C H A P T E R 9 BUILD YUR VCABULARY Chapter 9 This is an alphabetical list of new vocabular terms ou will learn in Chapter 9. As ou complete the stud notes for the chapter, ou will see Build Your Vocabular reminders to complete each term s definition or description on these pages. Remember to add the tetbook page number in the second column for reference when ou stud. Vocabular Term Found on Page Definition Description or Eample arithmetic sequence common difference constant of variation direct variation domain function function table line of fit linear function (continued on the net page) Math Connects, Course 3 221
Chapter 9 BUILD YUR VCABULARY Vocabular Term Found on Page Definition Description or Eample range rise run scatter plot sequence slope slope-intercept form sstem of equations sstem of inequalities term -intercept 222 Math Connects, Course 3
9 1 Sequences MAIN IDEA Write algebraic epressions to determine an term in an arithmetic sequence. BUILD YUR VCABULARY (pages 221 222) A sequence is an of numbers. Each number in a is called a term. An arithmetic sequence is a sequence in which the the same. between an two consecutive terms is The difference between an two in an sequence is called the common difference. EXAMPLE Identif Arithmetic Sequences State whether the sequence 23, 15, 7, -1, -9,... is arithmetic. If it is, state the common difference. Write the net three terms of the sequence. 23, 15, 7, -1, -9 Notice that 15-23 = -8, 7-15 = -8, and so on. -8-8 -8-8 The terms have a common is. Continue the pattern to find the net three terms. -9,,, -8-8 -8 of -8, so the sequence The net three terms are,, and. Math Connects, Course 3 223
9 1 Check Your Progress State whether the sequence 29, 27, 25, 23, 21,... is arithmetic. If it is, state the common difference. Write the net three terms of the sequence. EXAMPLE Describe an Arithmetic Sequence Write an epression that can be used to find the nth term of the sequence 0.6, 1.2, 1.8, 2.4,... Then write the net three terms. Use a table to eample the sequence. Term Number (n) 1 2 3 4 Term 0.6 1.2 1.8 2.4 The terms have a common difference of 0.6. Also, each term is times its term number. An epression that can be used to find the nth term is. The net three terms are,, and. Check Your Progress Write an epression that can be used to find the nth term of the sequence 1.5, 3, 4.5, 6,... Then write the net three terms. 224 Math Connects, Course 3
9 1 EXAMPLE TRANSPRTATIN This arithmetic Miles Cost ($) sequence shows the cost of a tai 1 5.25 ride for 1, 2, 3, and 4 miles. What would be the cost of 2 7.00 a 9-mile ride? 3 8.75 The common difference between the 4 10.50 costs is. This implies that the epression for the nth mile is. Compare each cost to the value of for each number of miles. Each cost is 3.50 more than Miles Cost ($) 1.75n. So, the epression is the cost of a tai ride for n miles. To find the cost of a 9-mile ride, let c represent the cost. Then write and solve an equation for n = 9. c = 1.75n + 3.50 1 5.25 1.75 2 7.00 3.50 3 8.75 5.25 4 10.50 7.00 Write the equation. HMEWRK ASSIGNMENT Page(s): Eercises: c = 1.75 + 3.50 Replace n with. c = + 3.50 or Simplif. It would cost for a 9-mile tai ride. Check Your Progress TRANSPRTATIN This arithmetic sequence shows the cost of a tai ride for 1, 2, 3, and 4 miles. What would be the cost of a 15-mile ride? Miles Cost ($) 1 6.00 2 7.50 3 9.00 4 10.50 Math Connects, Course 3 225
