3-1 Practice. Graphing Linear Equations

Transcription

1 NAME DATE PERID - Practice Graphing Linear Equations Determine whether each equation is a linear equation. Write es or no. If es, write the equation in standard form and determine the - and -intercepts.. + = 9. - = =. - =. - =. - = Graph each equation.. - =. - = = 9 0. CMMUNICATINS A telephone compan charges $.9 per month for long distance calls plus $0.0 per minute. The monthl cost c of long distance calls can be described b the equation c = 0.0m +.9, where m is the number of minutes. a. Find the -intercept of the graph of the equation. b. Graph the equation. c. If ou talk 0 minutes, what is the monthl cost?. MARINE BILGY Killer whales usuall swim at a rate of. 9. kilometers per hour, though the can travel up to. kilometers per hour. Suppose a migrating killer whale is swimming at an average rate of. kilometers per hour. The distance d the whale has traveled in t hours can be predicted b the equation d =.t. a. Graph the equation. b. Use the graph to predict the time it takes the killer whale to travel 0 kilometers. Cost ($) 0 Long Distance Time (minutes) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

2 NAME DATE PERID - Word Problem Practice Graphing Linear Equations. FTBALL ne football season, the Carolina Panthers won more games than the lost. This can be represented b = +, where is the number of games lost and is the number of games won. Write this linear equation in standard form.. TWING Pick-M-Up Towing Compan charges $0 to hook a car and $.0 for each mile that it is towed. The equation =. + 0 represents the total cost for miles towed. Determine the -intercept. Describe what the value means in this contet.. BUSINESS The equation = represents the monthl profits of a start-up dr cleaning compan. Time in months is and profit in dollars is. The first date of operation is when time is zero. However, preparation for opening the business began months earlier with the purchase of equipment and supplies. Graph the linear function for -values from - to. 000 Lesson Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. SHIPPING The CL Shenzhen, one of the world s largest container ships, carries 0 TEUs (0 cubic feet containers). Workers can unload a ship at a rate of a TEU ever minute. Using this rate, write and graph an equation to determine how man hours it will take the workers to unload half of the containers from the Shenzhen. TEUs on Ship (thousands) Time (hours) BNE GRWTH The height of a woman can be predicted b the equation h =. +.r, where h is her height in centimeters and r is the length of her radius bone in centimeters. a. Is this is a linear function? Eplain. b. What are the r- and h-intercepts of the equation? Do the make sense in the situation? Eplain. c. Use the function to find the approimate height of a woman whose radius bone is centimeters long. Chapter 9 Glencoe Algebra

3 NAME DATE PERID - Word Problem Practice Rate of Change and Slope. HIGHWAYS Roadwa signs such as the one below are used to warn drivers of an upcoming steep down grade that could lead to a dangerous situation. What is the grade, or slope, of the hill described on the sign?. REAL ESTATE The median price of an eisting single-famil home in the United States was $9,00 in 00. The median price had risen to $,900 b 00. Find the average annual rate of change in median home price from 00 to 00.. CAL EXPRTS The graph shows the annual coal eports from U.S. mines in millions of short tons.. AMUSEMENT PARKS The SheiKra roller coaster at Busch Gardens in Tampa, Florida, features a -foot vertical drop. What is the slope of the coaster track at this part of the ride? Eplain.. CENSUS The table shows the population densit for the state of Teas in various ears. Find the average annual rate of change in the population densit from 000 to 00. Population Densit Year People Per Square Mile Source: Bureau of the Census, U.S. Dept. of Commerce Million Short Tons Total Eports Source: Energ Information Association a. What was the rate of change in coal eports between 00 and 00? 00 b. How does the rate of change in coal eports from 00 to 00 compare to that of 00 to 00? c. Eplain the meaning of the part of the graph with a slope of zero. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

4 NAME DATE PERID - Enrichment Treasure Hunt with Slopes Using the definition of slope, draw segments with the slopes listed below in order. A correct solution will trace the route to the treasure. Treasure Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc no slope. 9. Start Here Lesson - Chapter Glencoe Algebra

5 NAME DATE PERID - Word Problem Practice Direct Variation. ENGINES The engine of a chainsaw requires a miture of engine oil and gasoline. According to the directions, oil and gasoline should be mied as shown in the graph below. What is the constant of variation for the line graphed? il (fl oz) 0 9. SALARY Henr started a new job in which he is paid $9.0 an hour. Write and solve an equation to determine Henr s gross salar for a 0-hour work week.. SALES TAX Amelia received a gift card to a local music shop for her birthda. She plans to use the gift card to bu some new CDs. a. Amelia chose CDs that each cost $. The sales ta on the three CDs is $.9. Write a direct variation equation relating sales ta to the price. 0 Gasoline (gal) b. Graph the equation ou wrote in part a.. RACING In 00, English driver Lewis Hamilton won the United States Grand Pri at the Indianapolis Motor Speedwa. His speed during the race averaged. miles per hour. Write a direct variation equation for the distance d that Hamilton drove in h hours at that speed.. CURRENCY The echange rate from U.S. dollars to British pound sterling ( ) was approimatel $.0 to in 00. Write and solve a direct variation equation to determine how man pounds sterling ou would receive in echange for $90 of U.S. currenc. c. What is the sales ta rate that Amelia is paing on the CDs? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

