe Algebra

37 - NAME DATE PERID Enrichment 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 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 enders. Find,, and, the medians of the values in groups,, and, respectivel. Find,, and, the medians of the values in groups,, and, respectivel. 98, 988, 994;,, 9. Find an equation of the line through (, ) and (, ) Find Y, the -coordinate of the point on the line in Eercise with an -coordinate of.. The median-fit line is parallel to the line in Eercise, but is one-third closer to (, ). This means it passes through, Y.Find this ordered pair. about (988,.67) 6. Write an equation of the median-fit line Use the median-fit line to predict the percentage of juvenile violent crime offenders in 00 and : about %; 00: about6% Glencoe/McGraw-Hill 86 Glencoe Algebra

38 -6 NAME DATE PERID Stud Guide and Intervention 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 Lesson -6 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. b. f() f() a constant function a step function Eercises 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 Glencoe/McGraw-Hill 87 Glencoe Algebra

39 -6 NAME DATE PERID Stud Guide and Intervention (continued) Special Functions Absolute Value and Piecewise Functions Another special function is the absolute value function, which is also called a piecewise function. Absolute Value Function f() two ras that are mirror images of each other and meet at a point, the verte To graph a special function, use its definition and our knowledge of the parent graph. Find several ordered pairs, if necessar. Eample Graph f() 4. Find several ordered pairs. Graph the points and connect them. You would epect the graph to look similar to its parent function, f() f() Eample Graph f() if if. First, graph the linear function f() for. Since does not satisf this inequalit, stop with a circle at (, 4). Net, graph the linear function f() for. Since does satisf this inequalit, begin with a dot at (, ). f() Eercises Graph each function. Identif the domain and range.. g(). h(). h( ) if 0 6 if 0 if domain: all real domain: all real domain: all real numbers; range: numbers; range: numbers; range: all integers { 0} { } Glencoe/McGraw-Hill 88 Glencoe Algebra

40 -6 NAME DATE PERID Skills Practice Special Functions Identif each function as S for step, C for constant, A for absolute value, or P for piecewise.... Lesson -6 S C A Graph each function. Identif the domain and range. 4. f(). f() f() f() D all reals, R all integers D all reals, R all integers 6. g() 7. f() g() f() D all reals, D all reals, R { } R nonnegative reals 8. f() if 0 9. h() if if 0 if > f() h() D all reals, D { or }, R { 0 or } R { } Glencoe/McGraw-Hill 89 Glencoe Algebra

41 -6 NAME DATE PERID Practice (Average) Special Functions Graph each function. Identif the domain and range.. f() 0.. f() f() f() D all reals, R all integers D all reals, R all integers. g() 4. f() g() f() D all reals, D all reals, R nonpositive reals R nonnegative reals. f() if 6. h() 4 if 0 if if 0 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 $.00 $40 per hour or an fraction thereof per pound for less than 0 pounds of cand and for labor. Draw a graph of the step $.0 per pound for 0 or more pounds. Draw a function that represents this situation. Labor Costs graph of the function that represents this situation. Total Cost ($) Hours Glencoe/McGraw-Hill 90 Glencoe Algebra

42 -6 NAME DATE PERID Reading to Learn Mathematics Special Functions 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 0. ounce? $0.4 or 4 cents Give three different weights of letters that would each cost cents to mail. Answers will var. Sample answer:. ounces,.9 ounces,.0 ounces Lesson -6 Reading the Lesson. Find the value of each epression. a. b c 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 0 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. Glencoe/McGraw-Hill 9 Glencoe Algebra

43 -6 NAME DATE PERID Enrichment 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 Glencoe/McGraw-Hill 9 Glencoe Algebra

44 -7 NAME DATE PERID Stud Guide and Intervention 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. (0, 0) 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 4. The boundar is the graph of 4. Use the slope-intercept form,, to graph the boundar line. The boundar line should be solid. Now test the point (0, 0). Lesson -7 0 (0)? 4 (, ) (0, 0) 0 4 false Shade the region that does not contain (0, 0). Eercises Graph each inequalit Glencoe/McGraw-Hill 9 Glencoe Algebra

45 NAME DATE PERID -7 Stud Guide and Intervention (continued) Graphing Inequalities Graph Absolute Value Inequalities Graphing absolute value inequalities is similar to graphing linear inequalities. The graph of the related absolute value equation is the boundar. This boundar is graphed as a solid line if the inequalit is or, and dashed if the inequalit is or. Choose a test point not on the boundar to determine which region to shade. Eample Graph. First graph the equation. Since the inequalit is, the graph of the boundar is solid. Test (0, 0). 0? 0 (, ) (0, 0) 0? 0 true Shade the region that contains (0, 0). Eercises Graph each inequalit Glencoe/McGraw-Hill 94 Glencoe Algebra

46 -7 NAME DATE PERID Skills Practice Graphing Inequalities Graph each inequalit Lesson Glencoe/McGraw-Hill 9 Glencoe Algebra

