2.1. Functions and Their Graphs. What you should learn

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1 Page of 8. Functions and Their Graphs What ou should learn GOAL Represent relations and functions. GOAL Graph and evaluate linear functions, as applied in Es. 55 and 56. Wh ou should learn it To model real-life quantities, such as the distance a hot air balloon travels in Eample 6. GOAL REPRESENTING RELATIONS AND FUNCTIONS A relation is a mapping, or pairing, of input values with output values. The set of input values is the domain, and the set of output values is the range. A relation is a function provided there is eactl one output for each input. It is not a function if at least one input has more than one output. Relations (and functions) between two quantities can be represented in man was, including mapping diagrams, tables, graphs, equations, and verbal descriptions. EXAMPLE Identifing Functions Identif the domain and range. Then tell whether the relation is a function. a. Input Output b. Input Output a. The domain consists of º,, and 4, and the range consists of º,,, and 4. The relation is not a function because the input is mapped onto both º and. b. The domain consists of º,,, and 4, and the range consists of º,, and. The relation is a function because each input in the domain is mapped onto eactl one output in the range A relation can be represented b a set of ordered pairs of the form (, ). In an ordered pair the first number is the -coordinate and the second number is the -coordinate. To graph a relation, plot each of its ordered pairs in a coordinate plane, such as the one shown. A coordinate plane is divided into four quadrants b the -ais and the -ais. The aes intersect at a point called the origin. Stud Tip Although the origin O is not usuall labeled, it is understood to be the point (0, 0). 4 -ais Quadrant II Quadrant I 0, 0 0, 0 -ais O origin Quadrant III 0, 0 (0, 0) Quadrant IV 0, 0 4. Functions and Their Graphs 67

2 Page of 8 EXAMPLE Graphing Relations Graph the relations given in Eample. Skills Review For help with plotting points in a coordinate plane, see p. 9. a. Write the relation as a set of b. Write the relation as a set of ordered pairs: (º, ), (, º), ordered pairs: (º, ), (, ), (, ), (4, 4). Then plot the (, ), (4, º). Then plot the points in a coordinate plane. points in a coordinate plane. (, ) (4, 4) (, ) (, ) (, ) (, ) (, ) (4, ) In Eample notice that the graph of the relation that is not a function (the graph on the left) has two points that lie on the same vertical line. You can use this propert as a graphical test for functions. VERTICAL LINE TEST FOR FUNCTIONS A relation is a function if and onl if no vertical line intersects the graph of the relation at more than one point. FOCUS ON CAREERS Variables other than and are often used when working with relations in real-life situations, as shown in the net eample. EXAMPLE Using the Vertical Line Test in Real Life FORESTER A forester manages, develops, and protects natural resources. To measure the diameter of trees, a forester uses a special tool called diameter tape. CAREER LINK INTERNET FORESTRY The graph shows the ages a and diameters d of several pine trees at Lundbreck Falls in Canada. Are the diameters of the trees a function of their ages? Eplain. Source: National Geographical Data Center The diameters of the trees are not a function of their ages because there is at least one vertical line that intersects the graph at more than one point. For eample, a vertical line intersects the graph at the points (75,.) and (75,.58). So, at least two trees have the same age but different diameters. Diameter (meters) d Pine Trees Age (ears) a 68 Chapter Linear Equations and Functions

3 Page of 8 GOAL GRAPHING AND EVALUATING FUNCTIONS Man functions can be represented b an equation in two variables, such as = º 7. An ordered pair (, ) is a solution of such an equation if the equation is true when the values of and are substituted into the equation. For instance, (, º) is a solution of = º 7 because º = () º 7 is a true statement. In an equation, the input variable is called the independent variable. The output variable is called the dependent variable and depends on the value of the input variable. For the equation = º 7, the independent variable is and the dependent variable is. The graph of an equation in two variables is the collection of all points (, ) whose coordinates are solutions of the equation. GRAPHING EQUATIONS IN TWO VARIABLES To graph an equation in two variables, follow these steps: STEP STEP STEP Construct a table of values. Graph enough solutions to recognize a pattern. Connect the points with a line or a curve. EXAMPLE 4 Graphing a Function Graph the function = +. Begin b constructing a table of values. Choose. º º 0 Evaluate. º 0 Plot the points. Notice the five points lie on a line. Draw a line through the points Stud Tip When ou see function notation ƒ(), remember that it means the value of ƒ at. It does not mean ƒ times. The function in Eample 4 is a linear function because it is of the form = m + b Linear function where m and b are constants. The graph of a linear function is a line. B naming a function ƒ ou can write the function using function notation. ƒ() = m + b Function notation The smbol ƒ() is read as the value of ƒ at, or simpl as ƒ of. Note that ƒ() is another name for. The domain of a function consists of the values of for which the function is defined. The range consists of the values of ƒ() where is in the domain of ƒ. Functions do not have to be represented b the letter ƒ. Other letters such as g or h can also be used.. Functions and Their Graphs 69

4 Page 4 of 8 EXAMPLE 5 Evaluating Functions Decide whether the function is linear. Then evaluate the function when = º. a. ƒ() = º º + 5 b. g() = + 6 HOMEWORK HELP Visit our Web site for etra eamples. INTERNET a. ƒ() is not a linear function because it has an -term. ƒ()= º º + 5 Write function. ƒ(º)= º(º) º (º) + 5 Substitute º for. = 7 Simplif. b. g() is a linear function because it has the form g() = m + b. g()= + 6 Write function. g(º)= (º) + 6 Substitute º for = Simplif. In Eample 5 the domain of each function is all real numbers. In real-life problems the domain is restricted to the numbers that make sense in the real-life contet. EXAMPLE 6 Using a Function in Real Life FOCUS ON PEOPLE BALLOONING In March of 999, Bertrand Piccard and Brian Jones attempted to become the first people to fl around the world in a balloon. Based on an average speed of 97.8 kilometers per hour, the distance d (in kilometers) that the traveled can be modeled b d = 97.8t where t is the time (in hours). The traveled a total of about 478 hours. The rules governing the record state that the minimum distance covered must be at least 6,700 kilometers. Source: Breitling a. Identif the domain and range and determine whether Piccard and Jones set the record. b. Graph the function. Then use the graph to approimate how long it took them to travel 0,000 kilometers. Distance Traveled PICCARD AND JONES are the first pilots to fl around the world in a balloon. Piccard is a medical doctor in Switzerland specializing in pschiatr, and Jones is a member of the Roal Air Force in the United Kingdom. a. Because their trip lasted 478 hours, the domain is 0 t 478. The distance the traveled was d = 97.8(478) 46,700 kilometers, so the range is 0 d 46,700. Since 46,700 > 6,700, the did set the record. b. The graph of the function is shown. Note that the graph ends at (478, 46,700). To find how long it took them to travel 0,000 kilometers, start at 0,000 on the d-ais and move right until ou reach the graph. Then move down to the t-ais. It took them about 00 hours to travel 0,000 kilometers. Distance (km) d 50,000 40,000 0,000 0,000 0, (478, 46,700) Time (hours) t 70 Chapter Linear Equations and Functions

5 Page 5 of 8 GUIDED PRACTICE Vocabular Check Concept Check Skill Check. What are the domain and range of a relation?. Eplain wh a vertical line, rather than a horizontal line, is used to determine if a graph represents a function.. Eplain the process for graphing an equation. 4. Identif the domain and range of the relation shown. Then tell whether the relation is a function. Graph the function. E = º 6. = 4 7. = = 9. = º 0. = º + 9 Evaluate the function when =.. ƒ() =. ƒ() = 6. ƒ() = 4. g() = h() = º j() = º 7 HIGHWAY DRIVING In Eercises 7 and 8, use the following information. A car has a 6 gallon gas tank. On a long highwa trip, gas is used at a rate of about gallons per hour. The gallons of gas g in the car s tank can be modeled b the equation g = 6 º t where t is the time (in hours). 7. Identif the domain and range of the function. Then graph the function. 8. At the end of the trip there are gallons of gas left. How long was the trip? PRACTICE AND APPLICATIONS DOMAIN AND RANGE Identif the domain and range. Etra Practice to help ou master skills is on p Input Output 0. Input Output Input 4 Output 4 GRAPHS Graph the relation. Then tell whether the relation is a function. HOMEWORK HELP Eample : Es. 9 7 Eample : Es. 7 Eample : Es. 0, 5 54 Eample 4: Es. 4 4 Eample 5: Es Eample 6: Es º4 4 º º º5 º4 º º6 º4 º º º º4 º6 º º 0.5 º º.5. Functions and Their Graphs 7

6 Page 6 of 8 MAPPING DIAGRAMS Use a mapping diagram to represent the relation. Then tell whether the relation is a function Skills Review For help with if-then statements, see p Writing Is a function alwas a relation? Is a relation alwas a function? Eplain our reasoning. 9. LOGICAL REASONING Rewrite the vertical line test as two if-then statements. VERTICAL LINE TEST Use the vertical line test to determine whether the relation is a function CRITICAL THINKING Wh does = represent a function, but = does not? GRAPHING FUNCTIONS Graph the function. 4. = º 5. = º = = º = º 4 9. = º º 40. = 0 4. = 5 4. = º + 4 EVALUATING FUNCTIONS Decide whether the function is linear. Then evaluate the function for the given value of. 4. ƒ() = º ; ƒ(4) 44. ƒ() = ; ƒ(º4) 45. ƒ() = º 5; ƒ(º6) 46. ƒ() = 9 º + ; ƒ() 47. ƒ() = º º + 5; ƒ(6) 48. ƒ() = º + 4; ƒ º 49. GEOMETRY CONNECTION The volume of a cube with side length s is given b the function V(s) = s. Find V(5). Eplain what V(5) represents. 50. GEOMETRY CONNECTION The volume of a sphere with radius r is given b the function V(r) = 4 πr. Find V(). Eplain what V() represents. 5. BOSTON MARATHON The graph shows the ages and finishing places of the top three competitors in each of the four categories of the 00th Boston Marathon. Is the finishing place of a competitor a function of his or her age? Eplain our reasoning. Source: Boston Athletic Association Place 00th Boston Marathon p Age a 7 Chapter Linear Equations and Functions

7 Page 7 of 8 5. HOUSE OF REPRESENTATIVES The graph shows the number of Independent representatives for the 00th 05th Congresses. Is the number of Independent representatives a function of the Congress number? Eplain our reasoning. Source: The Office of the Clerk, United States House of Representatives Independents House of Representatives r c Congress STATISTICS CONNECTION In Eercises 5 and 54, use the table which shows the number of shots attempted and the number of shots made b 9 members of the Utah Jazz basketball team in Game of the 998 NBA Finals. Source: NBA Plaer Shots attempted, Shots made, Bron Russell 6 Karl Malone 5 9 Greg Foster 5 Jeff Hornacek 0 John Stockton 9 Howard Eisle 6 4 Chris Morris 6 Greg Ostertag Shandon Anderson 5 FOCUS ON APPLICATIONS 5. Identif the domain and range of the relation. Then graph the relation. 54. Is the relation a function? Eplain. WATER PRESSURE In Eercises 55 and 56, use the information below and in the caption to the photo. Water pressure can be measured in atmospheres, where atmosphere equals 4.7 pounds per square inch. At sea level the water pressure is atmosphere, and it increases b atmosphere for ever feet in depth. Therefore, the water pressure p can be modeled as a function of the depth d b this equation: p = d +, 0 d 0 WATER PRESSURE Scuba divers must equalize the pressure on the inside of their bodies with the water pressure on the outside of their bodies. The maimum safe depth for recreational divers is 0 feet. 55. Identif the domain and range of the function. Then graph the function. 56. What is the water pressure at a depth of 00 feet? CAP SIZES In Eercises 57 and 58, use the following information. Your cap size is based on our head circumference (in inches). For head circumferences from inches to 5 inches, cap size s can be modeled as a function of head circumference c b this equation: s = c º 57. Identif the domain and range of the function. Then graph the function. 58. If ou wear a size 7 cap, what is our head circumference?. Functions and Their Graphs 7