6 NAME DATE PERID -. PSTAGE In 00, the price for first class mail was raised to cents for the first ounce and cents for each additional ounce. The table below shows the cost for weights up to ounces. Weight (ounces) Postage (cents) Word Problem Practice Arithmetic Sequences as Linear Functions Source: United States Postal Servcie. NUMBER THERY ne of the most famous sequences in mathematics is the Fibonacci sequence. It is named after Leonardo de Pisa (0 0) or Filius Bonacci, alias Leonardo Fibonacci. The first several numbers in the Fibonacci sequence are:,,,,,,,,,, 9,... Does this represent an arithmetic sequence? Wh or wh not? How much did a letter weigh that cost $. to send?. SPRTS Wanda is the manager for the soccer team. ne of her duties is to hand out cups of water at practice. Each cup of water is ounces. She begins practice with a -ounce cooler of water. How much water is remaining after she hands out the th cup?. THEATER A theater has 0 seats in the first row, in the second row, in the third row, and so on for rows. How man seats are in the last row?. SAVINGS Inga s grandfather decides to start a fund for her college education. He makes an initial contribution of $000 and each month deposits an additional $00. After one month he will have contributed $00. a. Write an equation for the n th term of the sequence. b. How much mone will Inga s grandfather have contributed after months? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

7 Chapter Resource Masters

8 NAME DATE PERID - Skills Practice Graphing Linear Equations Determine whether each equation is a linear equation. Write es or no. If es, write the equation in standard form.. =. = -. = -. = +. = - +. = +. - = 0. + = + 9. = Find the - and -intercepts of each linear function. Lesson Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Graph each equation b making a table.. =. =. = + Graph each equation b using the - and -intercepts.. - =. 0 = -. = + Chapter Glencoe Algebra

9 NAME DATE PERID - Enrichment Translating Linear Graphs Linear graphs can be translated on the coordinate plane. This means that the graph moves up, down, right, or left without changing its direction. Translating the graphs up or down affects the -coordinate for a given value. Translating the graph right or left affects the -coordinate for a given -value. Eample Translate the graph of = +, units up. = Add to each -value. Translation - 0 = + 9 Eercises Graph the function and the translation on the same coordinate plane.. = +, units down. =, units left. = - +, unit right. = - -, units up Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 Glencoe Algebra

10 NAME DATE PERID - Spreadsheet Activit Linear Equations In addition to organizing data, a spreadsheet can be used to represent data graphicall. Eample An internet retailer charges $.99 per order plus $0.99 per item to ship books and CDs. Graph the equation = , where is the number of items ordered and is the shipping cost. Step Use column A for the numbers of items and column B for the shipping costs. Step Create a graph from the data. Select the data in columns A and B and select Chart from the Insert menu. Select an XY (Scatter) chart to show the data points connected with line segments. Shipping.ls A B Items Shipping Cost $.9 $.9 $.9 $.9 $.9 $.9 $.9 $9.9 $0.90 $.9 Sheet Sheet Sheet Lesson - Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Eercises. A photo printer offers a subscription for digital photo finishing. The subscription costs $.99 per month. Each standard size photo a subscriber prints costs $0.9. Use a spreadsheet to graph the equation = , where is the number of photos printed and is the total monthl cost.. A long distance service plan includes a $.9 per month fee plus $0.0 per minute of calls. Use a spreadsheet to graph the equation = , where is the number of minutes of calls and is the total monthl cost. Chapter Glencoe Algebra

11 NAME DATE PERID - Stud Guide and Intervention Solving Linear Equations b Graphing Solve b Graphing You can solve an equation b graphing the related function. The solution of the equation is the -intercept of the function. Eample Solve the equation - = - b graphing. First set the equation equal to 0. Then replace 0 with f(). Make a table of ordered pair solutions. Graph the function and locate the -intercept. - = = = 0 f() = + riginal equation Add to each side. Simplif. Replace 0 with f(). To graph the function, make a table. Graph the ordered pairs. f() = + f() [, f()] f() = () + (, ) - f(-) = (-) + 0 (-, 0) - f(-) = (-) + - (-, -) The graph intersects the -ais at (-, 0). The solution to the equation is = -. Eercises Solve each equation.. - = = = 0. 0 = -. - =. - + = 0 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

12 NAME DATE PERID - Stud Guide and Intervention (continued) Solving Linear Equations b Graphing Estimate Solutions b Graphing Sometimes graphing does not provide an eact solution, but onl an estimate. In these cases, solve algebraicall to find the eact solution. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Eample WALKING You and our cousin decide to walk the -mile trail at the state park to the ranger station. The function d =.t represents our distance d from the ranger station after t hours. Find the zero of this function. Describe what this value means in this contet. Make a table of values to graph the function. t d = -.t d (t, d) 0 d = -.(0) (0, ) d = -.(). (,.) d = -.() 0. (, 0.) The graph intersects the t ais between t = and t =, but closer to t =. It will take ou and our cousin just over two hours to reach the ranger station. You can check our estimate b solving the equation algebraicall. Eercises. MUSIC Jessica wants to record her favorite songs to one CD. The function C = 0 -.n represents the recording time C available after n songs are recorded. Find the zero of this function. Describe what this value means in this contet.. GIFT CARDS Enrique uses a gift card to bu coffee at a coffee shop. The initial value of the gift card is $0. The function n = 0.c represents the amount of mone still left on the gift card n after purchasing c cups of coffee. Find the zero of this function. Describe what this value means in this contet. Miles from Ranger Station 0 Value Left on Card ($) Time (hours) Time Available (min) Number of Songs 0 Coffees Bought Lesson - Chapter Glencoe Algebra