47 -7 NAME DATE PERID Practice (Average) Graphing Inequalities Graph each inequalit CMPUTERS For Eercises 0, use the following information. A school sstem is buing new computers. The will bu desktop computers costing $000 per unit, and notebook computers costing $00 per unit. The total cost of the computers cannot eceed $80, Write an inequalit that describes this situation. 000d 00n 80,000. Graph the inequalit.. If the school wants to bu 0 of the desktop computers and of the notebook computers, will the have enough mone? es Notebooks Computers Purchased Desktops Glencoe/McGraw-Hill 96 Glencoe Algebra

48 -7 NAME DATE PERID Reading to Learn Mathematics Graphing Inequalities Pre-Activit How do inequalities appl to fantas football? Reading the Lesson 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?. 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. Lesson -7. 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 4. 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. Glencoe/McGraw-Hill 97 Glencoe Algebra

49 -7 NAME DATE PERID Enrichment Algebraic Proof The following paragraph states a result ou might be asked to prove in a mathematics course. Parts of the paragraph are numbered. 0 Let n be a positive integer. 0 Also, let n s(n ) be the sum of the squares of the digits in n. 0 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. 04 In general, n k s(n k ) is the sum of the squares of the digits of n k. 0 Consider the sequence: n, n, n, n,, n k,. 06 In this sequence either all the terms from some k on have the value, 07 or some term, sa n j, has the value 4, so that the eight terms 4, 6, 7, 8, 89, 4, 4, and 0 keep repeating from that point on. Use the paragraph to answer these questions.. Use the sentence in line 0. List the first five values of n.. Use 946 for n and give an eample to show the meaning of line 0.. In line 0, which smbol shows a function? Eplain the function in a sentence. 4. For n 946, find n and n as described in sentence 0.. How do the first four sentences relate to sentence 0? 6. Use n and find the first four terms of the sequence. 7. Which sentence of the paragraph is illustrated b n? 8. Use n 6 and find the first ten terms. 9. Which sentence is illustrated b n 6? Glencoe/McGraw-Hill 98 Glencoe Algebra

50 Answers (Lesson -) Lesson - - NAME DATE PERID Stud Guide and Intervention 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. 0 Eercises Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.. {(, ), (, ),. {(, 4), (, 0),. {(0, 4), (, ), (, ), (, )} (, ), (, )} (, ), (, )} D {,,, }, D {,, }, D {, 0,, }, R {, }; es R { 4,, 0, }; no R {,,, 4}; es D all reals, D all reals, D all reals, R { }; es R all reals; es R all reals; es Glencoe/McGraw-Hill 7 Glencoe Algebra - NAME DATE PERID Stud Guide and Intervention (continued) Relations and Functions Equations of Functions and Relations Equations that represent functions are often written in functional notation. For eample, 0 8 can be written as f() 0 8. This notation emphasizes the fact that the values of, the dependent variable, depend on the values of, the independent variable. To evaluate a function, or find a functional value, means to substitute a given value in the domain into the equation to find the corresponding element in the range. Eample Given the function f(), find each value. a. f() f() riginal function f() () Substitute. Simplif. b. f(a) f() riginal function f(a) (a) (a) Substitute. a 0a Simplif. Eercises Find each value if f() 4.. f() 0. f(6) 8. f(b) 4b 4 Find each value if g(). 4. g() 0. g( ) 6 6. g(7c) 4c 7c Find each value if f() and g() f(0.) 8. f( 8) g().4 0. g(.).. f(4a) 8a. g.. f 6 4. g(0) 8.8. f(00) a b b 0 Let f(). 6. Find the values of f() and f(). f () 7, f () Compare the values of f() f() and f( ). f () f () 4, f ( ) 99 Glencoe/McGraw-Hill 8 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

51 Answers (Lesson -) - NAME DATE PERID Skills Practice Relations and Functions Glencoe/McGraw-Hill 9 Glencoe Algebra Answers Lesson - Determine whether each relation is a function. Write es or no.. D R es. D R no es 4. no 4 6 Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.. {(, ), (, 4), (, )} 6. {(, 6), (6, )} (, 4) (, 6) (6, ) (, ) (, ) D {}, R {,, 4}; no D {, 6}, R {, 6}; es 7. {(, 4), (, 4), (, ), (, )} 8. (, 4) (, 4) (, ) (, ) D {,,, }, D { }, R all reals; no R {, 4}; es Find each value if f() and g(). 9. f(0) 0. f(). g(4) 4. f( ). g( ) 4. f(d) d NAME DATE PERID - Practice (Average) Relations and Functions Determine whether each relation is a function. Write es or no.. D R no. D R es es 4. no Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function.. {( 4, ), (4, 0), (0, ), (, 0)} 6. (0, ) (4, 0) (, 0) ( 4, ) D { 4, 0,, 4}, D all reals, R all reals; es R {, 0, }; es Find each value if f() and g(). 7. f() 8. f( 4) 9. g 0. f( ) undefined. g( 6). f(m ) m. MUSIC The ordered pairs (, 6), (, 6), (, ), (4, ), and (, 48) 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 {,,,4,},R {6,, 48}; es 4. 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 calculations. How long would it take the computer to do billion calculations? 7. s Glencoe/McGraw-Hill 60 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