2 1 9 2 Functions MAIN IDEA Complete function tables. BUILD YUR VCABULARY (pages 221 222) A where one thing another is called a function. EXAMPLE Find a Function Value Find each function value. f (4) if f () = - 8 f () = - 8 Write the function. f ( ) = - 8 Substitute for into the function rule. = Simplif. So, f (4) =. RGANIZE IT In our Foldable, write how ou would find the value of a function. You ma wish to include an eample. f (-6) if f () = 3 + 4 f () = 3 + 4 Write the function. f ( ) = 3 ( ) + 4 Substitute for into f ( ) = + 4 Multipl. the function rule. = Simplif. So, f (-6) =. Check Your Progress Find each function value. a. f (2) if f () = - 7 b. f (-2) if f () = 2 + 6 226 Math Connects, Course 3
9 2 BUILD YUR VCABULARY (pages 221 222) The set of values in a function is called the domain. The set of values in a function is called the range. You can use a function table to organize the input,, and output. EXAMPLE Make a Function Table Complete the function table for f () = 4-1. Then state the domain and the range of the function. Substitute each value of, or, into the function rule. Input -3-2 -1 0 1 Rule 4-1 utput f () Then simplif to find the. f () = 4-1 f (-3) = f (-2) = f (-1) = f (0) = f (1) = or or or or or Input -3-2 -1 The domain is. 0 1 Rule 4-1 utput f () The range is. Math Connects, Course 3 227
9 2 Check Your Progress Complete the function table for f () = 3-2. Then state the domain and the range of the function. Input -3-2 Rule 3-2 utput f () -1 0 1 EXAMPLE Functions with Two Variables PARKING FEES The price for parking at a cit lot is $3.00 plus $2.00 per hour. Write a function to represent the price of parking for h hours. Then determine how much would it cost to park at the lot for 2 hours. Words Function p = + Cost of parking equals $3.00 plus $2.00 per hour. HMEWRK ASSIGNMENT Page(s): Eercises: The function p = Substitute p = + p = 3 + 2 or It will cost for h into the function rule. to park for 2 hours. represents the situation. Check Your Progress TAXI The price of a tai ride is $5.00 plus $20.00 per hour. Write a function using two variables to represent the price of riding a tai for h hours. Then determine how much would it cost for a 3-hour tai ride. 228 Math Connects, Course 3
9 3 Representing Linear Functions EXAMPLE MAIN IDEA Represent linear functions using function tables and graphs. MUSIC During a clearance sale, a music store is selling CDs for $3 and tapes for $1. Graph the function 3 + = 6 to find how man CDs and tapes Bill can bu with $6. First, rewrite the equation b solving for. 3 + = 6 Write the equation. 3 - + = 6 - Subtract from each side. = 6-3 Simplif. Choose values for and substitute them to find. Then graph the ordered pairs. = 6-3 (, ) RGANIZE IT In our Foldable, include a linear function and its graph. 0 = 6-3 1 = 6-3 2 = 6-3 6 5 4 3 2 1 1 2 3 4 He cannot bu negative numbers of CDs or tapes, so the solutions are CDs and tapes, CD and tapes, or CDs and tapes. Check Your Progress BAKE SALE During a bake sale, a plate of brownies is sold for $2 and a plate of cookies is sold for $1. Graph the function 2 + = 4 to find how man plates of brownies and cookies Craig can bu with $4. Math Connects, Course 3 229
9 3 EXAMPLE Graph a Function Graph = - 3. Step 1 Choose some values for. Make a function table. Include a column of ordered pairs of the form (, ). - 3 (, ) 0-3 1-3 2-3 3-3 Step 2 Graph each ordered pair. Draw a line that passes through each point. Note that the ordered pair for an point on this line is a solution of = - 3. The line is the complete graph of the function. Check It appears from the graph that (-1, -4) is also a solution. Check this b substitution. = - 3 Write the function. - 3 Replace and. = Simplif. Check Your Progress Graph = - 2. 230 Math Connects, Course 3