13 NAME DATE PERID - Skills Practice Solving Linear Equations b Graphing Solve each equation.. - = = 0. + = -. - = +. - = 0. 0 = +. 0 = = = 0 0. GIFT CARDS You receive a gift card for trading cards from a local store. The function d = 0.9c represents the remaining dollars d on the gift card after obtaining c packages of cards. Find the zero of this function. Describe what this value means in this contet. Amount Remaining on Gift Card ($) d c Packages of Cards Bought Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

14 NAME DATE PERID - Practice Solve each equation. Solving Linear Equations b Graphing. - = = -. - = -. + = -. + =. + = - Lesson - Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Solve each equation b graphing. Verif our answer algebraicall. + = = = - 0. DISTANCE A bus is driving at 0 miles per hour toward a bus station that is 0 miles awa. The function d = 0 0t represents the distance d from the bus station the bus is t hours after it has started driving. Find the zero of this function. Describe what this value means in this contet. Distance from Bus Station (miles) Time (hours) Chapter Glencoe Algebra

15 NAME DATE PERID - Word Problem Practice Solving Linear Equations b Graphing. PET CARE You bu a.-pound bag of dr cat food for our cat. The function c =. 0.p represents the amount of cat food c remaining in the bag when the cat is fed the same amount each da for p das. Find the zero of this function. Describe what this value means in this contet.. SAVINGS Jessica is saving for colle ge using a direct deposit from her pacheck into a savings account. The function m = 0 -.0t represents the amount of mone m still needed after t weeks. Find the zero of this function. What does this value mean in this contet?. DENTAL HYGIENE You are packing our suitcase to go awa to a -da summer camp. The store carries three sizes of tubes of toothpaste. Tube Size (ounces) Size (grams) A 0.. B 0.9. C.0.0 Source: National Academ of Sciences a. The function n =. - 0.b represents the number of remaining brushings n using b grams per brushing using Tube A. Find the zero of this function. Describe what this value means in this contet.. FINANCE Michael borrows $00 from his dad. The function v = 00 -.p represents the outstanding balance v after p weekl paments. Find the zero of this function. Describe what this value means in this contet.. BAKE SALE Ashle has $ in the Pep Club treasur to pa for supplies for a chocolate chip cookie bake sale. The function d = 0.0c represents the dollars d left in the club treasur after making c cookies. Find the zero of this function. What does this value represent in this contet? b. The function n =. 0.b represents the number of remaining brushings n using b grams per brushing using Tube B. Find the zero of this function. Describe what this value means in this contet. c. Write a function to represent the number of remaining brushings n using b grams per brushing using Tube C. Find the zero of this function. Describe what this value means in this contet. d. If ou will brush our teeth twice each da while at camp, which is the smallest tube of toothpaste ou can choose? Eplain our reasoning. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

16 NAME DATE PERID - Enrichment Composite Functions Three things are needed to have a function a set called the domain, a set called the range, and a rule that matches each element in the domain with onl one element in the range. Here is an eample. Rule: f() = + f() - - f() = + f() = () + = + = f() = () + = + = f(-) = (-) + = + = - Suppose we have three sets A, B, and C and two functions described as shown below. Rule: f() = + Rule: g( ) = - A B C f() g[f()] g( ) = - g() = () - = Lesson - Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Let s find a rule that will match elements of set A with elements of set C without finding an elements in set B. In other words, let s find a rule for the composite function g[ f()]. Since f() = +, g[ f()] = g( + ). Since g( ) = -, g( + ) = ( + ) -, or -. Therefore, g[ f()] = -. Find a rule for the composite function g[f()].. f() = and g( ) = +. f() + and g( ) =. f() = - and g( ) = -. f() = and g( ) = - -. Is it alwas the case that g[ f()] = f[ g()]? Justif our answer. Chapter Glencoe Algebra

17 NAME DATE PERID - Stud Guide and Intervention Rate of Change and Slope Rate of Change The rate of change tells, on average, how a quantit is changing over time. Eample PPULATIN The graph shows the population growth in China. a. Find the rates of change for 90 9 and for : 000 0: change in population = change in time 9-90 = 0. or 0.0 change in population = change in time = 0. or 0.00 Source: United Nations Population Division b. Eplain the meaning of the rate of change in each case. From 90 9, the growth was 0.0 billion per ear, or. million per ear. From 000 0, the growth is epected to be 0.00 billion per ear, or. million per ear. c. How are the different rates of change shown on the graph? There is a greater vertical change for 90 9 than for Therefore, the section of the graph for 90 9 has a steeper slope. Eercises. LNGEVITY The graph shows the predicted life epectanc for men and women born in a given ear. a. Find the rates of change for women from and b. Find the rates of change for men from and c. Eplain the meaning of our results in Eercises and. d. What pattern do ou see in the increase with each -ear period? Population Growth in China Predicting Life Epectanc 000 0* 00* Year Born Women Men *Estimated e. Make a prediction for the life epectanc for Eplain how ou arrived at our prediction. People (billions) Age * Year *Estimated Source: USA TDAY Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