52 Answers (Lesson -) Lesson - - NAME 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, 0) tell ou? For a deer, the average longevit is 8 ears and the maimum longevit is 0 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.. a. Write the domain and range of the relation shown in the graph. (, ) (0, 4) (, ) (, 0) (, ) (, 4) D: {,,, 0, }; R: {, 4, 0,,, 4} 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. Glencoe/McGraw-Hill 6 Glencoe Algebra - NAME DATE PERID Enrichment Mappings 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. Into mapping Amapping from set A to set B where ever element of A is mapped to one or more elements of set B, but never to an element not in B. nto mapping Amapping from set A to set B where each element of set B has at least one element of set A mapped to it. ne-to-one Amapping 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 a g k l q into, onto into, onto into, onto, into, onto one-to-one into into, onto into, onto into, onto, one-to-one 9. Can a set be mapped onto a set with fewer elements than it has? es 0. Can a set be mapped into a set that has more elements than it has? es. If a mapping from set A into set B is a one-to-one correspondence, what can ou conclude about the number of elements in A and B? The sets have the same number of elements. Glencoe/McGraw-Hill 6 Glencoe Algebra Glencoe/McGraw-Hill A4 Glencoe Algebra

53 Answers (Lesson -) - NAME DATE PERID Stud Guide and Intervention Linear Equations Glencoe/McGraw-Hill 6 Glencoe Algebra Answers Lesson - 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 Eample Is f() 0. a linear function? Eplain. Yes; it is a linear function because it can be written in the form f() 0.. Eample Is 0 a linear function? Eplain. No; it is not a linear function because the variables and are multiplied together in the middle term. Find the -intercept and the -intercept of the graph of 4 0. Then graph the equation. The -intercept is the value of when riginal equation 4 (0) 0 Substitute 0 for. Simplif. So the -intercept is. Similarl, the -intercept is 4. 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: 0; -int: -int: ; -int:. - NAME DATE PERID Stud Guide and Intervention (continued) Linear Equations Standard Form The standard form of a linear equation is A B C, where A, B, and C are integers whose greatest common factor is. Eample Write each equation in standard form. Identif A, B, and C. a. 8 8 riginal equation 8 Subtract 8 from each side. 8 Multipl each side b. So A 8, B, and C. b riginal equation 4 7 Add 7 to each side. Divide each side b 7. So A, B, and C. Eercises Write each equation in standard form. Identif A, B, and C ; A, ; A, ; A, B 4, C B, C B, C ; A 8, ; A 8, 4 ; A 4, B 6, C B 9, C 60 B, C ; A, 4 7; A, 6; A, B, C 0 B 4, C 7 B, C ; A 6, ; A, 6; A, B, C 0 B, C B 0, C ; A 9, 6; A, 4 0; A, B, C 6 B, C 6 B 4, C ; A 8, 0 ; A 0, 8 ; A 8, B, C B, C B, C ; A, ; A, 00; A, B, C 4 B, C B, C 00 Glencoe/McGraw-Hill 64 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

54 Answers (Lesson -) Lesson - - NAME DATE PERID Skills Practice Linear Equations State whether each equation or function is linear. Write es or no. If no, eplain our reasoning... es es f() 4 es No; the eponent of is not No; is in a denominator. es 7. g() 8 8. h() es No; is inside a square root. Write each equation in standard form. Identif A, B, and C. 9. 0;,, 0 0. ;,, ;, 7, 4. ;,, ; 0,, ; 4,, 7 Find the -intercept and the -intercept of the graph of each equation. Then graph the equation.. 6, , 0 (, 0) (0, 0) (0, 6) 7., 8. 0, (0, ) (0, ) (, 0) (, 0) Glencoe/McGraw-Hill 6 Glencoe Algebra NAME DATE PERID - Practice (Average) Linear Equations State whether each equation or function is linear. Write es or no. If no, eplain our reasoning.. h() es. es. No; is a denominator No; and are multiplied. Write each equation in standard form. Identif A, B, and C ; 7,, ;, 8, ; 0,, ; 8, 8, Find the -intercept and the -intercept of the graph of each equation. Then graph the equation. 9. 4, , (0, 4) (0, ) (, 0) (7, 0). 4, , (, 0) (0, ) (0, 4) (, 0). MEASURE The equation.4 gives the length in centimeters corresponding to a length in inches. What is the length in centimeters of a -foot ruler? 0.48 cm LNG DISTANCE For Eercises 4 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 0.0t, where t is the number of minutes talked. 4. What is the total cost of talking 8 hours? of talking 0 hours? $0; $66. What is the effective cost per minute (the total cost divided b the number of minutes talked) of talking 8 hours? of talking 0 hours? $0.06; $0.0 Glencoe/McGraw-Hill 66 Glencoe Algebra Glencoe/McGraw-Hill A6 Glencoe Algebra