9 3 BUILD YUR VCABULARY (pages 221 222) A function in which the graph of solutions forms a is called a linear function. EXAMPLE TEST EXAMPLE Which line graphed below best represents the table of values for the ordered pairs (, )? A C 0 1 1 3 2 5 3 7 B D Read the Item You need to decide which of the four graphs represents the data in the table. Solve the Item The values in the table represent the ordered pairs, graph. Graph, and. Test the ordered pairs with each ordered pairs. The answer is. is the onl graph which contains all these Math Connects, Course 3 231
2 1 9 3 Check Your Progress MULTIPLE CHICE Which line graphed below best represents the table of values for the ordered pairs (, )? 0 3 F H 1 0 2-3 3-6 G J HMEWRK ASSIGNMENT Page(s): Eercises: 232 Math Connects, Course 3
9 4 Slope MAIN IDEA Find the slope of a line using the slope formula. BUILD YUR VCABULARY (pages 221 222) Slope is the of the rise, or change, to the run, or change. EXAMPLE ACCESS RAMPS The access ramp from the sidewalk to the door of a hotel rises 8 inches for ever horizontal change of 96 inches. What is the slope of the access ramp? slope = Defi nition of slope = rise = inches, run = inches = Simplif. The slope of the access ramp is. Check Your Progress ACCESS RAMPS The access ramp from the sidewalk to the door of an office building rises 14 inches for ever horizontal change of 210 inches. What is the slope of the access ramp? EXAMPLE Find Slope Using a Graph Find the slope of the line. Choose two points on the line. The vertical change is -3 units while the horizontal change is 2 units. X Math Connects, Course 3 233
9 4 slope = _ rise run Defi nition of slope = rise =, run = The slope of the line is. Check Your Progress Find the slope of the line. 3 2 3 2 1 2 2 3 EXAMPLE Find Slope Using a Table The points given in the table lie on a line. Find the slope of the line. Then graph the line. change in slope = change in = -3-1 1-2 1 4 = The slope is. Check Your Progress The points given in the table below lie on a line. Find the slope of the line. Then graph the line. 2 5 5 7 8 9 11 11 234 Math Connects, Course 3
9 4 EXAMPLE Positive Slope Find the slope of the line that passes through A (3, 3) and B (2, 0). m = 2-1 2 - Defi nition of slope 1 m = _ 0-3 ( 1, 1 ) = (3, 3) 2-3 ( 2, 2 ) = (2, 0) m = _ 3 or 3 Simplif. 1 A(3, 3) B(2, 0) EXAMPLE Negative Slope Find the slope of the line that passes through X (-2, 3) and Y (3, 0). m = 2-1 2-1 Defi nition of slope m = ( 1, 1 ) = (-2, 3) m = _ -3 5 or - 3_ 5 ( 2, 2 ) = (3, 0) Simplif. HMEWRK ASSIGNMENT Page(s): Eercises: Check Your Progress Find the slope of the line that passes through each pair of points. a. A(4, 3) and B(1, 0) b. X(-3, 3) and Y(1, 0) Math Connects, Course 3 235
9 5 Direct Variation MAIN IDEA Use direct variation to solve problems. BUILD YUR VCABULARY (pages 221 222) When two variable quantities have a, their relationship is called a direct variation. The constant is called the constant variation. EXAMPLE Find a Constant Ratio EARNINGS The amount of mone Serena earns at her job varies directl as the number of hours she works. Determine the amount Serena earns per hour. Since the graph of the data forms a line, the rate of change. Use the graph to find. amount earned hours worked Serena earns. Check Your Progress EARNINGS The amount of mone Elizabeth earns at her job varies directl as the number of hours she works. Determine the amount Elizabeth earns per hour. Dollars Earned 70 60 50 40 30 20 10 0 1 2 3 4 5 6 Time (hours) 236 Math Connects, Course 3