18 NAME DATE PERID - Stud Guide and Intervention (continued) Rate of Change and Slope Find Slope The slope of a line is the ratio of change in the - coordinates (rise) to the change in the - coordinates (run) as ou move in the positive direction. Slope of a Line m = rise run or m = - -, where (, ) and (, ) are the coordinates of an two points on a nonvertical line Eample Find the slope of the Eample line that passes through (-, ) and (, -). Let (-, ) = (, ) and m = - (, -) = (, ). - m = - - Slope formula - = - r = = -, =, =, = - = - r (-) = - = - Simplif. Find the value of r so that the line through (0, r) and (, ) has a slope of -. Slope formula m = -, =, = r, =, = 0 Simplif. -(-) = ( - r) Cross multipl. = - r Distributive Propert - = -r Subtract from each side. = r Divide each side b -. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Eercises Find the slope of the line that passes through each pair of points.. (, 9), (, ). (-, -), (-, -). (-, -), (-, -). (, ), (, 9). (, -), (, -). (, -), (, -). (, -), (, ). (, ), (, ) 9. (,.), (-,.) Find the value of r so the line that passes through each pair of points has the given slope. 0. (, ), (r, -), m =. (-, -), (, r), m =. (, -), (, r), m = 0. (r, ), (, ), m =. (, ), (r, -) m = -. (, ), (r, 9), m = Lesson - Chapter 9 Glencoe Algebra

19 NAME DATE PERID - Skills Practice Rate of Change and Slope Find the slope of the line that passes through each pair of points.. (, ).. (0, ) (0, ) (, ) (0, 0) (, -). (, ), (, ). (, ), (-, ). (, ), (, ). (, ), (, -). (, ), (-, -) 9. (9, ), (, -) 0. (-, -), (-, ). (-, 0), (-, ). (, ), (, ). (-, -), (, ). (0, 0), (-, ). (, -), (-, -). (, -9), (, -). (, ), (, -). (-, ), (-, -) 9. (-, ), (, -) Find the value of r so the line that passes through each pair of points has the given slope. 0. (r, ), (, 9), m =. (, 9), (r, -), m = -. (r, ), (, ), m =. (r, ), (, ), m = Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. (, ), (r, -), m =. (, r), (, ), m = 0 Chapter 0 Glencoe Algebra

20 NAME DATE PERID - Practice Rate of Change and Slope Find the slope of the line that passes through each pair of points.. (, ).. (, ) (, ) (, ) (, 0) (, ). (, ), (, -). (-9, -), (-, -). (, -), (, -). (, -), (, ). (-, ), (-, ) 9. (, 9), (, 9) 0. (, ), (-, ). (, 9), (-, ). (-, -), (, ). (, 0), (, ) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. (0., -0.9), (0., -0.9). (,, ) (-, ) Find the value of r so the line that passes through each pair of points has the given slope.. (-, r), (, ), m =. (-, -), (-, r), m = - 9. (-, ), (r, ), m = 9. (-, r), (, ), m = 0. (, ), (r, ), m undefined. (-, ), (-, r), m = -. (r, ), (, ), m = -. (r, ), (, r), m = 0. RFING The pitch of a roof is the number of feet the roof rises for each feet horizontall. If a roof has a pitch of, what is its slope epressed as a positive number? Lesson -. SALES A dail newspaper had, subscribers when it began publication. Five ears later it had 0,00 subscribers. What is the average earl rate of change in the number of subscribers for the five-ear period? Chapter Glencoe Algebra

21 NAME DATE PERID - Stud Guide and Intervention Direct Variation Direct Variation Equations A direct variation is described b an equation of the form = k, where k 0. We sa that varies directl as. In the equation = k, k is the constant of variation. Eample Name the constant of Eample Suppose varies directl variation for the equation. Then find as, and = 0 when =. the slope of the line that passes a. Write a direct variation equation through the pair of points. that relates and. Find the value of k. = = k Direct variation equation (, ) 0 = k() Replace with 0 and with. (0, 0) = k Divide each side b. Therefore, the equation is =. For =, the constant of variation is. m = - - = = The slope is. Eercises Slope formula (, ) = (0, 0), (, ) = (, ) Simplif. b. Use the direct variation equation to find when =. = Direct variation equation = Replace with. = Divide each side b. Therefore, = when =. Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points... (, ) (0, 0) = (, ) = (0, 0) Suppose varies directl as. Write a direct variation equation that relates to. Then solve.. If = when =, find when =.. If = 9 when = -, find when =.. If = -. when = -., find when = -.. If = when =, find when =.. (, ) (0, 0) = Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