55 Answers (Lesson -) - NAME DATE PERID Reading to Learn Mathematics Linear Equations Glencoe/McGraw-Hill 67 Glencoe Algebra Answers Lesson - 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 c. 7 es 4 d. 7 No; B is not an integer. e No; A and B are both 0. f. 4 8 No; The greatest common factor of, 4, 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 0, then the graph is a horizontal line. If B 0, 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 0 to find the -intercept and let 0 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 0, so let 0 to find an -intercept. The -intercept is the -coordinate of a point on the -ais. Ever point on the -ais has -coordinate 0, so let 0 to find a -intercept. - NAME DATE PERID Enrichment 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 0. The process can be written as follows. 88 4() 6 () 4 6() 0 () 6 0() 6 () 0 6() (4) 6 () The last nonzero remainder is the GCF of the two coefficients. If the constant term 7 is divisible b the GCF, then integers and do eist that satisf the equation. To find and, work backward in the following manner [0 6()] Substitute for using (4) 6(0) 8(6) 6(0) 8[6 0()] Substitute for 6 using () 8(6) 4(0) 8(6) 4[4 6()] Substitute for 0 using () 4(4) 66(6) 4(4) 66[88 4()] Substitute for 6 using () 88( 66) 4(90) Thus, 66 and 90. Find integers and, if the eist, that satisf each equation and and no integral solutions eist and and 8 and 98 Glencoe/McGraw-Hill 68 Glencoe Algebra Glencoe/McGraw-Hill A7 Glencoe Algebra

56 Answers (Lesson -) Lesson - - Slope NAME DATE PERID Stud Guide and Intervention Slope change in Slope m of a Line For points (, ) and (, ), where, m change in Eample Eample Determine the slope of the line that passes through (, ) and ( 4, ). m Slope formula ( ) (, ) (, ), (, ) ( 4, ) Simplif. The slope of the line is. Graph the line passing through (, ) with a slope of. Graph the ordered pair (, ). Then, according to the slope, go up 4 units and right units. Plot the new point (4,). Connect the points and draw the line. 4 Eercises Find the slope of the line that passes through each pair of points.. (4, 7) and (6, ). (6, 4) and (, 4) 0. (, ) and (7, ) 4. (, ) and ( 4, ). (, 0) and (, ) 6. (, 4) and (, ) 7. (7, ) and (, ) 4 8. (, 9) and (, ) 9. (4, ) and ( 4, 8) 4 Graph the line passing through the given point with the given slope. 0. slope. slope. slope 0 passes through (0, ) passes through (, 4) passes through (, ). slope 4. slope 4. slope passes through ( 4, 6) passes through (, 0) passes through (0, 0) Glencoe/McGraw-Hill 69 Glencoe Algebra - NAME DATE PERID Stud Guide and Intervention (continued) Slope Parallel and Perpendicular Lines In a plane, nonvertical lines with the same slope are parallel. All vertical lines are parallel. slope m In a plane, two oblique lines are perpendicular if and onl if the product of their slopes is. An vertical line is perpendicular to an horizontal line. slope m slope m slope m Eample Are the line passing through (, 6) and (, ) and the line passing through (, 0) and (0, 4) parallel, perpendicular, or neither? Find the slopes of the two lines. 6 The slope of the first line is. ( ) The slope of the second line is. 0 The slopes are not equal and the product of the slopes is not, so the lines are neither parallel nor perpendicular. Eercises Are the lines parallel, perpendicular, or neither?. the line passing through (4, ) and (. ) and the line passing through (, ) and (, ) perpendicular. the line passing through (, 8) and (, ) and the line passing through (0, 9) and (6, 0) neither. the line passing through (, 9) and (, ) and the graph of parallel 4. the line with -intercept and -intercept and the line with -intercept and -intercept parallel. the line with -intercept and -intercept and the line with -intercept and -intercept neither 6. the line passing through (, ) and (, ) and the graph of 0 perpendicular 7. the line passing through ( 4, 8) and (6, 4) and the graph of parallel Glencoe/McGraw-Hill 70 Glencoe Algebra Glencoe/McGraw-Hill A8 Glencoe Algebra