9 5 KEY CNCEPT In a direct variation, the ratio of to is constant. This can be stated as varies directl with. A direct variation can be represented algebraicall as k = _ or = k where k 0. EXAMPLE Solve a Direct Variation SHPPING The total cost for cans of soup varies directl as the number of cans purchased. If 4 cans of soup cost $5, how much would it cost to bu 8 cans? METHD 1 Use an equation. Write an equation of direct variation. Let represent the number of cans and let represent the cost. = k Direct variation = k =, = 1.25 = k Simplif. = Substitute for. Use the equation to find when = 8. = 1.25 = 1.25 = = Multipl. METHD 2 Use a proportion. cans cost It would cost 4_ 5 = 8 _ cans cost = Find the cross products. 4 = 40 _ 4 4 = _ 40 4 = to bu 8 cans. Multipl. Divide each side b 4. Simplif. Check Your Progress SHPPING A grocer store sells 6 apples for $2.70. How much would it cost to bu 10 apples? Math Connects, Course 3 237
9 5 EXAMPLES Identif Direct Variation Determine whether each linear function is a direct variation. If so, state the constant of variation. Das, 2 4 6 8 Hours worked, 16 32 54 72 Compare the ratios to check for a common ratio. _ hours das The ratios are, so the function is. Hours, 3 6 9 12 Miles, 25.5 51 76.5 102 Compare the ratios to check for a common ratio. _ miles hours HMEWRK ASSIGNMENT Page(s): Eercises: Since the ratios are, the function is a direct variation. The constant of variation is. Check Your Progress Determine whether the linear function is a direct variation. If so, state the constant of variation. a. b. Das, 1 2 3 4 Hours worked, 8 16 24 32 Hours, 2 4 6 8 Miles, 12 25 35 45 238 Math Connects, Course 3
9 6 Slope-Intercept Form MAIN IDEA Graph linear equations using the slope and -intercept. BUILD YUR VCABULARY (pages 221 222) Slope-intercept form is when an equation is written in the form, where m is the and b is the -intercept. EXAMPLES Find the Slopes and -intercepts of Graphs State the slope and the -intercept of the graph of each equation. = 3 _ 4-5 = 3 _ 4 + ( ) Write the equation in the form = m + b. = m + b m = 3 _ 4, b = The slope of the graph is is. 2 + = 8 2 + = 8, and the -intercept Write the original equation. - - Subtract from each side. = = Simplif. Write the equation in the form = m + b. = m + b m =, b = The slope of the graph is and the -intercept is. Math Connects, Course 3 239
9 6 Check Your Progress State the slope and the -intercept of the graph of each equation. a. = 1_ - 2 b. 3 + = 5 4 EXAMPLE Graph an Equation Graph = _ 2 + 2 using the slope and -intercept. 3 Step 1 Find the slope and -intercept. = 2_ 3 + 2 slope = 2_ 3 -intercept = 2 Step 2 Graph the -intercept. Step 3 Use the slope to locate a second point on the line. HMEWRK ASSIGNMENT Page(s): Eercises: m = 2_ 3 change in : up 2 units change in : right 3 units Step 4 Draw a line through the two points. Check Your Progress Graph = 1_ + 3 using the slope 3 and -intercept. 240 Math Connects, Course 3
9 7 Sstems of Equations MAIN IDEA Solve sstems of equations b graphing. BUILD YUR VCABULARY (pages 221 222) A sstem of equations consists of two two. and EXAMPLE ne Solution Solve the sstem = 3-2 and = + 1 b graphing. Graph each equation on the same coordinate plane. 2 4 3 2 1 4 1 2 3 4 2 4 The graphs appear to intersect at. Check in both equations b replacing with and with. Check = 3-2 = + 1 3-2 + 1 2.5-2 2.5 = 2.5 2.5 = 2.5 The solution of the sstem is (1.5, 2.5). Math Connects, Course 3 241
9 7 EXAMPLE No Solution Solve the sstem = 2-1 and = 2 + 1 b graphing. 4 3 2 1 4 2 1 2 3 4 2 4 The graphs appear to be lines. Since there is no coordinate point that is a solution of both questions, there is for the sstem of equations. EXAMPLE Infinitel Man Solutions Solve the sstem = 3-2 and - 2 = - 2 b graphing. Write - 2 = - 2 in slope-intercept form. - 2 = - 2 Write the equation. - 2 + = - 2 + Add to both sides. = - Simplif. 4 3 2 1 4 2 1 2 3 4 2 4 The solution of the sstem is all pairs of the on the line = 3-2. 242 Math Connects, Course 3