22 NAME DATE PERID - Stud Guide and Intervention (continued) Direct Variation Direct Variation Problems The distance formula d = rt is a direct variation equation. In the formula, distance d varies directl as time t, and the rate r is the constant of variation. Eample TRAVEL A famil drove their car miles in hours. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. a. Write a direct variation equation to find the distance traveled for an number of hours. Use given values for d and t to find r. d = rt riginal equation = r() d = and t = = r Divide each side b. Therefore, the direct variation equation is d = t. b. Graph the equation. The graph of d = t passes through the origin with slope. m = rise run CHECK (, ) lies on the graph. c. Estimate how man hours it would take the famil to drive 0 miles. d = t riginal equation 0 = t Replace d with 0. t = Divide each side b. Therefore, it will take hours to drive 0 miles. Eercises. RETAIL The total cost C of bulk jell beans is $.9 times the number of pounds p. a. Write a direct variation equation that relates the variables. b. Graph the equation on the grid at the right. c. Find the cost of pound of jell beans.. CHEMISTRY Charles s Law states that, at a constant pressure, volume of a gas V varies directl as its temperature T. A volume of cubic feet of a certain gas has a temperature of 00 degrees Kelvin. a. Write a direct variation equation that relates the variables. b. Graph the equation on the grid at the right. c. Find the volume of the same gas at 0 degrees Kelvin. Distance (miles) d Automobile Trips d = t (, ) 90 (, ) 0 t Time (hours) Cost (dollars) Volume (cubic feet) Cost of Jell Beans C V 0 w Weight (pounds) Charles s Law T Temperature (K) Lesson - Chapter Glencoe Algebra

23 NAME DATE PERID - Skills Practice Direct Variation Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points.. (, ) (0, 0).. (-, ) (0, 0) (, ) (0, 0) = = - = Graph each equation.. =. = -. = Suppose varies directl as. Write a direct variation equation that relates and. Then solve.. If = - when = -, find. If = when =, find when =. when =. 9. If = - when, find 0. If = -9 when =, find when = -. when = -.. If = when =, find. If = when =, find when =. when = -. Write a direct variation equation that relates the variables. Then graph the equation.. TRAVEL The total cost C of gasoline. SHIPPING The number of delivered tos T is $.00 times the number of gallons g. is times the total number of crates c. Cost ($) C 0 Gasoline Cost Tos T 9 Tos Shipped Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0 g Gallons 0 c Crates Chapter Glencoe Algebra

24 NAME DATE PERID - Practice Direct Variation Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points.... = (, ) (, ) (, ) = - (0, 0) (0, 0) = (0, 0) Graph each equation.. = -. =. = - Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Suppose varies directl as. Write a direct variation equation that relates and. Then solve.. If =. when = 0., find when = If = 0 when =, find when = If = when =, find when =. Write a direct variation equation that relates the variables. Then graph the equation. 0. MEASURE The width W of a. TICKETS The total cost C of tickets is rectangle is two thirds of the length l. $.0 times the number of tickets t. Width Rectangle Dimensions W l Length. PRDUCE The cost of bananas varies directl with their weight. Miguel bought pounds of bananas for $.. Write an equation that relates the cost of the bananas to their weight. Then find the cost of pounds of bananas. Chapter Glencoe Algebra Lesson -

25 NAME DATE PERID - Enrichment nth Power Variation An equation of the form = k n, where k 0, describes an nth power variation. The variable n can be replaced b to indicate the second power of (the square of ) or b to indicate the third power of (the cube of ). Assume that the weight of a person of average build varies directl as the cube of that person s height. The equation of variation has the form w = kh. The weight that a person s legs will support is proportional to the cross-sectional area of the leg bones. This area varies directl as the square of the person s height. The equation of variation has the form s = kh. Answer each question.. For a person feet tall who weighs 00 pounds, find a value for k in the equation w = kh.. Use our answer from Eercise to predict the weight of a person who is feet tall.. Find the value for k in the equation w = kh for a bab who is 0 inches long and weighs pounds. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. How does our answer to Eercise demonstrate that a bab is significantl fatter in proportion to its height than an adult?. For a person feet tall who weighs 00 pounds, find a value for k in the equation s = kh.. For a bab who is 0 inches long and weighs pounds, find an infant value for k in the equation s = kh.. According to the adult equation ou found (Eercise ), how much would an imaginar giant 0 feet tall weigh?. According to the adult equation for weight supported (Eercise ), how much weight could a 0-foot tall giant s legs actuall support? Lesson - 9. What can ou conclude from Eercises and? Chapter 9 Glencoe Algebra

26 NAME DATE PERID - Stud Guide and Intervention Arithmetic Sequences as Linear Functions Recognize Arithmetic Sequences A sequence is a set of numbers in a specific order. If the difference between successive terms is constant, then the sequence is called an arithmetic sequence. Arithmetic Sequence Terms of an Arithmetic Sequence a numerical pattern that increases or decreases at a constant rate or value called the common difference If a is the first term of an arithmetic sequence with common difference d, then the sequence is a, a + d, a + d, a + d,.... nth Term of an Arithmetic Sequence a n = a + (n - )d Eample Determine whether the Eample sequence,,,, 9,,... is an arithmetic sequence. Justif our answer. If possible, find the common difference between the terms. Since - =, - =, and so on, the common difference is. Since the difference between the terms of,,,, 9,,... is constant, this is an arithmetic sequence. Eercises Write an equation for the nth term of the sequence,,,,.... In this sequence, a is. Find the common difference The common difference is. Use the formula for the nth term to write an equation. a n = a + (n - )d Formula for the nth term a n = + (n - ) a =, d = a n = + n - a n = n + 9 Distributive Propert Simplif. The equation for the nth term is a n = n + 9. Determine whether each sequence is an arithmetic sequence. Write es or no. Eplain..,, 9,,,....,, 0, -, -,....,, 9,,,... Find the net three terms of each arithmetic sequence.. 9,,,,,...., 0, -, -, -,.... 9,,,,... Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc..,,,,.... -, -, -, -0, , -9, -, -9,... Chapter 0 Glencoe Algebra