57 Answers (Lesson -) - NAME DATE PERID Skills Practice Slope Glencoe/McGraw-Hill 7 Glencoe Algebra Answers Lesson - Find the slope of the line that passes through each pair of points.. (, ), (, ) 4. (0, ), (, 0). (, 9), (0, 6) 4. (8, ), (4, ) 4. (, ), (, ) undefined 6. (, ), (0, ) (4, ), (, 7) 8. (, 4), (, ) 9. (, ), (, ) 0 Graph the line passing through the given point with the given slope. 0. (0, 4), m. (, 4), m (0, 4) (, 4). (, ), m. (, ), m (, ) (, ) Graph the line that satisfies each set of conditions. 4. passes through (0, ), perpendicular to. passes through (0, ), parallel to the graph of a line whose slope is (0,) (0, ) 6. HIKING Naomi left from an elevation of 7400 feet at 7:00 A.M. and hiked to an elevation of 9800 feet b :00 A.M. What was her rate of change in altitude? 600 ft/h NAME DATE PERID - Practice (Average) Slope Find the slope of the line that passes through each pair of points.. (, 8), (, ) 4. ( 0, ), (7, ) 7. ( 7, 6), (, 6) 0 4. (8, ), (8, ) undefined. (4, ), (7, ) 6. ( 6, ), ( 8, 4) 7 Graph the line passing through the given point with the given slope. 7. (0, ), m 8. (, ), m 4 (0, ) (, ) 9. (0, ), m 0 0. (, ), m 4 (0, ) (, ) Graph the line that satisfies each set of conditions.. passes through (, 0), perpendicular. passes through (, ), parallel to a line whose slope is to a line whose slope is (, ) (, 0) DEPRECIATIN For Eercises, use the following information. A machine that originall cost $,600 has a value of $700 at the end of ears. The same machine has a value of $800 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. $700 per ear 4. Find the average rate of change in value of the machine between the end of ears and the end of 8 ears. $940 per ear. Interpret the sign of our answers. It is negative because the value is decreasing. Glencoe/McGraw-Hill 7 Glencoe Algebra Glencoe/McGraw-Hill A9 Glencoe Algebra

58 Answers (Lesson -) Lesson - - NAME 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 40 feet over a horizontal distance of 000 feet? 4% What is the grade of a road that rises meters over a horizontal distance of 0 kilometers? ( kilometer 000 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. Zero The line is horizontal. 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. 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. Glencoe/McGraw-Hill 7 Glencoe Algebra - NAME DATE PERID Enrichment 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, ) Step Find D, the sum of the downward diagonal products (from left to right). D ( ) ( ) ( ) (6 7) 6 4 or 7 (, 7) (, ) Step Find U, the sum of the upward diagonal products (from left to right). U ( 7) ( ) (6 ) ( ) or 4 Step 4 Use the formula A (D U) to find the area. A (7 4) (0) or (, ) (6, ) (, 7) The area is square units. Count the number of square units enclosed b the polgon. Does this result seem reasonable? Use the coordinate method to find the area of each region in square units units 4 units 4 units Glencoe/McGraw-Hill 74 Glencoe Algebra Glencoe/McGraw-Hill A0 Glencoe Algebra

59 Answers (Lesson -4) -4 NAME DATE PERID Stud Guide and Intervention Writing Linear Equations Glencoe/McGraw-Hill 7 Glencoe Algebra Answers Lesson -4 Forms of Equations Slope-Intercept Form of a Linear Equation m b, where m is the slope and b is the -intercept Point-Slope Form m( ), where (, ) are the coordinates of a point on the line and of a Linear Equation m is the slope of the line Eample Eample Write an equation in slope-intercept form for the line that has slope and passes through the point (, 7). Substitute for m,, and in the slope-intercept form. m b Slope-intercept form 7 ( )() b (, ) (, 7), m 7 6 b Simplif. b Add 6 to both sides. The -intercept is. The equation in slope-intercept form is. Eercises Write an equation in slope-intercept form for the line that has slope and -intercept. m b Slope-intercept form 0 () b (, ) (, 0), m 0 b Simplif. b Subtract from both sides. The -intercept is. The slope-intercept form is. Write an equation in slope-intercept form for the line that satisfies each set of conditions.. slope, passes through ( 4, 6). slope, -intercept 4 4. slope, passes through (, ) 4. slope, passes through (, 7) 6 Write an equation in slope-intercept form for each graph (, 6) (, ) (4, ) ( 4, ) (0, 0) (, 0) NAME DATE PERID Stud Guide and Intervention (continued) Writing Linear Equations Parallel and Perpendicular Lines Use the slope-intercept or point-slope form to find equations of lines that are parallel or perpendicular to a given line. Remember that parallel lines have equal slope. The slopes of two perpendicular lines are negative reciprocals, that is, their product is. Eample Eample Write an equation of the line that passes through (8, ) and is perpendicular to the line whose equation is. The slope of the given line is. Since the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line is. Use the slope and the given point to write the equation. m( ) Point-slope form ( 8) (, ) (8, ), m 6 Distributive Prop. 4 Add to each side. An equation of the line is 4. Write an equation of the line that passes through (, ) and is parallel to the graph of. The slope of the given line is. Since the slopes of parallel lines are equal, the slope of the parallel line is also. Use the slope and the given point to write the equation. m( ) Point-slope form ( ( )) (, ) (, ), m Distributive Prop. 8 Add to each side. An equation of the line is 8. Eercises Write an equation in slope-intercept form for the line that satisfies each set of conditions.. passes through ( 4, ), parallel to the line whose equation is 4. passes through (, ), perpendicular to the graph of. passes through (, ), parallel to the line that passes through (4, ) and (, ) 4. passes through (4, 7), perpendicular to the line that passes through (, 6) and (, ) 7. passes through (8, 6), perpendicular to the graph of 4 6. passes through (, ), perpendicular to the graph of 6 7. passes through (6, ), parallel to the line with -intercept and -intercept 9 8. passes through (, ), perpendicular to the line 4 4 Glencoe/McGraw-Hill 76 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