9 7 Check Your Progress Solve each sstem of equations b graphing. a. = - 4 and = 2-6 b. = -3-2 and = -3 + 4 c. = 2-5 and + 2 = 2-3 Math Connects, Course 3 243
9 8 Problem-Solving Investigation: Use a Graph MAIN IDEA Solve problems b using a graph. EXAMPLE Use a Graph The graph shows how man boes of cookies were sold b five students for a school fundraiser. How man boes did the students sell altogether? UNDERSTAND The graph shows ou how man boes were sold b each of five students. You want to know the total number of boes sold b the students. PLAN Use the graph to add the numbers of boes sold. SLVE + + + + = The students sold altogether. HMEWRK ASSIGNMENT Page(s): Eercises: CHECK Look at the numbers at the top of each bar. Double check our sum. Check Your Progress PETS The graph shows how man dogs Edmond walked each da this week. How man dogs did he walk altogether during the week? 244 Math Connects, Course 3
9 9 Scatter Plots MAIN IDEA Construct and interpret scatter plots. BUILD YUR VCABULARY (pages 221 222) A scatter plot is a graph that shows the between sets of data. A line of fit is a line that is ver close to of the data points in a scatter plot. EXAMPLES Identif a Relationship Eplain whether the scatter plot of the data for each of the following shows a positive, negative, or no relationship. cups of hot chocolate sold at a concession stand and the outside temperature As the temperature decreases, the number of cups of hot chocolate sold. Therefore, the scatter plot might show a relationship. birthda and number of sports plaed The number of sports plaed does not depend on our birthda. Therefore, the scatter plot shows relationship. Check Your Progress Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. a. number of cups of lemonade sold at a concession stand and the outside temperature b. age and the color of our hair Math Connects, Course 3 245
9 9 EXAMPLES Line of Fit ZS The table at the right shows the average and maimum longevit of various animals in captivit. Make a scatter plot using the data. Then draw a line that best seems to represent the data. Maimum 80 70 60 50 40 30 20 10 Animal Longevit (Years) 0 5 10 15 20 25 30 35 40 45 Average Longevit (ears) Average Maimum 12 47 25 50 15 40 8 20 35 70 40 77 41 61 20 54 Source: Walker s Mammals of the World Write an equation for this line of fit. The line passes through points at and. Use these points to find the slope of the line. m = 2-1 2-1 Defi nition of slope m = ( 1, 1 ) =, ( 2, 2 ) = m = Simplif. The slope is, and the -intercept is. Use the slope and the -intercept to write the equation. = m + b Slope-intercept form = + m =, b = The equation for the line of fit is. 246 Math Connects, Course 3
9 9 Use the equation to predict the maimum longevit for an animal with an average longevit of 33 ears. = _ 3 + 17.5 Equation for the line of fi t 2 = _ 3 + 17.5 or 2 The maimum longevit is about. Check Your Progress The table shows the average hourl earnings of production workers since 2000. a. Make a scatter plot using Production Workers Earnings the data. Year Since Average Hourl b. Write an equation for the 2000 Earnings best-fit line using points (0, 11.43) and (5, 13.76). 0 $11.43 c. Use the equation to predict the average hourl earnings of production workers in 2009. 1 $11.82 2 $12.28 3 $12.78 4 $13.24 5 $13.76 6 $14.32 HMEWRK ASSIGNMENT Page(s): Eercises: Hourl Earnings (dollars) 14 13 12 11 0 Production Workers Average Hourl Earnings 1 2 3 4 5 6 Years Since 2000 Math Connects, Course 3 247