8 Page 8 of 8 Test Preparation QUANTITATIVE COMPARISON In Eercises 59 6, choose the statement that is true about the given quantities. A The quantit in column A is greater. B The quantit in column B is greater. C The two quantities are equal. D The relationship cannot be determined from the given information Column A Column B ƒ() = + 0 when = 0 ƒ() = º 4 when = 7 ƒ() = º 4 º when = 6 ƒ() = º + 5 when = 4 ƒ() = º 7 + when = º ƒ() = º º 4 when = 6. ƒ() = + 8 when = ƒ() = º8 + 9 when = º 4 Challenge EXTRA CHALLENGE 6. TELEPHONE KEYPADS For the numbers through 9 on a telephone kepad, draw two mapping diagrams: one mapping numbers onto letters, and the other mapping letters onto numbers. Are both relations functions? Eplain. MIXED REVIEW EVALUATING EXPRESSIONS Evaluate the epression for the given values of and. (Review. for.) 64. º 6 when = º and = º 65. º when = º4 and = 5 º 9 º 66. º (º 5) when = and = 5 º 67. º ( º ) when = 6 and = 4 º ( º 4) º when = and = º when = 6 and = 8 º 4 º SOLVING EQUATIONS Solve the equation. Check our solution. (Review.) = 7. º = = 0 º 7. 5 º 7 = º ( º 5) = º = 0.5( + 6) º 4 CHECKING S Decide whether the given number is a solution of the inequalit. (Review.6) 76. º 4 < 0; º 8 0; º 6; > º5; º 80. º5 + 8 < 5; 8. º.7 < º or > 6.9;.5 74 Chapter Linear Equations and Functions

9 Page of 7. Slope and Rate of Change What ou should learn GOAL Find slopes of lines and classif parallel and perpendicular lines. GOAL Use slope to solve real-life problems, such as how to safel adjust a ladder in Eample 5. Wh ou should learn it To model real-life quantities, such as the average rate of change in the temperature of the Grand Canon in E. 5. GOAL FINDING SLOPES OF LINES The slope of a nonvertical line is the ratio of vertical change (the rise) to horizontal change (the run). The slope of a line is represented b the letter m. Just as two points determine a line, two points are all that are needed to determine a line s slope. The slope of a line is the same regardless of which two points are used. THE SLOPE OF A LINE (, ) (, ) rise run The slope of the nonvertical line passing through the points (, ) and (, ) is: m = º = r ise º run When calculating the slope of a line, be careful to subtract the coordinates in the correct order. EXAMPLE Finding the Slope of a Line Find the slope of the line passing through (º, 5) and (, ). Let (, ) = (º, 5) and (, ) = (, ). Look Back For help with evaluating epressions, see p.. m = º º º 5 = º (º ) º 4 = Rise: Difference of -values Run: Difference of -values Substitute values. Simplif. = º 4 Simplif. 5 (, 5) 5 4 (, ) In Eample notice that the line falls from left to right and that the slope of the line is negative. This suggests one of the important uses of slope to decide whether decreases, increases, or is constant as increases.. Slope and Rate of Change 75

10 Page of 7 CONCEPT SUMMARY CLASSIFICATION OF LINES BY SLOPE A line with a positive slope rises from left to right. (m > 0) A line with a negative slope falls from left to right. (m < 0) A line with a slope of zero is horizontal. (m = 0) A line with an undefined slope is vertical. (m is undefined.) Positive slope Negative slope Zero slope Undefined slope EXAMPLE Classifing Lines Using Slope Without graphing tell whether the line through the given points rises, falls, is horizontal, or is vertical. a. (, º4), (, º6) b. (, º), (, 5) a. m = º6 º (º º 4) = º º = b. m = 5 º (º ) = 6 º Because m > 0, the line rises. Because m is undefined, the line is vertical. Stud Tip You can think of horizontal lines as flat and vertical lines as infinitel steep. The slope of a line tells ou more than whether the line rises, falls, is horizontal, or is vertical. It also tells ou the steepness of the line. For two lines with positive slopes, the line with the greater slope is steeper. For two lines with negative slopes, the line with the slope of greater absolute value is steeper. m m m m m m EXAMPLE Comparing Steepness of Lines Tell which line is steeper. Line : through (, ) and (4, 7) Line : through (º, ) and (4, 5) The slope of line is m = 7 º 5 º = and the slope of line is m 4 º = = 4 º (º ) 5. Because the lines have positive slopes and m > m, line is steeper than line. 76 Chapter Linear Equations and Functions

11 Page of 7 Two lines in a plane are parallel if the do not intersect. Two lines in a plane are perpendicular if the intersect to form a right angle. Slope can be used to determine whether two different (nonvertical) lines are parallel or perpendicular. SLOPES OF PARALLEL AND PERPENDICULAR LINES Consider two different nonvertical lines l and l with slopes m and m. L L PARALLEL LINES The lines are parallel if and onl if the have the same slope. m = m PERPENDICULAR LINES The lines are perpendicular if and onl if their slopes are negative reciprocals of each other. m = º or m m m = º L L EXAMPLE 4 Classifing Parallel and Perpendicular Lines HOMEWORK HELP Visit our Web site for etra eamples. INTERNET Tell whether the lines are parallel, perpendicular, or neither. a. Line : through (º, ) and (, º) b. Line : through (º, ) and (, 4) Line : through (º, º) and (, ) Line : through (º4, º) and (4, ) a. The slopes of the two lines are: º º m = = º 4 = º º ( º ) 6 m = º ( º ) = 6 º ( º ) 4 = Because m m = º = º, m and m are negative reciprocals of each other. Therefore, ou can conclude that the lines are perpendicular. (, ) Line (, ) Line (, ) (, ) b. The slopes of the two lines are: 4 º m = = º (º ) 6 = m = º ( º ) = 4 4 º ( º 4) 8 = Because m = m (and the lines are different), ou can conclude that the lines are parallel. Line (, ) ( 4, ) (, 4) (4, ) Line. Slope and Rate of Change 77

12 Page 4 of 7 GOAL USING SLOPE IN EXAMPLE 5 Geometrical Use of Slope Ladder Safet Skills Review For help with solving proportions, see p. 90. FOCUS ON APPLICATIONS In a home repair manual the following ladder safet guideline is given. Adjust the ladder until the distance from the base of the ladder to the wall is at least one quarter of the height where the top of the ladder hits the wall. For eample, a ladder that hits the wall at a height of feet should have its base at least feet from the wall. a. Find the maimum recommended slope for a ladder. b. Find the minimum distance a ladder s base should be from a wall if ou need the ladder to reach a height of 0 feet. a. A ladder that hits the wall at a height of feet with its base about feet from the wall has slope m = r ise = = 4. The maimum recommended slope is 4. run b. Let represent the minimum distance that the ladder s base should be from the wall for the ladder to safel reach a height of 0 feet. r ise = 4 run Write a proportion. 0 4 = The rise is 0 and the run is. 0 = 4 Cross multipl. 5 = Solve for. The ladder s base should be at least 5 feet from the wall Not drawn to scale In real-life problems slope is often used to describe an average rate of change. These rates involve units of measure, such as miles per hour or dollars per ear. 0 ft EXAMPLE 6 Slope as a Rate of Change DESERTS In the Mojave Desert in California, temperatures can drop quickl from da to night. Suppose the temperature drops from 00 F at P.M. to 68 F at 5 A.M. Find the average rate of change and use it to determine the temperature at 0 P.M. DESERTS Animals in the Mojave Desert must cope with etreme temperatures. Man reptiles burrow into the ground to escape high temperatures. APPLICATION LINK INTERNET Change in temperature Average rate of change = Change in time 68 F º 00 F º F = = º F per hour 5 A.M. º P.M. 5 hours Because 0 P.M. is 8 hours after P.M., the temperature changed 8(º F) = º6 F. That means the temperature at 0 P.M. was about 00 F º 6 F = 84 F. 78 Chapter Linear Equations and Functions

13 Page 5 of 7 GUIDED PRACTICE Vocabular Check Concept Check. Describe what is meant b the slope of a nonvertical line. Eplain how our description relates to the definition of slope.. What tpe of line has a slope of zero? What tpe of line has a slope that is undefined? Skill Check. How can ou decide, using slope, whether two nonvertical lines are parallel? whether two nonvertical lines are perpendicular? Find the slope of the line passing through the given points. Then tell whether the line rises, falls, is horizontal, or is vertical. 4. (4, ), (4, ) 5. (8, 4), (8, ) 6. (º, 4), (, º5) 7. (º, 4), (º6, 8) 8. (º7, ), (4, ) 9. (6, 9), (º, º7) Tell which line is steeper. 0. Line : through (º5, 0) and (, 4). Line : through (, 4) and (, 7) Line : through (0, 4) and (, 6) Line : through (5, ) and (, ) Tell whether the lines are parallel, perpendicular, or neither.. Line : through (, 5) and (º4, º). Line : through (, º) and (º, 7) Line : through (, 0) and (º, º7) Line : through (4, º5) and (5, ) 4. Line : through (, 6) and (, º) 5. Line : through (9, 0) and (, 4) Line : through (º, ) and (6, ) Line : through (º5, 6) and (4, 0) 6. AVERAGE SPEED You are driving through Europe. At 9:00 A.M. ou are 40 kilometers from Rome. At :00 P.M. ou are 08 kilometers from Rome. Find our average speed. PRACTICE AND APPLICATIONS ESTIMATING SLOPE Estimate the slope of the line. Etra Practice to help ou master skills is on p HOMEWORK HELP Eample : Es. 7 Eample : Es. 0 Eample : Es. 5, 7 40 Eample 4: Es Eample 5: Es Eample 6: Es , 5, 5 FINDING SLOPE Find the slope of the line passing through the given points. Then tell whether the line rises, falls, is horizontal, or is vertical. 0. (, ), (º4, ). (, º4), (, 6). (4, º), (4, ). (º0, º), (, º6) 4. (º7, ), (º, ) 5. (6, º6), (º6, 6) 6. (4, ), (º8, ) 7. (º9, 8), (º9, ) 8. (, 4),, º , 7,, , º, 5, º. 4, º 9 5, 4, º 8 5. Slope and Rate of Change 79

14 Page 6 of 7 MATCHING SLOPES AND LINES Match the given slopes with the given lines.. º º 6. LOGICAL REASONING Use the formula for slope to verif that a horizontal line has a slope of zero and that a vertical line has an undefined slope. b c d a DETERMINING STEEPNESS Tell which line is steeper. 7. Line : through (º, 6) and (, 8) 8. Line : through (4, ) and (º8, 6) Line : through (0, º4) and (5, º) Line : through (º, 4) and (º, º8) 9. Line : through (, º0) and (, º0) 40. Line : through (º5, 6) and (º, º9) Line : through (º6, 8) and (, ) Line : through, and 5 4, TYPES OF LINES Tell whether the lines are parallel, perpendicular, or neither. 4. Line : through (º, 9) and (º6, º6) 4. Line : through (4, º) and (º8, ) Line : through (º7, º) and (0, º) Line : through (5, ) and (8, 0) 4. Line : through (0, ) and (0, º7) 44. Line : through (, 0) and (5, 5) Line : through (º6, º4) and (, º4) Line : through, and (4, ) AVERAGE RATE OF CHANGE Find the average rate of change in for the given -pairs. State the unit of measure for the average rate of change. 45. (4, ) and (8, 7) is measured in hours and is measured in dollars 46. (0, 5) and (, 7) is measured in seconds and is measured in meters 47. (, 0) and (4, 6) is measured in ears and is measured in inches JACQUES COUSTEAU was famous for his work in oceanograph, which is discussed in E. 5. Cousteau invented the aqua-lung, the one-man submarine, and the first underwater diving station. FOCUS ON PEOPLE 48. HISTORY CONNECTION Aqueducts were once used to carr water from rivers using gravit. Water flowing too quickl might damage an aqueduct, but water flowing too slowl might not keep the aqueduct clear. One of the best and most common designs for an aqueduct was to raise it meters for ever kilometer in length. What is the slope of an aqueduct built with this design? Source: Roman Aqueducts and Water Suppl 49. LEANING TOWER OF PISA The top of the Leaning Tower of Pisa is about 55.9 meters above the ground. As of 997 its top was leaning about 5. meters off-center. Approimate the slope of the tower. Source: Ende Engineering 50. PITCH OF A ROOF Building codes require the minimum slope, or pitch, of a roof with asphalt shingles to be such that it rises at least 4 feet for ever feet of horizontal distance. A 7 foot wide apartment building has a foot high roof. Does it meet the building code? Eplain. 5. OCEANOGRAPHY Loihi is the name of an underwater volcano that has formed twent miles off the coast of Hawaii. The peak of the volcano is currentl 00 feet below sea level. Oceanographers estimate that it will take about 50,000 ears before the peak breaks the water. If this holds true, what will be the rate of change in the volcano s height? Eplain. Source: United States Geological Surve 80 Chapter Linear Equations and Functions