60 Answers (Lesson -4) Lesson -4-4 NAME DATE PERID Skills Practice Writing Linear Equations State the slope and -intercept of the graph of each equation.. 7 7,.,., , , , 7., 8. 6, Write an equation in slope-intercept form for each graph (, ) (0, ) (, ) (4, ) (, 4) (, ) Write an equation in slope-intercept form for the line that satisfies each set of conditions.. slope, passes through (, ). slope, passes through (0, 0) 6 4. slope, passes through (0, ). slope, passes through (, 0) 6 6. passes through (, ) and (, ) 7. passes through (, 4) and (, 8) intercept, -intercept intercept, -intercept 6 0. passes through (, ), perpendicular to the graph of 4. 0 Glencoe/McGraw-Hill 77 Glencoe Algebra NAME DATE PERID -4 Practice (Average) Writing Linear Equations State the slope and -intercept of the graph of each equation.. 8 8, ,., ,., 6. 0, Write an equation in slope-intercept form for each graph (4, 4) (, ) (0, ) (, ) (0, ) Write an equation in slope-intercept form for the line that satisfies each set of conditions slope, passes through (, 8). slope, passes through (0, ) 4. slope 0, passes through (0, 0). 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 RESERVIRS The surface of Grand Lake is at an elevation of 648 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 $0,000 profit if it manufactures 00,000 drives, and $,70,000 profit if it manufactures 00,000 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 4n 0,000 Glencoe/McGraw-Hill 78 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

61 Answers (Lesson -4) -4 NAME DATE PERID Reading to Learn Mathematics Writing Linear Equations Glencoe/McGraw-Hill 79 Glencoe Algebra Answers Lesson -4 Pre-Activit How do linear equations appl to business? Read the introduction to Lesson -4 at the top of page 7 in our tetbook. If the total cost of producing a product is given b the equation 400.7, what is the fied cost? What is the variable cost (for each item produced)? $400; $.7 Write a linear equation that describes the following situation: A compan that manufactures computers has a fied cost of $8,70 and a variable cost of $8 to produce each computer. 8,70 8 Reading the Lesson. 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. 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 4. 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 4. 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. -4 NAME DATE PERID Enrichment 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 (, 0) and (0, 6), then draw a straight line through them. Eample 4 4 Simplif. Write 4 in two-intercept form. Divide b to obtain on the right side. The -intercept is 4; the -intercept is. Use the given intercepts a and b, to write an equation in two-intercept form. Then draw the graph. See students graphs.. a, b 4. a, b a, b 4. a 6, b Write each equation in two-intercept form. Then draw the graph Glencoe/McGraw-Hill 80 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

62 Glencoe/McGraw-Hill A4 Glencoe Algebra - NAME DATE PERID Stud Guide and Intervention 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 990s. Make a scatter plot of the data. Home Runs Source: New York Times Almanac Make a scatter plot for the data in each table below.. FUEL EFFICIENCY The table below shows the average fuel efficienc in miles per gallon of new cars manufactured during the ears listed. Year Runs Batted In Eercises Fuel Efficienc (mpg) Year 990 Source: New York Times Almanac. CNGRESS The table below shows the number of women serving in the United States Congress during the ears Congressional Session Source: Wall Street Journal Almanac Runs Batted In 0 00 Number of Women MVP HRs and RBIs Home Runs Glencoe/McGraw-Hill 8 Glencoe Algebra Miles per Gallon Average Fuel Efficienc Number of Women Women in Congress Session of Congress Lesson - - Prediction Equations A line of fit is a line that closel approimates a set of data graphed in a scatter plot. The equation of a line of fit is called a prediction equation because it can be used to predict values not given in the data set. To find a prediction equation for a set of data, select two points that seem to represent the data well. Then to write the prediction equation, use what ou know about writing a linear equation when given two points on the line. Eample STRAGE CSTS According to a certain prediction equation, the cost of 00 square feet of storage space is $60. The cost of square feet of storage space is $60. a. Find the slope of the prediction equation. What does it represent? Since the cost depends upon the square footage, let represent the amount of storage space in square feet and represent the cost in dollars. The slope can be found using the formula m So,m The slope of the prediction equation is 0.8. This means that the price of storage increases 80 for each one-square-foot increase in storage space. b. Find a prediction equation. Using the slope and one of the points on the line, ou can use the point-slope form to find a prediction equation. m( ) Point-slope form ( 00) (, ) (00, 60), m Distributive Propert Add 60 to both sides. A prediction equation is Eercises NAME DATE PERID Stud Guide and Intervention (continued) Modeling Real-World Data: Using Scatter Plots SALARIES The table below shows the ears of eperience for eight technicians at Lewis Techomatic and the hourl rate of pa each technician earns. Use the data for Eercises and. Eperience (ears) Hourl Rate of Pa $7 $0 $0 $7 $9 $ $0 $. Draw a scatter plot to show how ears of eperience are related to hourl rate of pa. Draw a line of fit. See graph.. Write a prediction equation to show how ears of eperience () are related to hourl rate of pa (). Sample answer using (, 7) and (9, 7): Eperience (ears) Glencoe/McGraw-Hill 8 Glencoe Algebra Hourl Pa ($) Technician Salaries Answers (Lesson -)