C H A P T E R 9 BRINGING IT ALL TGETHER STUDY GUIDE VCABULARY PUZZLEMAKER BUILD YUR VCABULARY Use our Chapter 9 Foldable to help ou stud for our chapter test. To make a crossword puzzle, word search, or jumble puzzle of the vocabular words in Chapter 9, go to: glencoe.com You can use our completed Vocabular Builder (pages 221 222) to help ou solve the puzzle. 9-1 Sequences State whether each sequence is arithmetic. Write es or no. If it is, state the common difference. Write the net three terms of the sequence. 1. 3, 7, 11, 15, 19,... 2. 5, -15, 45, -135, 405,... 3. 5, -1, -7, -13, -19,... 4. 4 1_, 3, 1 1_, 0, -1 1_ 2 2 2,... 9-2 Functions Match each description with the word it describes. 5. an output value of a function 6. the set of values of the dependent variable 7. the underlined letter in f () = 2 + 5 8. Complete the function table for f = 2 + 2. Then give the domain and range. Domain: Range: a. independent variable b. dependent variable c. domain d. range 2 + 2 f () -2 0 1 3 248 Math Connects, Course 3
Chapter 9 BRINGING IT ALL TGETHER 9-3 Representing Linear Functions 9. Complete the function table. Then graph = - + 2. - + 2 (,) -2 0 1 3 9-4 Slope Find the slope of the line that passes through each pair of points. 10. A (1, -2), B (4, 4) 11. C (1, 2), D (3, -2) 12. E (-1, 2), F (2, 2) 9-5 Direct Variation Determine whether each linear function is a direct variation. If so, state the constant of variation. 13. hours, 1 2 3 4 wages, $6 $12 $18 $24 15. hours, 5 6 7 8 miles, 480 415 350 285 14. length, 1 3 5 7 width, 2 6 10 14 16. minutes, 3 6 8 12 pages, 66 132 176 264 Math Connects, Course 3 249
Chapter 9 BRINGING IT ALL TGETHER 9-6 Slope-Intercept Form State the slope and the -intercept for the graph of each equation. 17. = -3 + 4 18. = 2_ - 7 19. 1_ 3 2 + = 8 9-7 Sstems of Equations 20. Solve the sstem = 2-4 and = - - 1 b graphing. 21. Solve the sstem = 4-4 and = 4 + 3 b graphing. 250 Math Connects, Course 3
Chapter 9 BRINGING IT ALL TGETHER 9-8 Problem-Solving Investigation: Use a Graph 22. SHPPING The Bu nline Compan charges $1.50 per pound plus $2 for shipping and handling. The Best Catalog Compan charges $1 per pound plus $5 for shipping and handling. Use a graph to determine the weight at which the shipping and handling will be the same for both companies. Shipping and Handling Weight 9-9 Scatter Plots 23. Complete. A scatter plot that shows a negative relationship will have a pattern of data points that go. Write whether a scatter plot of the data for the following might show a positive, negative, or no relationship. 24. favorite color and tpe of pet Math Connects, Course 3 251
C H A P T E R 9 Checklist ARE YU READY FR THE CHAPTER TEST? Check the one that applies. Suggestions to help ou stud are given with each item. Visit glencoe.com to access our tetbook, more eamples, self-check quizzes, and practice tests to help ou stud the concepts in Chapter 9. I completed the review of all or most lessons without using m notes or asking for help. You are probabl read for the Chapter Test. You ma want to take the Chapter 9 Practice Test on page 523 of our tetbook as a final check. I used m Foldable or Stud Notebook to complete the review of all or most lessons. You should complete the Chapter 9 Stud Guide and Review on pages 518 522 of our tetbook. If ou are unsure of an concepts or skills, refer to the specific lesson(s). You ma also want to take the Chapter 9 Practice Test on page 523. I asked for help from someone else to complete the review of all or most lessons. You should review the eamples and concepts in our Stud Notebook and Chapter 9 Foldable. Then complete the Chapter 9 Stud Guide and Review on pages 518 522 of our tetbook. If ou are unsure of an concepts or skills, refer to the specific lesson(s). You ma also want to take the Chapter 9 Practice Test on page 523. Student Signature Parent/Guardian Signature Teacher Signature 252 Math Connects, Course 3