15 Page 7 of 7 Test Preparation Skills Review For help with the Pthagorean theorem, see p. 97. Challenge 5. GRAND CANYON You are camping at the Grand Canon. When ou pitch our tent at :00 P.M. the temperature is 8 F. When ou wake up at 6:00 A.M. the temperature is 47 F. What is the average rate of change in the temperature? Estimate the temperature when ou went to sleep at 9:00 P.M. 5. CRITICAL THINKING Does it make a difference what two points on a line ou choose when finding slope? Does it make a difference which point is (, ) and which point is (, ) in the formula for slope? Draw a line and calculate its slope using several pairs of points to support our answer. 54. MULTI-STEP PROBLEM You are in charge of building a wheelchair ramp for a doctor s office. Federal regulations require that the ramp must etend inches for ever inch of rise. The ramp needs to rise to a height of 8 inches. Source: Uniform Federal Accessibilit Standards a. How far should the end of the ramp be from the base of the building? b. Use the Pthagorean theorem to determine the length of the ramp. c. Some northern states require that outdoor ramps etend 0 inches for ever inch of rise because of the added problems of winter weather. Under this regulation, what should be the length of the ramp? d. Writing How does changing the slope of the ramp affect the required length of the ramp? MISSING COORDINATES Find the value of k so that the line through the given points has the given slope. Check our solution. 55. (5, k) and (k, 7), m = 56. (º, k) and (k, 6), m = (º, k) and (k, 4), m = 58. (9, ºk) and (k, º), m = º 8 in. MIXED REVIEW IDENTIFYING PROPERTIES Identif the propert shown. (Review.) (º) = (6 + 5) + 0 = 6 + (5 + 0) 6. 8( + ) = = REWRITING EQUATIONS Solve the equation for. (Review.4 for.) = º º = = º6 + 4 = 0 5 SOLVING EQUATIONS Solve the equation. (Review.7) = º 6 = 69. º + = º 9 = 6 7. MIXED NUTS A 6 ounce can of mied nuts costs $5.8, but peanuts cost onl $.5 per ounce. The can contains 7 ounces of peanuts and 9 ounces of other nuts. What is the cost per ounce of the other nuts? (Review.5 for.). Slope and Rate of Change 8

16 Page of 8. Quick Graphs of Linear Equations What ou should learn GOAL Use the slopeintercept form of a linear equation to graph linear equations. GOAL Use the standard form of a linear equation to graph linear equations, as applied in Eample 5. Wh ou should learn it To identit relationships between real-life variables, such as the sales of student and adult basketball tickets in E. 6. GOAL SLOPE-INTERCEPT FORM In Lesson. ou graphed a linear equation b creating a table of values, plotting the corresponding points, and drawing a line through the points. In this lesson ou will stud two quicker was to graph a linear equation. If the graph of an equation intersects the -ais at the point (0, b), then the number b is the -intercept of the graph. To find the -intercept of a line, let = 0 in an equation for the line and solve for. ACTIVITY Developing Concepts Investigating Slope and -intercept Equation Points on graph Slope -intercept of equation = + (0,?), (,?)?? = º + (0,?), (,?)?? = º 4 (0,?), (,?)?? = º (0,?), (,?)?? = 7 (0,?), (,?)?? Cop and complete the table. What do ou notice about each equation and the slope of the line? What do ou notice about each equation and the -intercept of the line? The slope-intercept form of a linear equation is = m + b. As ou saw in the activit, a line with equation = m + b has slope m and -intercept b. GRAPHING EQUATIONS IN SLOPE-INTERCEPT FORM The slope-intercept form of an equation gives ou a quick wa to graph the equation. STEP Write the equation in slope-intercept form b solving for. STEP STEP STEP 4 Find the -intercept and use it to plot the point where the line crosses the -ais. Find the slope and use it to plot a second point on the line. Draw a line through the two points. 8 Chapter Linear Equations and Functions

17 Page of 8 EXAMPLE Graphing with the Slope-Intercept Form Graph = º. 4 4 The equation is alread in slope-intercept form. The -intercept is º, so plot the point (0, º) where the line crosses the -ais. The slope is, so plot a second point on the line b moving 4 units to the 4 right and units up. This point is (4, ). Draw a line through the two points. (4, ) (4, ) (0, ) 4 (0, ) In a real-life contet the -intercept often represents an initial amount and, as ou saw in Lesson., the slope often represents a rate of change. EXAMPLE Using the Slope-Intercept Form Buing a Computer You are buing an $00 computer on laawa. You make a $50 deposit and then make weekl paments according to the equation a = 850 º 50t where a is the amount ou owe and t is the number of weeks. a. What is the original amount ou owe on laawa? b. What is our weekl pament? c. Graph the model. a. First rewrite the equation as a = º50t so that it is in slope-intercept form. Then ou can see that the a-intercept is 850. So, the original amount ou owe on laawa (the amount when t =0) is $850. b. From the slope-intercept form ou can also see that the slope is m = º50. This means that the amount ou owe is changing at a rate of º$50 per week. In other words, our weekl pament is $50. c. The graph of the model is shown. Notice that the line stops when it reaches the t-ais (at t = 7) so the computer is completel paid for at that point. Dollars owed Buing a Computer a 800 (0, 850) (7, 0) 6 t Weeks. Quick Graphs of Linear Equations 8

18 Page of 8 GOAL STANDARD FORM The standard form of a linear equation is A + B = C where A and B are not both zero. A quick wa to graph an equation in standard form is to plot its intercepts (when the eist). You found the -intercept of a line in Goal. The -intercept of a line is the -coordinate of the point where the line intersects the -ais. GRAPHING EQUATIONS IN STANDARD FORM The standard form of an equation gives ou a quick wa to graph the equation: STEP STEP STEP STEP 4 Write the equation in standard form. Find the -intercept b letting = 0 and solving for. Use the -intercept to plot the point where the line crosses the -ais. Find the -intercept b letting = 0 and solving for. Use the -intercept to plot the point where the line crosses the -ais. Draw a line through the two points. EXAMPLE Drawing Quick Graphs Graph + =. Method USE STANDARD FORM The equation is alread written in standard form. + (0) = Let = 0. = 6 Solve for. (0, 4) The -intercept is 6, so plot the point (6, 0). (0) + = Let = 0. = 4 Solve for. (6, 0) The -intercept is 4, so plot the point (0, 4). 4 Draw a line through the two points. Method USE SLOPE-INTERCEPT FORM Look Back For help with solving an equation for, see p = = º + = º + 4 Slope-intercept form The -intercept is 4, so plot the point (0, 4). The slope is º, so plot a second point b moving units to the right and units down. This point is (, ). (0, 4) (, ) 4 Draw a line through the two points. 84 Chapter Linear Equations and Functions

19 Page 4 of 8 The equation of a vertical line cannot be written in slope-intercept form because the slope of a vertical line is not defined. Ever linear equation, however, can be written in standard form even the equation of a vertical line. HORIZONTAL AND VERTICAL LINES HORIZONTAL LINES The graph of = c is a horizontal line through (0, c). VERTICAL LINES The graph of = c is a vertical line through (c, 0). EXAMPLE 4 Graphing Horizontal and Vertical Lines Graph (a) = and (b) = º. a. The graph of = is a horizontal line that passes through the point (0, ). Notice that ever point on the line has a -coordinate of. b. The graph of = º is a vertical line that passes through the point (º, 0). Notice that ever point on the line has an -coordinate of º. (, 0) (0, ) Fundraising EXAMPLE 5 Using the Standard Form The school band is selling sweatshirts and T-shirts to raise mone. The goal is to raise $00. Sweatshirts sell for a profit of $.50 each and T-shirts for $.50 each. Describe numbers of sweatshirts and T-shirts the band can sell to reach the goal. First write a model for the problem. PROBLEM SOLVING STRATEGY VERBAL MODEL Profit per sweatshirt Number of Profit per Number of sweatshirts + T-shirt T-shirts = Total Profit LABELS Profit per sweatshirt = $.50 Number of sweatshirts = Profit per T-shirt = $.50 Number of T-shirts = t Total profit = $00 s Stud Tip Finding the intercepts of a line before ou draw the line can help ou determine reasonable scales for the -ais and the -ais. ALGEBRAIC MODEL.5 s +.5 t = 00 The graph of.5s +.5t = 00 is a line that intersects the s-ais at (480, 0) and intersects the t-ais at (0, 800). Points with integer coordinates on the line segment joining (480, 0) and (0, 800) represent was to reach the goal. For instance, the band can sell 00 sweatshirts and 00 T-shirts. Number of T-shirts t (0, 800) Total Profit (00, 00) (480, 0) 600 Number of sweatshirts s. Quick Graphs of Linear Equations 85

20 Page 5 of 8 GUIDED PRACTICE Vocabular Check Concept Check. What are the slope-intercept and standard forms of a linear equation?. Which of the two quick-graph techniques discussed in the lesson would ou use to graph = º + 4? Eplain. Skill Check. Which of the two quick-graph techniques discussed in the lesson would ou use to graph + 4 = 4? Eplain. Find the slope and -intercept of the line. 4. = = º º 7 6. º = 8 Find the intercepts of the line. 7. º = 8. 5 º = 0 9. = 5 º 5 Graph the equation. 0. = +. = º 4. = 7. = º5 4. º 6 = = º5 PRACTICE AND APPLICATIONS Etra Practice to help ou master skills is on p. 94. MATCHING GRAPHS Match the equation with its graph. 6. = º = º º 5 8. = 4 º A. B. C USING SLOPE AND -INTERCEPT Draw the line with the given slope and -intercept. 9. m =, b = º 0. m = º, b = 0. m =, b =. m =, b = 5. m = 0, b = º7 4. m = º, b = 4 7 HOMEWORK HELP Eample : Es. 6 6, 5 57 Eample : Es Eample : Es Eample 4: Es Eample 5: Es. 6 6 SLOPE-INTERCEPT FORM Graph the equation. 5. = º = = 4 5 º 8. = º 5 9. = º º 5 0. = 5 º FINDING SLOPE AND -INTERCEPT Find the slope and -intercept of the line.. = = º9. = = º = = 7 86 Chapter Linear Equations and Functions

21 Page 6 of 8 MATCHING GRAPHS Match the equation with its graph. 7. º 4 = º = º9 9. º = º A. B. C. 4 USING INTERCEPTS Draw the line with the given intercepts intercept: 4. -intercept: 4. -intercept: º4 -intercept: 5 -intercept: º6 -intercept: º STANDARD FORM Graph the equation. Label an intercepts = = = º0 46. º = º 6 = º = 49. = = º5 5. = º CHOOSE A METHOD Graph the equation using an method. 5. = = º0 54. º 7 = = = = 5 º RAINFORESTS In Brazil the rate of rainforest destruction is. million hectares per ear. Brazil recentl passed a law giving its government the authorit to protect forests. FOCUS ON APPLICATIONS 58. IRS The amount a (in billions of dollars) of annual taes collected b the Internal Revenue Service can be modeled b a = 57.t where t represents the number of ears since 980. Graph the equation. Source: Statistical Abstract of the United States 59. PLACING AN AD The cost C (in dollars) of placing a color advertisement in a newspaper can be modeled b C = 7n + 0 where n is the number of lines in the ad. Graph the equation. What do the slope and C-intercept represent? 60. RAINFORESTS The area A (in millions of hectares) of land covered b rainforests can be modeled b A = 78. º 4.6t where t represents the number of ears since 990. Graph the equation. What are three predicted future areas of land covered b rainforests? Source: Food and Agriculture Organization 6. CAR WASH A car wash charges $8 per wash and $ per wash-and-wa. After a bus da sales totaled $464. Use the verbal model to write an equation that shows the different numbers of washes and wash-and-waes that could have been done. Then graph the equation. Price per wash Number Price per Number of of washes + wash-and-wa wash-and-waes = Total sales 6. SAILING The owner of a sailboat takes passengers to an island 5 miles awa to go snorkeling. A sailboat averages about 9 miles per hour when using its sails and about 4 miles per hour when using its motor. Write an equation that shows the numbers of minutes the sailboat can use its sails and its motor to get to the island. Then graph the equation.. Quick Graphs of Linear Equations 87