27 NAME DATE PERID - Stud Guide and Intervention (continued) Arithmetic Sequences as Linear Functions Arithmetic Sequences and Functions An arithmetic sequence is a linear function in which n is the independent variable, a n is the dependent variable, and the common difference d is the slope. The formula can be rewritten as the function a n = a + (n-)d, where n is a counting number. Eample SEATING There are 0 seats in the first row of the balcon of the auditorium. There are seats in the second row, and seats in the third row. a. Write a function to represent this sequence. The first term a is 0. Find the common difference. b. Graph the function. The rate of change is. Make a table and plot points. 0 n a n a n + + The common difference is. a n = a + (n - )d = 0 + (n - ) = 0 + n - = + n formula for the nth term a = 0 and d = Distributive Propert Simplif. 0 0 n Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The function is a n = + n. Eercises. KNITTING Sarah learns to knit from her grandmother. Two das ago, she measured the length of the scarf she is knitting to be inches. Yesterda, she measured the length of the scarf to be. inches. Toda it measures inches. Write a function to represent the arithmetic sequence.. REFRESHMENTS You agree to pour water into the cups for the Booster Club at a football game. The pitcher contains ounces of water when ou begin. After ou have filled cups, the pitcher is empt and must be refilled. a. Write a function to represent the arithmetic sequence. b. Graph the function. a n 0 n Lesson - Chapter Glencoe Algebra

28 NAME DATE PERID - Skills Practice Arithmetic Sequences as Linear Functions Determine whether each sequence is an arithmetic sequence. Write es or no. Eplain..,, 9,,....,,, 9,...., 0,,,.... -, -, -, -,.... -, -, -,, , -, -, -,.... 0,,, 0, , -, 0,,... Find the net three terms of each arithmetic sequence. 9.,,,,... 0., 0,,,.... -, -, -9, , -, -, -,.... 9,, 9,,....,, -, -,.....,,., 0,.....,.,.,.,... Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence..,, 9,,.... 0,,,, , -, -,,... a n n a n n n 0. VIDE DWNLADING Brian is downloading episodes of his favorite TV show to pla on his personal media device. The cost to download episode is $.99. The cost to download episodes is $.9. The cost to download episodes is $.9. Write a function to represent the arithmetic sequence. - - a n Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

29 NAME DATE PERID - Practice Arithmetic Sequences as Linear Functions Determine whether each sequence is an arithmetic sequence. Write es or no. Eplain..,,, -,.... -,, 9,, , -., 0.,.,....,, 9,,.... 9,,, 0, , 0.,.,.0,... Find the net three terms of each arithmetic sequence..,, 0,, , -, -, -,... 9.,,, 0, , -,,...., 0,,,....,,, -,... Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence.. 9,,,,.... -, -,,,.... 9,,,,... Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. a n n a n n. BANKING Chem deposited $.00 in a savings account. Each week thereafter, he deposits $.00 into the account. a. Write a function to represent the total amount Chem has deposited for an particular number of weeks after his initial deposit. b. How much has Chem deposited 0 weeks after his initial deposit?. STRE DISPLAYS Tamika is stacking boes of tissue for a store displa. Each row of tissues has fewer boes than the row below. The first row has boes of tissues. a. Write a function to represent the arithmetic sequence. b. How man boes will there be in the tenth row? - a n n Lesson - Chapter Glencoe Algebra

30 NAME DATE PERID - Enrichment Arithmetic Series An arithmetic series is a series in which each term after the first ma be found b adding the same number to the preceding term. Let S stand for the following series in which each term is more than the preceding one. S = The series remains the same if we S = reverse the order of all the terms. So S = let us reverse the order of the terms S = and add one series to the other, term S = () b term. This is shown at the right. S = () = () = Let a represent the first term of the series. Let l represent the last term of the series. Let n represent the number of terms in the series. In the preceding eample, a =, l = 0, and n =. Notice that when ou add the two series, term b term, the sum of each pair of terms is. That sum can be found b adding the first and last terms, + 0 or a + l. Notice also that there are, or n, such sums. Therefore, the value of S is (), or n(a + l) in the general case. Since this is twice the sum n(a + l) of the series, ou can use the formula S = to find the sum of an arithmetic series. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Eample Find the sum: ( + 9) 9 a =, l = 9, n = 9, so S = =. 0 = Eample Find the sum: -9 + (-) + (-) (-9 + ) a = 9, l =, n =, so S = =. = Eercises Find the sum of each arithmetic series (-) + (-) + (-) + (-) even whole numbers from through 00. odd whole numbers between 0 and 00 Lesson - Chapter Glencoe Algebra