63 Glencoe/McGraw-Hill A Glencoe Algebra - NAME DATE PERID Skills Practice Modeling Real-World 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.. a.. a.. a ? ? 6 6? b. Sample answer using (, ) and (8, ): c. Sample answer: b. Sample answer using (, 9) and (40, 44): 4 c. Sample answer: b. Sample answer using (, 6) and (7, 4): c. Sample answer: 9.6 Glencoe/McGraw-Hill 8 Glencoe Algebra Lesson - - 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 approimate weights in tons and estimates for overall fuel econom in miles per gallon for several cars. b. Sample answer using (.4, 4) and (.4, ): c. Sample answer: 8.6 mi/gal. 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). b. Sample answer using (700, 6) and (9700, 0): 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 NAME DATE PERID Practice (Average) Modeling Real-World Data: Using Scatter Plots Altitude (ft) ,400,000 Temperature ( F) ? Time (min) Calories Burned ? Burning Calories Time (min) Weight (tons) Miles per Gallon 9 4? 7 0 7,000 8,000 9,000 0,000 Altitude (ft) b. Sample answer using (4, 80) and (48, 440): c. Sample answer: about 0 calories Glencoe/McGraw-Hill 84 Glencoe Algebra Fuel Econom (mi/gal) Fuel Econom Versus Weight Weight (tons) Temperature ( F) Temperature Versus Altitude Answers (Lesson -) Answers

64 Answers (Lesson -) Lesson - - NAME DATE PERID Reading to Learn Mathematics 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.. Suppose that tuition at a state college was $00 per ear in 99 and has been increasing at a rate of $ per ear. a. Write a prediction equation that epresses this information. 00 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 007. $600 Helping You Remember 4. 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. Glencoe/McGraw-Hill 8 Glencoe Algebra - NAME DATE PERID Enrichment 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 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. Find,, and, the medians of the values in groups,, and, respectivel. Find,, and, the medians of the values in groups,, and, respectivel. 98, 988, 994;,, 9. Find an equation of the line through (, ) and (, ) Find Y, the -coordinate of the point on the line in Eercise with an -coordinate of.. The median-fit line is parallel to the line in Eercise, but is one-third closer to (, ). This means it passes through, Y. Find this ordered pair. about (988,.67) 6. Write an equation of the median-fit line Use the median-fit line to predict the percentage of juvenile violent crime offenders in 00 and : about %; 00: about6% Glencoe/McGraw-Hill 86 Glencoe Algebra Glencoe/McGraw-Hill A6 Glencoe Algebra

65 Answers (Lesson -6) -6 NAME DATE PERID Stud Guide and Intervention Special Functions Glencoe/McGraw-Hill 87 Glencoe Algebra Answers Lesson -6 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 a step function Eercises 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 DATE PERID Stud Guide and Intervention (continued) Special Functions Absolute Value and Piecewise Functions Another special function is the absolute value function, which is also called a piecewise function. Absolute Value Function f() two ras that are mirror images of each other and meet at a point, the verte To graph a special function, use its definition and our knowledge of the parent graph. Find several ordered pairs, if necessar. Eample Graph f() 4. Find several ordered pairs. Graph the points and connect them. You would epect the graph to look similar to its parent function, f() f() Eample Graph f() if if. First, graph the linear function f() for. Since does not satisf this inequalit, stop with a circle at (, 4). Net, graph the linear function f() for. Since does satisf this inequalit, begin with a dot at (, ). f() Eercises Graph each function. Identif the domain and range.. g(). h(). h( ) if 0 6 if 0 if domain: all real domain: all real domain: all real numbers; range: numbers; range: numbers; range: all integers { 0} { } Glencoe/McGraw-Hill 88 Glencoe Algebra Glencoe/McGraw-Hill A7 Glencoe Algebra