22 Page 7 of 8 HOMEWORK HELP Visit our Web site for help with problem solving in E. 6. INTERNET Test Preparation Challenge 6. TICKET PRICES Student tickets at a high school basketball game cost $.50 each. Adult tickets cost $6.00 each. The ticket sales at the first game of the season totaled $7000. Write a model that shows the numbers of student and adult tickets that could have been sold. Then graph the model and determine three combinations of student and adult tickets that satisf the model. 64. Writing Eplain how to find the intercepts of a line if the eist. What kind of line has no -intercept? What kind of line has no -intercept? 65. MULTIPLE CHOICE You have an individual retirement account (IRA). The amount a ou have deposited into our account after t ears can be modeled b a = t. How much mone do ou put into our IRA ever ear? A $000 B $000 C $500 D $4500 E $ MULTIPLE CHOICE What is the slope-intercept form of 4 º 6 = 8? A = + 9 B = º C º = D 6 = º4 + 8 E 4 = CALCULATING SLOPE For the line = 7 + 6, show that the slope is 7 regardless of the points (, ) and (, ) ou use to calculate the slope. (Hint: Substitute and into the equation to obtain epressions for and.) MIXED REVIEW SOLVING INEQUALITIES Solve the inequalit. Then graph our solution. (Review.6) º + < 70. º > 4 º º 9 º 7. º5 < º or º 0 EVALUATING FUNCTIONS Evaluate the function for the given value of. (Review.) 74. ƒ() = º ; ƒ(8) 75. ƒ() = º + ; ƒ(5) 76. ƒ() = º ; ƒ(º7) 77. ƒ() = 0 º ; ƒ() 78. ƒ() = + 7 ; ƒ(º5) 79. ƒ() = º 9; ƒ FINDING SLOPE Find the slope of the line passing through the given points. (Review. for.4) 80. (, ), (7, ) 8. (6, º), (, 9) 8. (º, º9), (, º8) 8. (º, º), (º, º5) 84. (5, º), (º, ) 85. (º4, 7), (, º5) 86. READING SPEED You can read a novel at a rate of pages per minute. Write a model that shows the number of pages ou can read in h hours. Then find how long it will take ou to read a 048 page novel. (Review.5 for.4) 88 Chapter Linear Equations and Functions

23 Page 8 of 8 QUIZ Self-Test for Lessons.. Identif the domain and range. Then tell whether the relation is a function. (Lesson.)... Evaluate the function for the given value of. (Lesson.) 4. ƒ() = º º ; ƒ(4) 5. ƒ() = 5 º + 9; ƒ(º5) Tell whether the lines are parallel, perpendicular, or neither. (Lesson.) 6. Line : through (, 0) and (, 5) 7. Line : through (4, 5) and (9, º) Line : through (, º7) and (8, º8) Line : through (6, º6) and (º, º) Graph the equation. (Lesson.) 8. = º = 0 0. = º. BICYCLING There is an annual seven da biccle ride across Iowa that covers about 468 miles. If a participant rides each da from 8:00 A.M. to 5:00 P.M., stopping onl hour for lunch, what is the rider s average speed in miles per hour? (Lesson.) Transatlantic Voages INTERNET APPLICATION LINK THEN NOW AT :00 P.M. ON APRIL, 9, the Titanic left Cobh, Ireland, on her maiden voage to New York Cit. At :40 P.M. on April 4, the Titanic struck an iceberg and sank, having covered onl about 00 miles of the approimatel 400 mile trip.. What was the total length of the Titanic s maiden voage in hours?. What was the Titanic s average speed in miles per hour?. Write an equation relating the Titanic s distance from New York Cit and the number of hours traveled. Identif the domain and range. 4. Graph the equation from Eercise. TODAY, ocean liners still cross the Atlantic Ocean. The Queen Elizabeth, or QE, is one of the fastest with a top speed of.5 knots (about 7 miles per hour). Titanic s maiden voage Benoit Lecomte swims across the Atlantic Charles Lindbergh makes the first solo transatlantic flight. QE s maiden voage 998. Quick Graphs of Linear Equations 89

24 Page of 8.4 Writing Equations of Lines What ou should learn GOAL Write linear equations. GOAL Write direct variation equations, as applied in Eample 7. Wh ou should learn it To model real-life quantities, such as the number of calories ou burn while dancing in E. 64. GOAL WRITING LINEAR EQUATIONS In Lesson. ou learned to find the slope and -intercept of a line whose equation is given. In this lesson ou will stud the reverse process. That is, ou will learn to write an equation of a line using one of the following: the slope and -intercept of the line, the slope and a point on the line, or two points on the line. CONCEPT SUMMARY WRITING AN EQUATION OF A LINE SLOPE-INTERCEPT FORM Given the slope m and the -intercept b, use this equation: = m + b POINT-SLOPE FORM Given the slope m and a point (, ), use this equation: º = m( º ) TWO POINTS Given two points (, ) and (, ), use the formula º m = º to find the slope m. Then use the point-slope form with this slope and either of the given points to write an equation of the line. Ever nonvertical line has onl one slope and one -intercept, so the slope-intercept form is unique. The point-slope form, however, depends on the point that is used. Therefore, in this book equations of lines will be simplified to slope-intercept form so a unique solution ma be given. EXAMPLE Writing an Equation Given the Slope and the -intercept Write an equation of the line shown. From the graph ou can see that the slope is m =. You can also see that the line intersects the -ais at the point (0, º), so the -intercept is b = º. (0, ) m Because ou know the slope and the -intercept, ou should use the slope-intercept form to write an equation of the line. = m + b Use slope-intercept form. = º Substitute } } for m and º for b. An equation of the line is = º..4 Writing Equations of Lines 9

25 Page of 8 EXAMPLE Writing an Equation Given the Slope and a Point Write an equation of the line that passes through (, ) and has a slope of º. Because ou know the slope and a point on the line, ou should use the point-slope form to write an equation of the line. Let (, ) = (, ) and m = º. º = m( º ) Use point-slope form. º = º ( º ) Substitute for m,, and. Once ou have used the point-slope form to find an equation, ou can simplif the result to the slope-intercept form. º = º ( º ) º = º + = º + 4 Write point-slope form. Distributive propert Write in slope-intercept form. CHECK You can check the result graphicall. Draw the line that passes through the point (, ) with a slope of º. Notice that the line has a -intercept of 4, which agrees with the slope-intercept form found above. (, ) EXAMPLE Writing Equations of Perpendicular and Parallel Lines HOMEWORK HELP Visit our Web site for etra eamples. INTERNET Write an equation of the line that passes through (, ) and is (a) perpendicular and (b) parallel to the line = º +. a. The given line has a slope of m = º. So, a line that is perpendicular to this line must have a slope of m = º =. Because ou know the slope and a m point on the line, use the point-slope form with (, ) = (, ) to find an equation of the line. º = m ( º ) Use point-slope form. º = ( º ) Substitute for m,, and. º = º Distributive propert = + Write in slope-intercept form. b. For a parallel line use m = m = º and (, ) = (, ). º = m ( º ) Use point-slope form. º = º( º ) Substitute for m,, and. º = º + 9 Distributive propert = º + Write in slope-intercept form. 9 Chapter Linear Equations and Functions

26 Page of 8 FOCUS ON PEOPLE EXAMPLE 4 Writing an Equation Given Two Points Write an equation of the line that passes through (º, º) and (, 4). The line passes through (, ) = (º, º) and (, ) = (, 4), so its slope is: m = º = 4 º ( º ) = 5 º ( º ) 5 = º Because ou know the slope and a point on the line, use the point-slope form to find an equation of the line. º = m( º ) Use point-slope form. BARBARA JORDAN was the first African-American woman elected to Congress from a southern state. She was a member of the House of Representatives from 97 to 979. º (º) = [ º (º)] Substitute for m,, and. + = + Simplif. = + Write in slope-intercept form. EXAMPLE 5 Writing and Using a Linear Model POLITICS In 970 there were 60 African-American women in elected public office in the United States. B 99 the number had increased to. Write a linear model for the number of African-American women who held elected public office at an given time between 970 and 99. Then use the model to predict the number of African-American women who will hold elected public office in 00. INTERNET DATA UPDATE of Joint Center for Political and Economic Studies data at PROBLEM SOLVING STRATEGY º 60 The average rate of change in officeholders is m = º 970 You can use the average rate of change as the slope in our linear model. VERBAL MODEL Number of officeholders Number Average rate = + in 970 of change Years since 970 LABELS Number of officeholders = Number in 970 = 60 Average rate of change = 94.4 Years since 970 = t (people) (people) (people per ear) (ears) ALGEBRAIC MODEL = t In 00, which is 40 ears since 970, ou can predict that there will be = (40) 96 African-American women in elected public office. You can graph the model to check our prediction visuall. Officeholders African-American Women in Elected Public Office (40, 96) (0, 90) (, ) 60 t Years since Writing Equations of Lines 9

27 Page 4 of 8 GOAL WRITING DIRECT VARIATION EQUATIONS Two variables and show direct variation provided = k and k 0. The nonzero constant k is called the constant of variation, and is said to var directl with. The graph of = k is a line through the origin. EXAMPLE 6 Writing and Using a Direct Variation Equation The variables and var directl, and = when = 4. a. Write and graph an equation relating and. b. Find when = 5. a. Use the given values of and to find the constant of variation. = k Write direct variation equation. = k(4) Substitute for and 4 for. (5, 5) (4, ) = k Solve for k. The direct variation equation is =. The graph of = is shown. b. When = 5, the value of is = (5) = The equation for direct variation can be rewritten as = k. This tells ou that a set of data pairs (, ) shows direct variation if the quotient of and is constant. EXAMPLE 7 Identifing Direct Variation Jewelr Tell whether the data show direct variation. If so, write an equation relating and. a. 4-karat Gold Chains ( gram per inch) Length, (inches) Price, (dollars) b. Loose Diamonds (round, colorless, ver small flaws) Weight, (carats) Price, (dollars) ,000 0,400 For each data set, check whether the quotient of and is constant. a. For the 4-karat gold chains, 88 6 = 4 8 = 60 0 = 4 = 5 40 = 8. The data do 4 0 show direct variation, and the direct variation equation is = 8. b. For the loose diamonds, = 4500, but 40 0 = The data do not show. 7 direct variation. 94 Chapter Linear Equations and Functions

28 Page 5 of 8 GUIDED PRACTICE Vocabular Check Concept Check Skill Check. Define the constant of variation for two variables and that var directl.. How can ou find an equation of a line given the slope and the -intercept of the line? given the slope and a point on the line? given two points on the line?. Give a real-life eample of two quantities that var directl. Write an equation of the line that has the given properties. 4. slope:, -intercept: 5. slope:, passes through (0, º4) 5 6. slope: º, passes through (5, ) 7. slope: º, passes through (º7, 0) 4 8. passes through (4, 8) and (, ) 9. passes through (0, ) and (º5, 0) 0. Write an equation of the line that passes through (, º6) and is perpendicular to the line = Write an equation of the line that passes through (, 9) and is parallel to the line =5 º 5.. LAW OF SUPPLY The law of suppl states that the quantit supplied of an item varies directl with the price of that item. Suppose that for $4 per tape 5 million cassette tapes will be supplied. Write an equation that relates the number c (in millions) of cassette tapes supplied to the price p (in dollars) of the tapes. Then determine how man cassette tapes will be supplied for $5 per tape. PRACTICE AND APPLICATIONS Etra Practice to help ou master skills is on p. 94. SLOPE-INTERCEPT FORM Write an equation of the line that has the given slope and -intercept.. m = 5, b = º 4. m = º, b = º4 5. m = º4, b = 0 6. m = 0, b = 4 7. m = 5, b = 6 8. m = º 4, b = 7 POINT-SLOPE FORM Write an equation of the line that passes through the given point and has the given slope. 9. (0, 4), m = 0. (, 0), m =. (º6, 5), m = 0. (9, ), m = º. (, º), m = º 4 4. (7, º4), m = 5 HOMEWORK HELP Eample : Es. 8 Eample : Es. 9 4 Eample : Es. 5 8 Eample 4: Es Eample 5: Es Eample 6: Es Eample 7: Es , Write an equation of the line that passes through (, º) and is perpendicular to the line = º Write an equation of the line that passes through (6, º0) and is perpendicular to the line that passes through (4, º6) and (, º4). 7. Write an equation of the line that passes through (, º7) and is parallel to the line = Write an equation of the line that passes through (4, 6) and is parallel to the line that passes through (6, º6) and (0, º4)..4 Writing Equations of Lines 95

29 Page 6 of 8 VISUAL THINKING Write an equation of the line WRITING EQUATIONS Write an equation of the line that passes through the given points. 5. (8, 5), (, 4) 6. (º5, 9), (º4, 7) 7. (º8, 8), (0, ) 8. (, 0), (4, º6) 9. (º0, º0), (5, 5) 40. (º, 0), (0, 6) 4. LOGICAL REASONING Redo Eample b substituting the given point and slope into = m + b. Then solve for b to write an equation of the line. Eplain wh using this method does not change the equation of the line. 4. LOGICAL REASONING Redo Eample 4 b substituting (, 4) for (, ) into º = m ( º ). Then rewrite the equation in slope-intercept form. Eplain wh using the point (, 4) does not change the equation of the line. RELATING VARIABLES The variables and var directl. Write an equation that relates the variables. Then find when = =, = = º6, = = º, = = 4, = =, = 48. = 0.8, =.6 RELATING VARIABLES The variables and var directl. Write an equation that relates the variables. Then find when = º = 6, = 50. = 9, = 5 5. = º5, = º 5. = 00, = 5. = 5, = = º0., =. 4 IDENTIFYING DIRECT VARIATION Tell whether the data show direct variation. If so, write an equation relating and º º6 º9 º º5 º5 º4 º º º Chapter Linear Equations and Functions

30 Page 7 of POPULATION OF OREGON From 990 to 996 the population of Oregon increased b about 60,00 people per ear. In 996 the population was about,04,000. Write a linear model for the population P of Oregon from 990 to 996. Let t represent the number of ears since 990. Then estimate the population of Oregon in 04. Source: Statistical Abstract of the United States 60. AIRFARE In 998 an airline offered a special airfare of $0 to fl from Cincinnati to Washington, D.C., a distance of 86 miles. Special airfares offered for longer flights increased b about $.8 per mile. Write a linear model for the special airfares a based on the total number of miles t of the flight. Estimate the airfare offered for a flight from Boston to Sacramento, a distance of 69 miles. 6. BOOKSTORE SALES In 990 retail sales at bookstores were about $7.4 billion. In 997 retail sales at bookstores were about $.8 billion. Write a linear model for retail sales s (in billions of dollars) at bookstores from 990 through 997. Let t represent the number of ears since 990. Then estimate the retail sales at bookstores in 0. Source: American Booksellers Association 6. SCIENCE CONNECTION The velocit of sound in dr air increases as the temperature increases. At 40 C sound travels at a rate of about 55 meters per second. At 49 C it travels at a rate of about 60 meters per second. Write a linear model for the velocit v (in meters per second) of sound based on the temperature T (in degrees Celsius). Then estimate the velocit of sound at 60 C. Source: CRC Handbook of Chemistr and Phsics 6. BREAKING WAVES The height h (in feet) at which a wave breaks varies directl with the wave length l (in feet), which is the distance from the crest of one wave to the crest of the net. A wave that breaks at a height of 4 feet has a wave length of 8 feet. Write a linear model that gives h as a function of l. Then estimate the wave length of a wave that breaks at a height of 5.5 feet. Source: Rhode Island Sea Grant crest wave height wave length crest HAILSTONES The largest hailstone ever recorded fell at Coffeville, Kansas. It weighed.67 pounds and had a radius of about.75 inches. FOCUS ON APPLICATIONS 64. DANCING The number C of calories a person burns performing an activit varies directl with the time t (in minutes) the person spends performing the activit. A 60 pound person can burn 7 Calories b dancing for 0 minutes. Write a linear model that gives C as a function of t. Then estimate how long a 60 pound person should dance to burn 48 Calories. Source: Health Journal 65. HAILSTONES Hailstones are formed when frozen raindrops are caught in updrafts and carried into high clouds containing water droplets. As a rule of thumb, the radius r (in inches) of a hailstone varies directl with the time t (in seconds) that the hailstone is in a high cloud. After a hailstone has been in a high cloud for 60 seconds, its radius is 0.5 inch. Write a linear model that gives r as a function of t. Then estimate how long a hailstone was in a high cloud if its radius measures.75 inches. Source: National Oceanic and Atmospheric Administration 66. GEOMETRY CONNECTION When the length of a rectangle is fied, the area A (in square inches) of the rectangle varies directl with its width w (in inches). When the width of a particular rectangle is inches, its area is 6 square inches. Write an equation that gives A as a function of w. Then find A when w is 7.5 inches..4 Writing Equations of Lines 97

31 Page 8 of 8 STATISTICS CONNECTION Tell whether the data show direct variation. If so, write an equation relating and. 67. Applesauce Ounces, Price, $.89 $.5 $.9 $.09 $ Fresh Apples Pounds,.5.5 Price, $.89 $.4 $.78 $. $.49 Test Preparation Challenge 69. MULTI-STEP PROBLEM Besides slope-intercept and point-slope forms, another form that can be used to write equations of lines is intercept form: + b = a a. Graph + =. b. Graph º + =. 5 9 c. Writing Geometricall, what do a and b represent in the intercept form of a linear equation? d. Write an equation of the line shown using intercept form. e. Write an equation of the line with -intercept º5 and -intercept º8 using intercept form. f. Write an equation of the line that passes through (0, º) and (, 0) using intercept form. 70. SLOPE-INTERCEPT FORM Derive the slope-intercept form of a linear equation from the slope formula using (0, b) as the coordinates of the point where the line crosses the -ais and an arbitrar point (, ). MIXED REVIEW SOLVING EQUATIONS Solve the equation. (Review.7) 7. º 0 = º = 5 7. º º 9 = = = =.8 FINDING SLOPE Find the slope of the line passing through the given points. (Review. for.5) 77. (, º7), (, 7) 78. (º, º), (º5, º4) 79. (, 4), (5, 0) 80. (5, º), (º, º) 8. (º, 4), (, 4) 8. (º4, º), (5, º4) 8. (0, º8), (º9, 0) 84. (6, ), (6, º5) 85. (º, 4), (º4, ) GRAPHING EQUATIONS Graph the equation. (Review. for.5) 86. = 4 º = º = º = º 8 = º5 9. º5 + = 0 9. = 0 9. = º 94. = 98 Chapter Linear Equations and Functions

32 Page of 7 EXPLORING DATA AND STATISTICS Correlation and Best-Fitting Lines GOAL SCATTER PLOTS AND CORRELATION.5 What ou should learn GOAL Use a scatter plot to identif the correlation shown b a set of data. GOAL Approimate the best-fitting line for a set of data, as applied in Eample. Wh ou should learn it To identif real-life trends in data, such as when and for how long Old Faithful will erupt in E.. A scatter plot is a graph used to determine whether there is a relationship between paired data. In man real-life situations, scatter plots follow patterns that are approimatel linear. If tends to increase as increases, then the paired data are said to have a positive correlation. If tends to decrease as increases, then the paired data are said to have a negative correlation. If the points show no linear pattern, then the paired data are said to have relativel no correlation. Positive correlation Negative correlation Relativel no correlation EXAMPLE Determining Correlation MUSIC Describe the correlation shown b each scatter plot. CD and Cassette Sales CD and CD Plaer Sales Cassettes sold (millions) CD plaers sold (millions) CDs sold (millions) CDs sold (millions) Source: Statistical Abstract of the United States Sources: Electronic Market Data Book, Recording Industr Association of America The first scatter plot shows a negative correlation, which means that as CD sales increased, the sales of cassettes tended to decrease. The second scatter plot shows a positive correlation, which means that as CD sales increased, the sales of CD plaers tended to increase. 00 Chapter Linear Equations and Functions

33 Page of 7 GOAL APPROXIMATING BEST-FITTING LINES When data show a positive or negative correlation, ou can approimate the data with a line. Finding the line that best fits the data is tedious to do b hand. (See page 07 for a description of how to use technolog to find the best-fitting line.) You can, however, approimate the best-fitting line using the following graphical approach. APPROXIMATING A BEST-FITTING LINE: GRAPHICAL APPROACH STEP STEP STEP STEP 4 Carefull draw a scatter plot of the data. Sketch the line that appears to follow most closel the pattern given b the points. There should be as man points above the line as below it. Choose two points on the line, and estimate the coordinates of each point. These two points do not have to be original data points. Find an equation of the line that passes through the two points from Step. This equation models the data. EXAMPLE Fitting a Line to Data Walking Speeds Researchers have found that as ou increase our walking speed (in meters per second), ou also increase the length of our step (in meters). The table gives the average walking speeds and step lengths for several people. Approimate the best-fitting line for the data. Source: Biomechanics and Energetics of Muscular Eercise Speed Step Speed Step Begin b drawing a scatter plot of the data. Net, sketch the line that appears to best fit the data. Then, choose two points on the line. From the scatter plot shown, ou might choose (0.9, 0.6) and (.5, ). Finall, find an equation of the line. The line that passes through the two points has a slope of: º 0.6 m = = 0. 4 = º Use the point-slope form to write the equation. º = m( º ) Use point-slope form. Step length (m) Walking Speeds Speed (m/sec) º 0.6 = 0.5( º 0.9) Substitute for m,, and. = Simplif..5 Correlation and Best-Fitting Lines 0

34 Page of 7 FOCUS ON CAREERS EXAMPLE Using a Fitted Line SLEEP REQUIREMENTS The table shows the age t (in ears) and the number h of hours slept per da b 4 infants who were less than one ear old. PEDIATRICIAN A pediatrician is a medical doctor who specializes in children s health. About 7% of all medical doctors are pediatricians. CAREER LINK INTERNET Infant Sleep Requirements Age, t Sleep, h Age, t Sleep, h Age, t Sleep, h a. Approimate the best-fitting line for the data. b. Use the fitted line to estimate the number of hours that a 6 month old infant sleeps per da. a. Draw a scatter plot of the data. Sketch the line that appears to best fit the data. Choose two points on the line. From the scatter plot shown, ou might choose: (0, 5.5) and (0.5, 4.4) Find an equation of the line. The line Infant Sleep Requirements h that passes through the two points has a 0 0 slope of: t m = 4. 4 º 5.5 º. Age (ears) = º º Because the h-intercept was chosen as one of the two points for determining the line, ou can use the slope-intercept form to approimate the best-fitting line as follows: h = mt + b Use slope-intercept form. Sleep (hours per da) h = º.t Substitute for m and b. An equation of the line is h = º.t Notice that a newborn infant sleeps about 5.5 hours per da and tends to sleep less as he or she gets older. b. To estimate the number of hours that a 6 month old infant sleeps, use the model from part (a) and the fact that 6 months = 0.5 ears. h = º.t Write linear model. h = º.(0.5) Substitute 0.5 for t. h 4.4 Simplif. A 6 month old infant sleeps about 4.4 hours per da. 0 Chapter Linear Equations and Functions

35 Page 4 of 7 GUIDED PRACTICE Vocabular Check Concept Check Skill Check. Eplain the meaning of the terms positive correlation, negative correlation, and relativel no correlation.. Suppose ou were given the shoe sizes s and the heights h of one hundred 5 ear old men. Do ou think that s and h would have a positive correlation, a negative correlation, or relativel no correlation? Eplain.. ERROR ANALYSIS Eplain wh the line shown at the right is not a good fit for the data. 4. Does the scatter plot at the right show a positive correlation, a negative correlation, or relativel no correlation? Eplain. E. 5. Look back at Eample. Estimate the step length of a person who walks at a speed of 4 meters per second. FM RADIO STATIONS In Eercises 6 and 7, use the table below which gives the number of FM radio stations from 989 to 995. Source: Statistical Abstract of the United States E. 4 Years since FM radio stations Approimate the best-fitting line for the data. 7. If the pattern continues, how man FM radio stations will there be in 00? PRACTICE AND APPLICATIONS Etra Practice to help ou master skills is on p. 94. DETERMINING CORRELATION Tell whether and have a positive correlation, a negative correlation, or relativel no correlation DRAWING SCATTER PLOTS Draw a scatter plot of the data. Then tell whether the data have a positive correlation, a negative correlation, or relativel no correlation. HOMEWORK HELP Eample : Es. 8 4,, Eample : Es. 6 Eample : Es Correlation and Best-Fitting Lines 0

36 Page 5 of 7 DRAWING SCATTER PLOTS Draw a scatter plot of the data. Then tell whether the data have a positive correlation, a negative correlation, or relativel no correlation LOGICAL REASONING Eplain how ou can determine the tpe of correlation for data b eamining the data in a table as opposed to drawing a scatter plot. APPROXIMATING BEST-FITTING LINES Cop the scatter plot. Then approimate the best-fitting line for the data FITTING A LINE TO DATA Draw a scatter plot of the data. Then approimate the best-fitting line for the data. 9. º º º º º.5 º.5 FOCUS ON APPLICATIONS 0. º4 º º º º º º º4 º º º.5 º HIGH ALTITUDE TEMPERATURES The table shows the temperature for various elevations based on a temperature of 59 F at sea level. Draw a scatter plot of the data and describe the correlation shown. OLD FAITHFUL, shown here between eruptions, is just one of the gesers located in Yellowstone National Park. There are more gesers and hot springs in Yellowstone than in the rest of the world combined. Elevation (ft) ,000 5,000 0,000 0,000 Temperature ( F) º5 º47. OLD FAITHFUL Old Faithful is a geser in Yellowstone National Park. The table shows the duration of eruptions and the time interval between eruptions for a tpical da. Draw a scatter plot of the data and describe the correlation shown. Duration (min) Interval (min) Chapter Linear Equations and Functions

37 Page 6 of 7 HOMEWORK HELP Visit our Web site for help with problem solving in Es INTERNET CITY YEAR In Eercises 4 and 5, use the table below which gives the enrollment for the Cit Year national outh service program from 989 to 998. Years since Enrollment Approimate the best-fitting line for the data. 5. If the pattern continues, how man people will enroll in Cit Year in 00? BIOLOGY CONNECTION In Eercises 6 and 7, use the table below which gives the average life epectanc (in ears) of a person based on various ears of birth. Source: National Center for Health Statistics Year of birth Life epectanc Year of birth Life epectanc Test Preparation 6. Approimate the best-fitting line for the data. 7. Predict the life epectanc for someone born in MULTI-STEP PROBLEM The table below gives the numbers (in thousands) of black-and-white and color televisions sold in the United States for various ears from Source: Electronic Industries Association Black-and-white Color TVs Year TVs sold (thousands) sold (thousands) 955 7, , ,75, ,704 5, ,955 6, ,684 0, ,684 6, ,4 0, ,600 Challenge a. Draw a scatter plot of the data pairs (ear, black-and-white TVs sold). Then describe the correlation shown b the scatter plot. b. Draw a scatter plot of the data pairs (ear, color TVs sold). Then describe the correlation shown b the scatter plot. c. CRITICAL THINKING Based on our answers to parts (a) and (b), are blackand-white television sales and color television sales positivel correlated, negativel correlated, or neither? Eplain. 9. BEST-FITTING LINES Describe a set of real-life data where the best-fitting line could not be used to make a prediction. Eplain..5 Correlation and Best-Fitting Lines 05

38 Page 7 of 7 MIXED REVIEW SOLVING INEQUALITIES Solve the inequalit. Then graph our solution. (Review.6 for.6) 0. º 9 4. ( + 7) < º º 7 9. º 4 < 0 or º 6 4 DETERMINING STEEPNESS Tell which line is steeper. (Review.) 4. Line : through (º, 4) and (, 6) 5. Line : through (6, ) and (º4, 4) Line : through (, º5) and (6, ) Line : through (º, ) and (, º6) 6. Line : through (, 4) and (, 7) 7. Line : through (4, ) and (, º9) Line : through (º5, 8) and (, 8) Line : through (º, º4) and (, º7) GRAPHING EQUATIONS Graph the equation. (Review. for.6) 8. = = º = 7 4. º + = º = 4. = QUIZ Self-Test for Lessons.4 and.5 Write an equation of the line that passes through the given point and has the given slope. (Lesson.4). (0, 6), m =. (º4, º), m =. (, º7), m = º 5 4. Write an equation of the line that passes through (, º) and is perpendicular to the line that passes through (4, ) and (0, 4). (Lesson.4) Tell whether and have a positive correlation, a negative correlation, or relativel no correlation. (Lesson.5) WAVES The water depth d (in feet) at which a wave breaks varies directl with the height h (in feet) of the wave. A 6.5 foot wave breaks at a water depth of 8.45 feet. Write a linear model that gives d as a function of h. If a wave breaks at a depth of 5. feet, what is its height? (Lesson.4) 9. HEIGHTS OF CHILDREN The table gives the average heights of children for ages 0. Draw a scatter plot of the data and approimate the best-fitting line for the data. (Lesson.5) Age (ears) Height (cm) Chapter Linear Equations and Functions

39 Page of 6.6 Linear Inequalities in Two Variables What ou should learn GOAL Graph linear inequalities in two variables. GOAL Use linear inequalities to solve real-life problems, such as finding the number of minutes ou can call relatives using a calling card in Eample 4. Wh ou should learn it To model real-life data, such as blood pressures in our arm and ankle in E. 45. GOAL GRAPHING LINEAR INEQUALITIES A linear inequalit in two variables is an inequalit that can be written in one of the following forms: A + B < C, A + B C, A + B > C, A + B C An ordered pair (, ) is a solution of a linear inequalit if the inequalit is true when the values of and are substituted into the inequalit. For instance, (º6, ) is a solution of º 9 because (º6) º 9 is a true statement. EXAMPLE Checking Solutions of Inequalities Check whether the given ordered pair is a solution of + 5. a. (0, ) b. (4, º) c. (, ) ORDERED PAIR SUBSTITUTE CONCLUSION a. (0, ) (0) + () = / 5 (0, ) is not a solution. b. (4, º) (4) + (º) = 5 5 (4, º) is a solution. c. (, ) () + () = 7 5 (, ) is a solution. ACTIVITY Developing Concepts Investigating the Graph of an Inequalit Cop the scatter plot. Test each circled point to see whether it is a solution of +. If it is a solution, color it blue. If it is not a solution, color it red. Graph the line + =. What relationship do ou see between the colored points and the line? 4 Describe a general strateg for graphing an inequalit in two variables. The graph of a linear inequalit in two variables is the graph of all solutions of the inequalit. The boundar line of the inequalit divides the coordinate plane into two half-planes: a shaded region which contains the points that are solutions of the inequalit, and an unshaded region which contains the points that are not. 08 Chapter Linear Equations and Functions

40 Page of 6 GRAPHING A LINEAR INEQUALITY The graph of a linear inequalit in two variables is a half-plane. To graph a linear inequalit, follow these steps: STEP Graph the boundar line of the inequalit. Use a dashed line for < or > and a solid line for or. STEP To decide which side of the boundar line to shade, test a point not on the boundar line to see whether it is a solution of the inequalit. Then shade the appropriate half-plane. EXAMPLE Graphing Linear Inequalities in One Variable Look Back For help with inequalities in one variable, see p. 4. Graph (a) < º and (b) in a coordinate plane. a. Graph the boundar line = º. b. Graph the boundar line =. Use a dashed line because < º. Use a solid line because. Test the point (0, 0). Because (0, 0) Test the point (0, 0). Because (0, 0) is not a solution of the inequalit, is a solution of the inequalit, shade shade the half-plane below the line. the half-plane to the left of the line. (0,0) (0,0) < EXAMPLE Graphing Linear Inequalities in Two Variables Stud Tip Because our test point must not be on the boundar line, ou ma not alwas be able to use (0, 0) as a convenient test point. In such cases test a different point, such as (, ) or (, 0). Graph (a) < and (b) º 5 0. a. Graph the boundar line =. b. Graph the boundar line º 5 = 0. Use a dashed line because <. Use a solid line because º 5 0. Test the point (, ). Because (, ) is a solution of the inequalit, shade the half-plane below the line. (, ) Test the point (0, 0). Because (0, 0) is not a solution of the inequalit, shade the half-plane below the line. (0, 0) < Linear Inequalities in Two Variables 09

41 Page of 6 GOAL USING LINEAR INEQUALITIES IN EXAMPLE 4 Writing and Using a Linear Inequalit Communication You have relatives living in both the United States and Meico. You are given a prepaid phone card worth $50. Calls within the continental United States cost $.6 per minute and calls to Meico cost $.44 per minute. a. Write a linear inequalit in two variables to represent the number of minutes ou can use for calls within the United States and for calls to Meico. b. Graph the inequalit and discuss three possible solutions in the contet of the real-life situation. PROBLEM SOLVING STRATEGY a. VERBAL MODEL United States rate United States Meico Meico time + rate time Value of card LABELS ALGEBRAIC MODEL United States rate = 0.6 United States time = Meico rate = 0.44 Meico time = Value of card = (dollars per minute) (minutes) (dollars per minute) (minutes) (dollars) HOMEWORK HELP Visit our Web site for etra eamples. INTERNET b. Graph the boundar line = 50. Use a solid line because Test the point (0, 0). Because (0, 0) is a solution of the inequalit, shade the half-plane below the line. Finall, because and cannot be negative, restrict the graph to points in the first quadrant. Possible solutions are points within the shaded region shown. Meico time (min) Calling Cards (65, 90) 90 (8, 8) 60 0 (50, 0) United States time (min) One solution is to spend 65 minutes on calls within the United States and 90 minutes on calls to Meico. The total cost will be $50. To split the time evenl, ou could spend 8 minutes on calls within the United States and 8 minutes on calls to Meico. The total cost will be $ You could instead spend 50 minutes on calls within the United States and onl 0 minutes on calls to Meico. The total cost will be $ Chapter Linear Equations and Functions

42 Page 4 of 6 GUIDED PRACTICE Vocabular Check Concept Check. Compare the graph of a linear inequalit with the graph of a linear equation.. Would ou use a dashed line or a solid line for the graph of A + B < C? for the graph of A + B C? Eplain. Tell whether the statement is true or false. Eplain.. The point 4, 0 is a solution of º > 4. Skill Check 4. The graph of < + 5 is the half-plane below the line = + 5. GRAPHING INEQUALITIES Graph the inequalit in a coordinate plane. 5. > 5 6. < º º 4 9. º > º. º < º0. CALLING CARDS Look back at Eample 4. Suppose ou have relatives living in China instead of Meico. Calls to China cost $.75 per minute. Write and graph a linear inequalit showing the number of minutes ou can use for calls within the United States and for calls to China. Then discuss three possible solutions in the contet of the real-life situation. PRACTICE AND APPLICATIONS Etra Practice to help ou master skills is on p. 94. CHECKING S Check whether the given ordered pairs are solutions of the inequalit. 4. º5; (0, ), (º5, ) 5. 7; (, º6), (0, 4) 6. < º9 + 7; (º, ), (, º8) º0.5; (, ), (º, 0) HOMEWORK HELP Eample : Es. 4 7 Eample : Es. 8, 44 Eample : Es Eample 4: Es INEQUALITIES IN ONE VARIABLE Graph the inequalit in a coordinate plane º º <. 8 > º4. < 0.75 MATCHING GRAPHS Match the inequalit with its graph. 4. º 4 5. º º < A. B. C. INEQUALITIES IN TWO VARIABLES Graph the inequalit > º4 º 9. < 0.75 º > 4. 9 º 9 > º6. + >.6 Linear Inequalities in Two Variables

43 Page 5 of 6 MATCHING GRAPHS Match the inequalit with its graph.. + > º + A. B. C. GRAPHING INEQUALITIES Graph the inequalit in a coordinate plane º º8 7. < º 8. 5 > º º > º > < HEALTH RISKS B comparing the blood pressure in our ankle with the blood pressure in our arm, a phsician can determine whether our arteries are becoming clogged with plaque. If the blood pressure in our ankle is less than 90% of the blood pressure in our arm, ou ma be at risk for heart disease. Write and graph an inequalit that relates the unacceptable blood pressure in our ankle to the blood pressure in our arm. NUTRITIONISTS A nutritionist plans nutrition programs and promotes health eating habits. Over one half of all nutritionists work in hospitals, nursing homes, or phsician s offices. CAREER LINK INTERNET FOCUS ON CAREERS NUTRITION In Eercises 46 and 47, use the following information. Teenagers should consume at least 00 milligrams of calcium per da. Suppose ou get calcium from two different sources, skim milk and cheddar cheese. One cup of skim milk supplies 96 milligrams of calcium, and one slice of cheddar cheese supplies 8 milligrams of calcium. Source: Nutrition in Eercise and Sport 46. Write and graph an inequalit that represents the amounts of skim milk and cheddar cheese ou need to consume to meet our dail requirement of calcium. 47. Determine how man cups of skim milk ou should drink if ou have eaten two slices of cheddar cheese. MOVIES In Eercises 48 and 49, use the following information. You receive a gift certificate for $5 to our local movie theater. Matinees are $4.50 each and evening shows are $7.50 each. 48. Write and graph an inequalit that represents the numbers of matinees and evening shows ou can attend. 49. Give three possible combinations of the numbers of matinees and evening shows ou can attend. FOOTBALL In Eercises 50 and 5, use the following information. In one of its first five games of a season, a football team scored a school record of 6 points. In all of the first five games, points came from touchdowns worth 7 points and field goals worth points. 50. Write and graph an inequalit that represents the numbers of touchdowns and field goals the team could have scored in an of the first five games. 5. Give five possible numbers of points scored, including the number of touchdowns and the number of field goals, for the first five games. Chapter Linear Equations and Functions

44 Page 6 of 6 Test Preparation Challenge EXTRA CHALLENGE 5. MULTI-STEP PROBLEM You want to open our own truck rental compan. You do some research and find that the majorit of truck rental companies in our area charge a flat fee of $9.99, plus $.9 for ever mile driven. You want to charge less so that ou can advertise our lower rate and get more business. a. Write and graph an equation for the cost of renting a truck from other truck rental companies. b. Shade the region of the coordinate plane where the amount ou will charge must fall. c. To charge less than our competitors, will ou offer a lower flat fee, a lower rate per mile, or both? Eplain our choice. d. Write and graph an equation for the cost of renting a truck from our compan in the same coordinate plane used in part (a). e. CRITICAL THINKING Wh can t ou offer a lower rate per mile but a higher flat fee and still alwas charge less? VISUAL THINKING In Eercises 5 55, use the graph shown. 5. Write the inequalit whose graph is shown. 54. Eplain how ou came up with the inequalit. 55. What real-life situation could the first quadrant portion of the graph represent? MIXED REVIEW SCIENTIFIC NOTATION Write the number in scientific notation. (Skills Review, p. 9) 56. 0,000, ,650,000, , GRAPHING EQUATIONS Graph the equation. (Review. for.7) 6. = 5 º 5 6. = º5 º 64. = º º = = º4 + = 4 WRITING EQUATIONS Write an equation of the line that passes through the given points. (Review.4 for.7) 68. (, ), (5, 5) 69. (0, 7), (5, ) 70. (º, 6), (8, º) 7. (, ), (, º4) 7. (, 9), (º0, º6) 7. (4, º8), (º7, º8) 74. GARDENING The horizontal middle of the United States is at about 40 N latitude. As a rule of thumb, plants will bloom earlier south of 40 N latitude and later north of 40 N latitude. The function w = (l º 40) gives the number 5 of weeks w (earlier or later) that plants at latitude l N will bloom compared with those at 40 N. The equation is valid from 5 N to 45 N latitude. Identif the domain and range of the function and then graph the function. (Review.).6 Linear Inequalities in Two Variables

45 Page of 7.7 Piecewise Functions What ou should learn GOAL Represent piecewise functions. GOAL Use piecewise functions to model real-life quantities, such as the amount ou earn at a summer job in Eample 6. Wh ou should learn it To solve real-life problems, such as determining the cost of ordering silkscreen T-shirts in Es. 54 and 55. GOAL REPRESENTING PIECEWISE FUNCTIONS Up to now in this chapter a function has been represented b a single equation. In man real-life problems, however, functions are represented b a combination of equations, each corresponding to a part of the domain. Such functions are called piecewise functions. For eample, the piecewise function given b ƒ() = is defined b two equations. One equation gives the values of ƒ() when is less than or equal to, and the other equation gives the values of ƒ() when is greater than. EXAMPLE Evaluating a Piecewise Function Evaluate ƒ() when (a) = 0, (b) =, and (c) = 4. ƒ() = º, if +, if > +, if < +, if a. ƒ() = + Because 0 <, use first equation. ƒ(0) = 0 + = Substitute 0 for. b. ƒ() = + Because, use second equation. ƒ() = () + = 5 Substitute for. c. ƒ() = + Because 4, use second equation. ƒ(4) = (4) + = 9 Substitute 4 for. EXAMPLE Graphing a Piecewise Function Graph this function: ƒ() = +, if < º +, if To the left of =, the graph is given b = +. To the right of and including =, the graph is given b = º +. (, ) The graph is composed of two ras with common initial point (, ). 4 Chapter Linear Equations and Functions

46 Page of 7 EXAMPLE Graphing a Step Function Graph this function: ƒ() =, if 0 <, if <, if < 4, if < 4 The graph of the function is composed of four line segments. For instance, the first line segment is given b the equation = and represents the graph when is greater than or equal to 0 and less than. The solid dot at (, ) indicates that ƒ(). The open dot at (, ) indicates that ƒ() The function in Eample is called a step function because its graph resembles a set of stair steps. Another eample of a step function is the greatest integer function. This function is denoted b g() =. For ever real number, g() is the greatest integer less than or equal to. The graph of g() is shown at the right. Note that in Eample the function ƒ could have been written as ƒ() = +, 0 < 4. EXAMPLE 4 Writing a Piecewise Function Write equations for the piecewise function whose graph is shown. (0, ) (, ) (, 0) (0, 0) To the left of = 0, the graph is part of the line passing through (º, 0) and (0, ). An equation of this line is given b: = + To the right of and including = 0, the graph is part of the line passing through (0, 0) and (, ). An equation of this line is given b: = The equations for the piecewise function are: +, if <0 ƒ() =, if 0 Note that ƒ() = + does not correspond to = 0 because there is an open dot at (0, ), but ƒ() = does correspond to = 0 because there is a solid dot at (0, 0)..7 Piecewise Functions 5

47 Page of 7 GOAL USING PIECEWISE FUNCTIONS IN Urban Parking EXAMPLE 5 Using a Step Function a. Write and graph a piecewise function for the parking charges shown on the sign. b. What are the domain and range of the function? Garage Rates (Weekends) $ per half hour $8 maimum for hours a. For times up to one half hour, the charge is $. For each additional half hour (or portion of a half hour), the charge is an additional $ until ou reach $8. Let t represent the number of hours ou park. The piecewise function and graph are: ƒ(t) =, if 0 < t 0.5 6, if 0.5 < t 8, if < t Cost (dollars) ƒ(t) Weeknight Rates t Time (hours) b. The domain is 0 < t, and the range consists of, 6, 8 Wages EXAMPLE 6 Using a Piecewise Function You have a summer job that pas time and a half for overtime. That is, if ou work more than 40 hours per week, our hourl wage for the etra hours is.5 times our normal hourl wage of $7. a. Write and graph a piecewise function that gives our weekl pa P in terms of the number h of hours ou work. b. How much will ou get paid if ou work 45 hours? a. For up to 40 hours our pa is given b 7h. For over 40 hours our pa is given b: 7(40) +.5(7)(h º 40) = 0.5h º 40 The piecewise function is: 7h, if 0 h 40 P(h) = 0.5h º 40, if h >40 The graph of the function is shown. Note that for up to 40 hours the rate of change is $7 per hour, but for over 40 hours the rate of change is $0.50 per hour. b. To find how much ou will get paid for working 45 hours, use the equation P(h) = 0.5h º 40. P(45) = 0.5(45) º 40 =.5 You will earn $.50. Pa (dollars) P(h) Summer Job Hours (40, 80) h 6 Chapter Linear Equations and Functions

48 Page 4 of 7 GUIDED PRACTICE Vocabular Check Concept Check. Define piecewise function and step function. Give an eample of each.. Look back at Eample. What does a solid dot on the graph of a step function indicate? What does an open dot indicate? Skill Check Tell whether the statement is True or False. Eplain.. In the graph of a piecewise function, the separate pieces are alwas connected., if < 4. ƒ() = 4, if < can be rewritten as ƒ() =, < 4. 6, if < 4 Evaluate ƒ() = º, if 4 for the given value of. + 7, if > 4 5. = 0 6. = º 7. = 4 8. = º Graph the function. +, if < 9. ƒ() = 0. ƒ() = º + 4, if. Write equations for the piecewise function whose graph is shown.. PARKING RATES The weekda parking rates for a garage are shown. Write and graph a piecewise function for the weekda parking charges at that garage. 4, if 0 < 5, if < 4 6, if 4 < 6 (0, 6) (, ) E. (8, 0) Garage Rates (Weekdas) $ per half hour $8 maimum for hours PRACTICE AND APPLICATIONS Etra Practice to help ou master skills is on p. 94. EVALUATING FUNCTIONS Evaluate the function for the given value of. ƒ() = 5 º, if <º º 9, if º. ƒ(º4) 4. ƒ(º) 5. ƒ(0) 6. ƒ(5) h() = º 0, if 6 º º, if >6 7. h() 8. h(º0) 9. h(6) 0. h(0) HOMEWORK HELP Eample : Es. 0 Eample : Es. 6 Eample : Es. 7 Eample 4: Es Eamples 5 and 6: Es GRAPHING FUNCTIONS Graph the function.. ƒ() =, if. ƒ() = º +, if < +, if º5. ƒ() = 4. ƒ() = +, if <º5 º 4, if 4 5. ƒ() = 6. ƒ() = º + 6, if >4 + 6, if º º º, if >º º, if > º 4, if º 8, if <9 º, if 9.7 Piecewise Functions 7

49 Page 5 of 7 GRAPHING STEP FUNCTIONS Graph the step function., if º < 5, if < 4 7. ƒ() = 8. ƒ() = 8, if 4 < 9 0, if 9 < 6.5, if º4 < º 4., if º < 0.9, if < º., if < 6 º, if 0 < º, if < 9. ƒ() = º5, if < 0. ƒ() = º7, if < 4 º9, if 4 < 5 4, if º0 < º8 6, if º8 < º6 8, if º6 < º4 9., if º4 < º 0, if º < 0 SPECIAL STEP FUNCTIONS Graph the special step function. Then eplain how ou think the function got its name.. CEILING FUNCTION. ROUNDING FUNCTION......, if 0 <, if 0.5 <.5 ƒ() = =, if < ƒ() = ROUND() =, if.5 <.5, if <, if.5 < CRITICAL THINKING Look back at Eample. How would the graph of the function change if < was replaced with and was replaced with >? Eplain our answer. 4. CRITICAL THINKING Look back at Eample. How would the graph of the function change if each was replaced with < and each < was replaced with? Eplain our answer. WRITING PIECEWISE FUNCTIONS Write equations for the piecewise function whose graph is shown KEYSTROKE HELP Visit our Web site to see kestrokes for several models of calculators. INTERNET GREATEST INTEGER FUNCTION On man graphing calculators is denoted b int(). Use a graphing calculator to graph the function. 4. g() = 4. g() = 4. g() = º 44. g() = g() = g() = g() = g() = º 49. g() = º Chapter Linear Equations and Functions

50 Page 6 of 7 POSTAL RATES In Eercises 50 and 5, use the following information. As of Januar 0, 999, the cost C (in dollars) of sending net-da mail using the United States Postal Service, depending on the weight (in ounces) of a package up to five pounds, is given b the function below. INTERNET DATA UPDATE of United States Postal Service data at Graph the function. C () =.75, if 0 < , if 8 < 8.50, if < 48.5, if 48 < , if 64 < Identif the domain and range of the function. HOMEWORK HELP Visit our Web site for help with problem solving in Es. 5 and 5. INTERNET FOCUS ON APPLICATIONS PHOTOCOPY RATES In Eercises 5 and 5, use the function given for the cost C (in dollars) of making photocopies at a cop shop. 5. Graph the function. C () = 5. VISUAL THINKING Use our graph to eplain wh it would not be costeffective to make 450 photocopies. SILK-SCREEN T-SHIRTS In Eercises 54 and 55, use the following silk-screen shop charges. An initial charge of $0 to create the silk screen $7.00 per shirt for orders of 50 or fewer shirts $5.80 per shirt for orders of more than 50 shirts 54. Write a piecewise function that gives the cost C for an order of shirts. 55. Graph the function. 0.5, if 0 < 5 0.0, if , if , if 50 SOCIAL SECURITY In Eercises 56 and 57, use the following information. The amount of Social Securit ta ou pa, part of our Federal Insurance Contributions Act (FICA) deductions, depends on our annual income. As of 999 ou pa 6.% of our income if it is less than $7,600. If our income is at least $7,600, ou pa a fied amount of $ INTERNET DATA UPDATE of Social Securit Administration data at SNOWSTORM B weighing snow at the end of a snowstorm ou can determine the water content of the snow. This information is one of the factors used to determine avalanche warnings. APPLICATION LINK INTERNET 56. Write and graph a piecewise function that gives the Social Securit ta. 57. How much Social Securit ta do ou pa if ou make $0,000 per ear? SNOWSTORM In Eercises 58 and 59, use the following information. During a nine hour snowstorm it snows at a rate of inch per hour for the first two hours, at a of rate of inches per hour for the net si hours, and at a rate of inch per hour for the final hour. 58. Write and graph a piecewise function that gives the depth of the snow during the snowstorm. 59. How man inches of snow accumulated from the storm?.7 Piecewise Functions 9

51 Page 7 of 7 Test Preparation QUANTITATIVE COMPARISON In Eercises 60 and 6, choose the statement that is true about the given quantities. A The quantit in column A is greater. B The quantit in column B is greater. C The two quantities are equal. D The relationship cannot be determined from the given information. Column A Column B º 7, if ƒ() where ƒ() = ƒ() where ƒ() = º + 9, if > 5 +, if < 9 ƒ(0) where ƒ() = ƒ(º4) where ƒ() = 6 º 4, if 9 +, if < 8 º, if 8 9, if º4, if > º4 Challenge EXTRA CHALLENGE 6. SCUBA DIVING The time t (in minutes) that a person ma safel scuba dive without having to decompress while surfacing is determined b the depth d (in feet) of the dive. Using the information below, write and graph a piecewise inequalit that describes the time limits for scuba divers at various depths. For depths from 40 feet (the minimum depth requiring decompression) to 5 feet, the time must not eceed 600 minutes minus ten times the depth. For depths greater than 5 feet to less than 90 feet, the time must not eceed 0 minutes minus the depth. For depths from 90 feet to 0 feet (the maimum safe depth for a recreational diver), the time must not eceed 75 minutes minus one half the depth. MIXED REVIEW SOLVING EQUATIONS Solve the equation. (Review.7 for.8) = = 65. º = = 67. º 5 = 68. º 4 = 6 SCATTER PLOTS Draw a scatter plot of the data. Then tell whether the data have a positive, a negative, or relativel no correlation. (Review.5) 69. º8 º8 º7 º6 º5 º4 º4 º º º º º8 º5 º7 º º4 º8 º º º SLEEPING BAGS To be comfortable, sleeping bags rated for º40 F have.5 inches of insulation, and those rated for 40 F have.5 inches. Write a linear model for the amount a of insulation needed to be comfortable at temperature T. How much insulation would ou need to be comfortable at 0 F? (Review.4) 0 Chapter Linear Equations and Functions

52 Page of 7.8 Absolute Value Functions What ou should learn GOAL Represent absolute value functions. GOAL Use absolute value functions to model real-life situations, such as plaing pool in Eample 4. Wh ou should learn it To solve real-life problems, such as when an orchestra should reach a desired sound level in Es. 44 and 45. GOAL REPRESENTING ABSOLUTE VALUE FUNCTIONS In Lesson.7 ou learned that the absolute value of is defined b:, if > 0 = 0, if = 0 º, if < 0 The graph of this piecewise function consists of two ras, is V-shaped, and opens up. The corner point of the graph, called the verte, occurs at the origin. To the left of 0, the graph is given b the line. (, ) (, ) To the right of 0, the graph is given b the line. Notice that the graph of = is smmetric in the -ais because for ever point (, ) on the graph, the point (º, ) is also on the graph. ACTIVITY Developing Concepts Graphs of Absolute Value Functions In the same coordinate plane, graph = a for a = º, º,, and. What effect does a have on the graph of = a? What is the verte of the graph of = a? In the same coordinate plane, graph = º h for h = º, 0, and. What effect does h have on the graph of = º h? What is the verte of the graph of = º h? In the same coordinate plane, graph = + k for k = º, 0, and. What effect does k have on the graph of = + k? What is the verte of the graph of = + k? Although in the activit ou investigated the effects of a, h, and k on the graph of = a º h + k separatel, these effects can be combined. For eample, the graph of = º4 +is shown in red along with the graph of = in blue. Notice that the verte of the red graph is (4, ) and that the red graph is narrower than the blue graph. 4 Chapter Linear Equations and Functions

53 Page of 7 GRAPHING ABSOLUTE VALUE FUNCTIONS The graph of = a º h + k has the following characteristics. The graph has verte (h, k) and is smmetric in the line = h. The graph is V-shaped. It opens up if a > 0 and down if a < 0. The graph is wider than the graph of = if a <. The graph is narrower than the graph of = if a >. Skills Review For help with smmetr, see p. 99. To graph an absolute value function ou ma find it helpful to plot the verte and one other point. Use smmetr to plot a third point and then complete the graph. EXAMPLE Graphing an Absolute Value Function Graph = º + +. To graph = º + +, plot the verte at (º, ). Then plot another point on the graph, such as (º, ). Use smmetr to plot a third point, (º, ). Connect these three points with a V-shaped graph. Note that a = º < 0 and a =, so the graph opens down and is the same width as the graph of =. (, ) (, ) 4 (, ) EXAMPLE Writing an Absolute Value Function Write an equation of the graph shown. The verte of the graph is (0, º), so the equation has the form: (, ) = a º 0 + (º) or = a º To find the value of a, substitute the coordinates of the point (, ) into the equation and solve. (0, ) = a º Write equation. = a º Substitute for and for. = a º Simplif. 4 = a Add to each side. = a Divide each side b. An equation of the graph is = º. CHECK Notice the graph opens up and is narrower than the graph of =, so is a reasonable value for a..8 Absolute Value Functions

54 Page of 7 GOAL USING ABSOLUTE VALUE FUNCTIONS IN Camping EXAMPLE Interpreting an Absolute Value Function The front of a camping tent can be modeled b the function = º.4 º where and are measured in feet and the -ais represents the ground. a. Graph the function. b. Interpret the domain and range of the function in the given contet. a. The graph of the function is shown. The verte is (.5,.5) and the graph opens down. It is narrower than the graph of =. b. The domain is 0 5, so the tent is 5 feet wide. The range is 0.5, so the tent is.5 feet tall. (.5,.5) (0, 0) (5, 0) EXAMPLE 4 Interpreting an Absolute Value Graph Billiards While plaing pool, ou tr to shoot the eight ball into the corner pocket as shown. Imagine that a coordinate plane is placed over the pool table. The eight ball is at 5, 5 4 and the pocket ou are aiming for is at (0, 5). You are going to bank the ball off the side at (6, 0). a. Write an equation for the path of the ball. 0 ft 5 ft b. Do ou make our shot? a. The verte of the path of the ball is (6, 0), so the equation has the form = a º 6. Substitute the coordinates of the point 5, 5 4 into the equation and solve for a. 5 4 = a 5 º 6 Substitute }5 } for and 5 for. 4 5 = a Solve for a. 4 An equation for the path of the ball is = 5 º 6. 4 b. You will make our shot if the point (0, 5) lies on the path of the ball º 6 Substitute 5 for and 0 for. 4 5 = 5 Simplif. The point (0, 5) satisfies the equation, so ou do make our shot. 4 Chapter Linear Equations and Functions

55 Page 4 of 7 GUIDED PRACTICE Vocabular Check Concept Check Skill Check. What do the coordinates (h, k) represent on the graph of = a º h + k?. How do ou know if the graph of = a º h + k opens up or down? How do ou know if it is wider, narrower, or the same width as the graph of =?. ERROR ANALYSIS Eplain wh the graph shown is not the graph of = + +. Graph the function. Then identif the verte, tell whether the graph opens up or down, and tell whether the graph is wider, narrower, or the same width as the graph of =. 4. = 5. = = º 0 7. = = + 6 º 0 9. = º º º 4 E. 0. Write an equation for the function whose graph is shown.. CAMPING Suppose that the tent in Eample is 7 feet wide and 5 feet tall. Write a function that models the front of the tent. Let the -ais represent the ground. Then graph the function and identif the domain and range of the function. E. 0 PRACTICE AND APPLICATIONS Etra Practice to help ou master skills is on p. 94. EXAMINING THE EFFECT OF a Match the function with its graph.. ƒ() =. ƒ() = º 4. ƒ() = A. B. C. EXAMINING THE EFFECTS OF h AND k Match the function with its graph. 5. = º 6. = º 7. = + A. B. C. HOMEWORK HELP Eample : Es. 5 Eample : Es. 4 9 Eample : Es Eample 4: Es Absolute Value Functions 5

56 Page 5 of 7 GRAPHING ABSOLUTE VALUE FUNCTIONS Graph the function. Then identif the verte, tell whether the graph opens up or down, and tell whether the graph is wider, narrower, or the same width as the graph of =. 8. = 6 º 7 9. = = º º 8 +. = º + +. = º + 4. = º = º 5 5. = º + 6 KEYSTROKE HELP Visit our Web site to see kestrokes for several models of calculators. INTERNET ABSOLUTE VALUE On man graphing calculators is denoted b ABS(). Use a graphing calculator to graph the absolute value function. Then use the Trace feature to find the corresponding -value(s) for the given -value. 6. = + 4; = 0 7. = + 4 ; = 9 8. = 5 ; = 9. = + 4 º 5; = = º º + 5; = 0.5. = º. + 7; = º. = º º 5; = º5. =.5 º + 6; = 8.5 WRITING EQUATIONS Write an equation of the graph shown FOCUS ON APPLICATIONS 5 0 MUSIC SINGLES A musical group s single will change position in the charts from week to week. The Beatles were at No. most often with a total of hit singles. APPLICATION LINK INTERNET MUSIC SINGLES In Eercises 40 and 4, use the following information. A musical group s new single is released. Weekl sales s (in thousands) increase steadil for a while and then decrease as given b the function s = º t º where t is the time (in weeks). 40. Graph the function. 4. What was the maimum number of singles sold in one week? RAINSTORMS In Eercises 4 and 4, use the following information. A rainstorm begins as a drizzle, builds up to a heav rain, and then drops back to a drizzle. The rate r (in inches per hour) at which it rains is given b the function r = º0.5 t º where t is the time (in hours). 4. Graph the function. 4. For how long does it rain and when does it rain the hardest? 6 Chapter Linear Equations and Functions

57 Page 6 of 7 HOMEWORK HELP Visit our Web site for help with problem solving in E. 48. INTERNET Test Preparation Challenge EXTRA CHALLENGE SOUND LEVELS In Eercises 44 and 45, use the following information. Suppose a musical piece calls for an orchestra to start at fortissimo (about 90 decibels), decrease in loudness to pianissimo (about 50 decibels) in four measures, and then increase back to fortissimo in another four measures. The sound level s (in decibels) of the musical piece can be modeled b the function s = 0 m º where m is the number of measures. 44. Graph the function for 0 m After how man measures should the orchestra be at the loudness of mezzo forte (about 70 decibels)? 46. MINIATURE GOLF You are tring to make a holein-one on the miniature golf green shown. Imagine that a coordinate plane is placed over the golf green. The golf ball is at (.5, ) and the hole is at (9.5, ). You are going to bank the ball off the side wall of the green at (6, 8). Write an equation for the path of the ball and determine if ou make our shot. 47. REFLECTING SUNLIGHT You are sitting in a boat on a lake. You can get a sunburn from sunlight that hits ou directl and from sunlight that reflects off the water. Sunlight reflects off the water at the point (, 0) and hits ou at the point (.5, ). Write and graph the function that shows the path of the sunlight. 48. TRANSAMERICA PYRAMID The Transamerica Pramid, shown at the right, is an office building in San Francisco. It stands 85 feet tall and is 45 feet wide at its base. Imagine that a coordinate plane is placed over a side of the building. In the coordinate plane, each unit represents one foot, and the origin is at the center of the building s base. Write an absolute value function whose graph is the V-shaped outline of the sides of the building, ignoring the shoulders of the building. 49. MULTIPLE CHOICE Which statement is true about the graph of the function = º + +? A Its verte is at (, ). B Its verte is at (º, º). C It opens down. D It is wider than the graph of =. 50. MULTIPLE CHOICE Which function is represented b the graph shown? A = º º 0 + B = º + 0 º C = º º º 0 D = º GRAPHING Graph the functions. 5. = and = 5. = º5 and = 5 5. = + 6 and = = + (º) and = Based on our answers to Eercises 5 54, do ou think ab = a b and a + b = a + b are true statements? Eplain. ft 8 ft.8 Absolute Value Functions 7