31 NAME DATE PERID - Stud Guide and Intervention Proportional and Nonproportional Relationships. Proportional Relationships If the relationship between the domain and range of a relation is linear, the relationship can be described b a linear equation. If the equation passes through (0, 0) and is of the form = k, then the relationship is proportional. Eample CMPACT DISCS Suppose ou purchased a number of packages of blank compact discs. If each package contains compact discs, ou could make a chart to show the relationship between the number of packages of compact discs and the number of discs purchased. Use for the number of packages and for the number of compact discs. Make a table of ordered pairs for several points of the graph. Number of Packages Number of CDs 9 The difference in the values is, and the difference in the values is. This pattern shows that is alwas three times. This suggests the relation =. Since the relation is also a function, we can write the equation in function notation as f() =. The relation includes the point (0, 0) because if ou bu 0 packages of compact disks, ou will not have an compact discs. Therefore, the relationship is proportional. Eercises. NATURAL GAS Natural gas use is often measured in therms. The total amount a gas compan will charge for natural gas use is based on how much natural gas a household uses. The table shows the relationship between natural gas use and the total cost. Gas Used (therms) Total Cost ($) $.0 $.0 $.90 $.0 a. Graph the data. What can ou deduce from the pattern about the relationship between the number of therms used and the total cost? b. Write an equation to describe this relationship. c. Use this equation to predict how much it will cost if a household uses 0 therms. Total Cost ($) 0 Gas Used (therms) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

32 NAME DATE PERID - Stud Guide and Intervention (continued) Proportional and Nonproportional Relationships Nonproportional Relationships If the ratio of the value of to the value of is different for select ordered pairs on the line, the equation is nonproportional. Eample in the graph. Write an equation in functional notation for the relation shown Lesson - Select points from the graph and place them in a table The difference between the values is, while the difference between the -values is. This suggests that =. If =, then = () or. But the value for = is is alwas more than - Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. This pattern shows that should be added to one side of the equation. Thus, the equation is = - +. Eercises Write an equation in function notation for the relation shown in the table. Then complete the table Write an equation in function notation for each relation Chapter Glencoe Algebra

33 NAME DATE PERID - Skills Practice Proportional and Nonproportional Relationships Write an equation in function notation for each relation.. f(). f().. f() f(). f().. GAMESHWS The table shows how man points are awarded for answering consecutive questions on a gameshow. Question answered Points awarded f() Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. a. Write an equation for the data given. b. Find the number of points awarded if 9 questions were answered. Chapter Glencoe Algebra

34 NAME DATE PERID - Practice Proportional and Nonproportional Relationships. BILGY Male fireflies flash in various patterns to signal location and perhaps to ward off predators. Different species of fireflies have different flash characteristics, such as the intensit of the flash, its rate, and its shape. The table below shows the rate at which a male firefl is flashing. Times (seconds) Number of Flashes 0 Lesson - a. Write an equation in function notation for the relation. b. How man times will the firefl flash in 0 seconds?. GEMETRY The table shows the number of diagonals that can be drawn from one verte in a polgon. Write an equation in function notation for the relation and find the number of diagonals that can be drawn from one verte in a -sided polgon. Sides Diagonals 0 Write an equation in function notation for each relation.... Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. For each arithmetic sequence, determine the related function. Then determine if the function is proportional or nonproportional. Eplain..,,,....,,,.... -, -, -9,... Chapter 9 Glencoe Algebra

35 NAME DATE PERID - Word Problem Practice Proportional and Nonproportional Relationships. NLINE SHPPING Ricardo is buing computer cables from an online store. If he bus cables, the total cost will be $. If he bus cables, the total cost will be $9. If the total cost can be represented b a linear function, will the function be proportional or nonproportional? Eplain.. FD It takes about four pounds of grapes to produce one pound of raisins. The graph shows the relation for the number of pounds of grapes needed,, to make pounds of raisins. Write an equation in function notation for the relation shown..0 f(). MUSIC A measure of music contains the same number of beats throughout the song. The table shows the relation for the number of beats counted after a certain number of measures have been plaed in the si-eight time. Write an equation to describe this relationship. Measures Plaed () Total Number of Beats () Source: Sheet Music USA 0. GEMETRY A fractal is a pattern containing parts which are identical to the overall pattern. The following geometric pattern is a fractal. Pounds of Raisins Pounds of Grapes. PARKING Palmer Township recentl installed parking meters in their municipal lot. The cost to park for h hours is represented b the equation C = 0.h. a. Make a table of values that represents this relationship. a. Complete the table. Term Number of Smaller Triangles b. What are the net three numbers in the pattern? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. b. Describe the relationship between the time parked and the cost. c. Write an equation in function notation for the pattern. Chapter 0 Glencoe Algebra

36 NAME DATE PERID - Enrichment Taicab Graphs You have used a rectangular coordinate sstem to graph equations such as = - on a coordinate plane. In a coordinate plane, the numbers in an ordered pair (, ) can be an two real numbers. A taicab plane is different from the usual coordinate plane. The onl points allowed are those that eist along the horizontal and vertical grid lines. You ma think of the points as taicabs that must sta on the streets. The taicab graph shows the equations = - and = -. Notice that one of the graphs is no longer a straight line. It is now a collection of separate points. = - = - Lesson - Graph these equations on the taicab plane at the right.. = +. = - +. =.. = - Use our graphs for these problems. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. Which of the equations has the same graph in both the usual coordinate plane and the taicab plane?. Describe the form of equations that have the same graph in both the usual coordinate plane and the taicab plane. In the taicab plane, distances are not measured diagonall, but along the streets. Write the tai-distance between each pair of points.. (0, 0) and (, ). (0, 0) and (-, ) 9. (0, 0) and (,.) 0. (, ) and (, ). (, ) and (-, ). (0, ) and (-, 0) Draw these graphs on the taicab grid at the right.. The set of points whose tai-distance from (0, 0) is units.. The set of points whose tai-distance from (, ) is units. Chapter Glencoe Algebra

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38 NAME DATE PERID Student Recording Sheet Use this recording sheet with pages of the Student Edition. Multiple Choice Read each question. Then fill in the correct answer.. A B C D. A B C D. A B C D Assessment. F G H J. F G H J. F G H J Short Response/Gridded Response Record our answer in the blank. For gridded response questions, also enter our answer in the grid b writing each number or smbol in a bo. Then fill in the corresponding circle for that number or smbol. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. (grid in) (grid in).. 9 Etended Response Record our answers for Question on the back of this paper Chapter Glencoe Algebra

39 NAME DATE PERID Rubric for Scoring Etended Response General Scoring Guidelines If a student gives onl a correct numerical answer to a problem but does not show how he or she arrived at the answer, the student will be awarded onl credit. All etendedresponse questions require the student to show work. A full correct answer for a multiple-part question requires correct responses for all parts of the question. For eample, if a question has three parts, the correct response to one or two parts of the question that required work to be shown is not considered a full correct response. Students who use trial and error to solve a problem must show their method. Merel showing that the answer checks or is correct is not considered a complete response for full credit. Eercise Rubric Score Specific Criteria For part a, the eplanation must demonstrate an understanding that because each element of the domain is paired with eactl one element of the range, the table must represent a function. For part b, the eplanation must demonstrate an understanding that it cannot be represented b an equation of the form = k. A generall correct solution, but ma contain minor flaws in reasoning or computation. A partiall correct interpretation and/or solution to the problem. A correct solution with no evidence or eplanation. 0 An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Glencoe Algebra

40 NAME DATE PERID Chapter Quiz (Lessons - and -). Determine whether = - is a linear equation. If so, write the equation in standard form.. MULTIPLE CHICE What is the -intercept when - = - is graphed? A - B - C D.. Assessment. The distance d in miles that a car travels in t hours at a rate of miles per hour is given b the equation d = t. What is the best estimate of how far a car travels in hours?.. Graph - =... Solve - = 0.. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID Chapter Quiz (Lesson -) Find the slope of the line passing through each pair of points.. (, ) and (-, ). (9, ) and (, -). MULTIPLE CHICE Which value of r gives the line passing through (, ) and (r, -) a slope of? A - B - C D. Determine whether the function is linear. Write linear or not linear In 00, there were approimatel students in the Delaware High School band. In 00, that number increased to 0. Find the annual rate of change in the number of students in the band.. Chapter Glencoe Algebra

41 NAME DATE PERID Chapter Quiz (Lessons - and -) SCRE. MULTIPLE CHICE Suppose varies directl as. If = when =, find when =.. A B 0 C D 0.. Chris s wages var directl as the time she works. If her wages for 0 hours are $0, what are her wages for hours?.. Determine whether,, 9,,... is an arithmetic sequence... Find the net three terms of the arithmetic sequence, 9,,, Find the th term of the arithmetic sequence if a = - and d =.. NAME DATE PERID Chapter Quiz (Lesson -) For Questions, find the function that represents the following relationship Is the following relationship proportional? Write es or no Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. What values for a and b will make this linear relationship proportional? 0 a 9 b. Chapter Glencoe Algebra

42 NAME DATE PERID SCRE Chapter Mid-Chapter Test (Lessons - through -) Part I Write the letter for the correct answer in the blank at the right of each question.. Find the slope of the line through (, -) and (, -). A - B C D -. Find the slope of the line through (0, ) and (, ). F 0 G H J undefined. In 00, quadrillion Btu (British thermal units) of energ were consumed worldwide. It is epected that quadrillion thermal units will be consumed in 0. Which of the following is the epected rate of change in consumption from 00 to 0? A about 0.09 quadrillion Btu per ear B about. quadrillion Btu per ear C about quadrillion Btu per ear D about 0.0 ears per quadrillion Btu... Assessment. The total price of a bag of peaches varies directl with the cost per pound. If pounds of peaches cost $.0, how much would. pounds cost? F $.0 G $.0 H $.00 J $.9. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.. A carpenter can build cabinets in hours. At this rate, how long will it take for the carpenter to build cabinets? A hours C hours B 9 hours minutes D 9 hours 0 minutes Part II. Solve - = +.. Find the value of r so the line that passes through (-, ) and (, r) has a slope of -.. Determine whether = - is a linear equation. If so, write the equation in standard form. 9. Graph - = b using the - and -intercepts Chapter Glencoe Algebra

43 NAME DATE PERID Chapter Vocabular Test arithmetic sequence constant constant of variation direct variation famil of functions inductive reasoning linear equation parent function rate of change root slope standard form term -intercept -intercept Choose from the terms above to complete each sentence.. In an arithmetic sequence the constant rate of increase or decrease between successive terms is called the.. The zero of a function is its.. The of an equation is an value that makes the equation true.. An equation in the form A + B = C is called the of a linear equation.. In a(n), the difference between terms is constant.. For a(n) A + B = C, the numbers A, B, and C cannot all be zero A specific number in a sequence is called a.. The act of using a pattern to find a general rule is known as. Define each term in our own words. 9. -intercept 0. rate of change.. 9. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0. Chapter Glencoe Algebra