66 Glencoe/McGraw-Hill A8 Glencoe Algebra -6 NAME DATE PERID Skills Practice Special Functions Identif each function as S for step, C for constant, A for absolute value, or P for piecewise.... S C A Graph each function. Identif the domain and range. 4. f(). f() D all reals, R all integers 6. g() 7. f() D all reals, R all integers D all reals, D all reals, R { } R nonnegative reals 8. f() if 0 9. h() if if 0 if > f() g() f() D all reals, D { or }, R { 0 or } R { } Glencoe/McGraw-Hill 89 Glencoe Algebra f() f() h() Lesson -6-6 Graph each function. Identif the domain and range.. f() 0.. f() D all reals, R all integers. g() 4. f() D all reals, R nonpositive reals D all reals, R all integers D all reals, R nonnegative reals. f() if 6. h() 4 if 0 if if 0 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 $.00 $40 per hour or an fraction thereof per pound for less than 0 pounds of cand and for labor. Draw a graph of the step $.0 per pound for 0 or more pounds. Draw a function that represents this situation. graph of the function that represents this Labor Costs situation. Cand Costs Total Cost ($) NAME DATE PERID Practice (Average) Special Functions f() g() f() Hours Pounds Glencoe/McGraw-Hill 90 Glencoe Algebra h() f() f() Cost ($) Answers (Lesson -6)

67 Answers (Lesson -6) -6 NAME DATE PERID Reading to Learn Mathematics Special Functions Glencoe/McGraw-Hill 9 Glencoe Algebra Answers Lesson -6 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 0. ounce? $0.4 or 4 cents Give three different weights of letters that would each cost cents to mail. Answers will var. Sample answer:. ounces,.9 ounces,.0 ounces Reading the Lesson. Find the value of each epression. a. b c 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 0 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. -6 NAME DATE PERID Enrichment 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 Glencoe/McGraw-Hill 9 Glencoe Algebra.. Glencoe/McGraw-Hill A9 Glencoe Algebra

68 Answers (Lesson -7) Lesson -7-7 NAME DATE PERID Stud Guide and Intervention 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. (0, 0) 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 4. The boundar is the graph of 4. Use the slope-intercept form,, to graph the boundar line. The boundar line should be solid. Now test the point (0, 0). 0 (0)? 4 (, ) (0, 0) 0 4 false Shade the region that does not contain (0, 0). Eercises Graph each inequalit Glencoe/McGraw-Hill 9 Glencoe Algebra -7 NAME DATE PERID Stud Guide and Intervention (continued) Graphing Inequalities Graph Absolute Value Inequalities Graphing absolute value inequalities is similar to graphing linear inequalities. The graph of the related absolute value equation is the boundar. This boundar is graphed as a solid line if the inequalit is or, and dashed if the inequalit is or. Choose a test point not on the boundar to determine which region to shade. Eample Graph. First graph the equation. Since the inequalit is, the graph of the boundar is solid. Test (0, 0). 0? 0 (, ) (0, 0) 0? 0 true Shade the region that contains (0, 0). Eercises Graph each inequalit Glencoe/McGraw-Hill 94 Glencoe Algebra Glencoe/McGraw-Hill A0 Glencoe Algebra

69 Answers (Lesson -7) -7 NAME DATE PERID Skills Practice Graphing Inequalities Glencoe/McGraw-Hill 9 Glencoe Algebra Answers Lesson -7 Graph each inequalit NAME DATE PERID -7 Practice (Average) Graphing Inequalities Graph each inequalit CMPUTERS For Eercises 0, use the following information. A school sstem is buing new computers. The will bu desktop computers costing $000 per unit, and notebook computers costing $00 per unit. The total cost of the computers cannot eceed $80,000. Notebooks 0. Write an inequalit that describes this situation. 000d 00n 80,000. Graph the inequalit.. If the school wants to bu 0 of the desktop computers and of the notebook computers, will the have enough mone? es Computers Purchased Desktops Glencoe/McGraw-Hill 96 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

70 Answers (Lesson -7) Lesson -7-7 NAME 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 4. 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. Glencoe/McGraw-Hill 97 Glencoe Algebra -7 NAME DATE PERID Enrichment Algebraic Proof The following paragraph states a result ou might be asked to prove in a mathematics course. Parts of the paragraph are numbered. 0 Let n be a positive integer. 0 Also, let n s(n ) be the sum of the squares of the digits in n. 0 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. 04 In general, n k s(n k ) is the sum of the squares of the digits of n k. 0 Consider the sequence: n, n, n, n,, n k,. 06 In this sequence either all the terms from some k on have the value, 07 or some term, sa n j, has the value 4, so that the eight terms 4, 6, 7, 8, 89, 4, 4, and 0 keep repeating from that point on. Use the paragraph to answer these questions.. Use the sentence in line 0. List the first five values of n.,,, 4,. Use 946 for n and give an eample to show the meaning of line 0. n s(946) 7, because In line 0, which smbol shows a function? Eplain the function in a sentence. s(n); the sum of the squares of the digits of a number is a function of the number 4. For n 946, find n and n as described in sentence 0. n 9, n 06. How do the first four sentences relate to sentence 0? The eplain how to compute the terms of the sequence. 6. Use n and find the first four terms of the sequence., 0,, 7. Which sentence of the paragraph is illustrated b n? sentence Use n 6 and find the first ten terms. 6, 7, 8, 89, 4, 4, 0, 4, 6, 7 9. Which sentence is illustrated b n 6? sentence 07 Glencoe/McGraw-Hill 98 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra