Analyzing Linear Equations

Transcription

1 Analzing Linear Equations Lesson 5-1 Find the slope of a line. Lesson 5- Write direct variation equations. Lessons 5- through 5-5 Write linear equations in slope-intercept and point-slope forms. Lesson 5-6 Write equations for parallel and perpendicular lines. Lesson 5-7 Draw a scatter plot and write the equations of a line of fit. Ke Vocabular slope (p. 56) rate of change (p. 58) direct variation (p. 64) slope-intercept form (p. 7) point-slope form (p. 86) Linear equations are used to model a variet of real-world situations. The concept of slope allows ou to analze how a quantit changes over time. You can use a linear equation to model the cost of the space program. The United States began its eploration of space in Januar, 1958, when it launched its first satellite into orbit. In the 1970s, NASA developed the space shuttle to reduce costs b inventing the first reusable spacecraft. You will use a linear equation to model the cost of the space program in Lesson Chapter 5 Analzing Linear Equations

2 Prerequisite Skills To be successful in this chapter, ou ll need to master these skills and be able to appl them in problem-solving situations. Review these skills before beginning Chapter 5. For Lesson 5-1 Simplif. (For review, see pages 798 and 799.) Simplif Fractions For Lesson 5- Evaluate Epressions Evaluate a b for each set of values. c d (For review, see Lesson 1-.) 9. a 6, b 5, c 8, d a 5, b 1, c, d a, b 1, c 4, d 0 1. a 8, b, c 1, d 1 1. a, b, c 4, d a 1, b, c 7, d 9 For Lessons 5- through 5-7 Write the ordered pair for each point. (For review, see Lesson 4-1.) 15. J 16. K 17. L 18. M 19. N 0. P K N Identif Points on a Coordinate Plane J P L M Make this Foldable to help ou organize information about writing linear equations. Begin with four sheets of grid paper. Fold and Cut Staple Fold each sheet of grid paper in half along the width. Then cut along the crease. Staple the eight half-sheets together to form a booklet. Cut Tabs Label Cut seven lines from the bottom of the top sheet, si lines from the second sheet, and so on. Label each of the tabs with a lesson number. The last tab is for the vocabular Vocabular Reading and Writing As ou read and stud the chapter, use each page to write notes and to graph eamples for each lesson. Chapter 5 Analzing Linear Equations 55

3 Slope Standards 1.0,.01,.0, 4.01 Vocabular slope rate of change Find the slope of a line. Use rate of change to solve problems. is slope important in architecture? The slope of a roof describes how steep it is. It is the number of units the roof rises for each unit of run. In the photo, the roof rises 8 feet for each 1 feet of run. slope r ise run 8 or 1 8 ft rise 1 ft run Section of roof FIND SLPE The of a line is a number determined b an two points on slope the line. This number describes how steep the line is. The greater the absolute value of the slope, the steeper the line. Slope is the ratio of the change in the -coordinates (rise) to the change in the -coordinates (run) as ou move from one point to the other. The graph shows a line that passes through (1, ) and (4, 5). slope r ise run change in -coordinates change in -coordinates 5 or 4 1 So, the slope of the line is. run:4 1 (4, 5) (1, ) rise: 5 Slope of a Line Stud Tip Reading Math In 1, the 1 is called a subscript. It is read sub 1. Words Smbols The slope of a line is the ratio of the rise to the run. The slope m of a nonvertical line through an two points, ( 1, 1 ) and (, ), can be found as follows. m 1 change in 1 change in Model 1 1 ( 1, 1 ) (, ) 56 Chapter 5 Analzing Linear Equations

4 Stud Tip Common Misconception It ma make our calculations easier to choose the point on the left as ( 1, 1 ). However, either point ma be chosen as ( 1, 1 ). Eample 1 Positive Slope Find the slope of the line that passes through (1, ) and (, 4). Let (1, ) ( 1, 1 ) and (, 4) (, ). m 1 r ise run 1 4 Substitute. (1) 4 or 1 Simplif. The slope is 1. (1, ) (, 4) TEACHING TIP Eample Negative Slope Find the slope of the line that passes through (1, ) and (4, 1). Let (1, ) ( 1, 1 ) and (4, 1) (, ). m 1 r ise run 1 1 () Substitute. 4 (1) or 1 Simplif. The slope is 1. (4, 1) (1, ) Eample Zero Slope Find the slope of the line that passes through (1, ) and (1, ). Let (1, ) ( 1, 1 ) and (1, ) (, ). m 1 r ise run 1 Substitute or 0 Simplif. The slope is zero. (1, ) (1, ) Eample 4 Undefined Slope Find the slope of the line that passes through (1, ) and (1, ). Let (1, ) ( 1, 1 ) and (1, ) (, ). m 1 r ise run 1 () or Since division b zero is undefined, the slope is undefined. (1, ) (1, ) Lesson 5-1 Slope 57

5 Positive Slope Negative Slope Slope of 0 Undefined Slope line slopes up from left to right line slopes down from left to right horizontal line Classifing Lines vertical line Stud Tip Look Back To review cross products, see Lesson -6. Log on for: Updated data More activities on rate of change usa_toda If ou know the slope of a line and the coordinates of one of the points on a line, ou can find the coordinates of other points on the line. Eample 5 RATE F CHANGE Find Coordinates Given Slope Find the value of r so that the line through (r, 6) and (10, ) has a slope of. Let (r, 6) ( 1, 1 ) and (10, ) (, ). m 1 1 Slope formula (r, 6) 6 10 r Substitute r (10 r) (9) 0 r 18 0 r Slope can be used to describe a rate of change. The rate of change tells, on average, how a quantit is changing over time. Eample 6 Subtract. Find a Rate of Change DINING UT The graph shows the amount spent on food and drink at U.S. restaurants in recent ears. a. Find the rates of change for and Use the formula for slope. r ise change in quantit billion S run change in time ears Find the cross products. Simplif. Add 0 to each side. r 1 Simplif. r 1 Divide each side b. r 4 Simplif. USA TDAY Snapshots Dining out Food and drink sales at U.S. restaurants b ear (in billions): 1990: $9 (10, ) 000: $76 $00 $00 $ : $ Source: National Restaurant Association B Hilar Wasson and Alejandro Gonzalez, USA TDAY 58 Chapter 5 Analzing Linear Equations

6 change in quantit : Substitute. change in time or 11.9 Simplif. 10 Spending on food and drink increased b $119 billion in a 10-ear period for a rate of change of $11.9 billion per ear. change in quantit : Substitute. change in time or 1.7 Simplif. 10 ver this 10-ear period, spending increased b $17 billion, for a rate of change of $1.7 billion per ear. b. Eplain the meaning of the slope in each case. For , on average, $11.9 billion more was spent each ear than the last. For , on average, $1.7 billion more was spent each ear than the last. c. How are the different rates of change shown on the graph? There is a greater vertical change for than for Therefore, the section of the graph for has a steeper slope. Concept Check 1. Eplain how ou would find the slope of the line at the right.. PEN ENDED Draw the graph of a line having each slope. a. positive slope b. negative slope c. slope of 0 d. undefined slope (1, ) (, 5). Eplain wh the formula for determining slope using the coordinates of two points does not appl to vertical lines. 4. FIND THE ERRR Carlos and Allison are finding the slope of the line that passes through (, 6) and (5, ). Carlos 6 = or 1 5 Allison 6 = or 1 5 Who is correct? Eplain our reasoning. Guided Practice GUIDED PRACTICE KEY Find the slope of the line that passes through each pair of points. 5. (1, 1), (, 4) 6. (0, 0), (5, 4) 7. (, ), (1, ) 8. (7, 4), (9, 1) 9. (, 5), (, 5) 10. (1, ), (1, 0) Find the value of r so the line that passes through each pair of points has the given slope. 11. (6, ), (r, 6), m 4 1. (9, r), (6, ), m 1 Lesson 5-1 Slope 59

7 Application CABLE TV For Eercises 1 and 14, use the graph at the right. 1. Find the rate of change for Without calculating, find a -ear period that had a greater rate of change than Eplain our reasoning. Number (millions) U.S. Cable TV Subscribers Year Practice and Appl Homework Help For Eercises See Eamples Etra Practice See page 81. Find the slope of the line that passes through each pair of points (, 4) (, 1) (0, ) (, ) 17. (4, 1), (, ) 18. (, ), (1, ) 19. (, 1), (, ) 0. (, ), (9, 7) 1. (5, 7), (, ). (, 6), (, 4). (, 4), (5, 1) 4. (, 1), (5, ) 5. (5, 4), (5, 1) 6. (, 6), (1, ) 7. (, ), (8, ) 8. (, 9), (7, 6) 9. (8, ), (6, ) 0. (, 0), (1, 1) 1. (4.5, 1), (5., ). (0.75, 1), (0.75, 1). 1, 11, 1, , 11 4, 1, 1 ARCHITECTURE Use a ruler to estimate the slope of each roof Chapter 5 Analzing Linear Equations 7. Find the slope of the line that passes through the origin and (r, s). 8. What is the slope of the line that passes through (a, b) and (a, b)? 9. PAINTING A ladder reaches a height of 16 feet on a wall. If the bottom of the ladder is placed 4 feet awa from the wall, what is the slope of the ladder as a positive number?

8 40. PART-TIME JBS In 1991, the federal minimum wage rate was $4.5 per hour. In 1997, it was increased to $5.15. Find the annual rate of change in the federal minimum wage rate from 1991 to Find the value of r so the line that passes through each pair of points has the given slope. 41. (6, ), (9, r), m 1 4. (4, 5), (, r), m 8 4. (5, r), (, ), m (, 7), (r, ), m , 1 4, r, 5 4, m 4 46., r, 1, 1, m (4, r), (r, ), m (r, 5), (, r), m CRITICAL THINKING Eplain how ou know that the slope of the line through (4, 5) and (4, 5) is positive without calculating. HEALTH For Eercises 50 5, use the table that shows Karen s height from age 1 to age 0. Age (ears) Height (inches) Make a broken-line graph of the data. 51. Use the graph to determine the two-ear period when Karen grew the fastest. Eplain our reasoning. 5. Eplain the meaning of the horizontal section of the graph. SCHL For Eercises 5 55, use the graph that shows public school enrollment. 5. For which 5-ear period was the rate of change the greatest? When was the rate of change the least? 54. Find the rate of change from 1985 to Eplain the meaning of the part of the graph with a negative slope. Number (millions) U.S. Public School Enrollment Grades Year 56. RESEARCH Use the Internet or other reference to find the population of our cit or town in 190, 1940,..., 000. For which decade was the rate of change the greatest? 57. CNSTRUCTIN The slope of a stairwa determines how eas it is to climb the stairs. Suppose the vertical distance between two floors is 8 feet 9 inches. Find the total run of the ideal stairwa in feet and inches. tread (ideal 11 in.) riser (ideal 7 in.) Lesson 5-1 Slope 61

9 58. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. Wh is slope important in architecture? Include the following in our answer: an eplanation of how to find the slope of a roof, and a comparison of the appearance of roofs with different slopes. NC Practice Standardized Test Practice 59. The slope of the line passing through (5, 4) and (5, 10) is A positive. B negative. C zero. D undefined. 60. The slope of the line passing through (a, b) and (c, d) is d c. b d. d b A B C. b a a c a c D a c. b d Etending the Lesson 61. Choose four different pairs of points from those labeled on the graph. Find the slope of the line using the coordinates of each pair of points. Describe our findings. (, ) (4, 0) (1, 1) (5, ) 6. MAKE A CNJECTURE Determine whether Q(, ), R(1, 1), and S(4, ) lie on the same line. Eplain our reasoning. Maintain Your Skills Mied Review Write an equation for each function. (Lesson 4-8) f() f() Determine whether each relation is a function. (Lesson 4-6) {(1, 0), (1, 4), (1, 1)} 68. {(6, ), (5, ), (, )} 69. Graph 0. (Lesson 4-4) 70. What number is 40% of 7.5? (Prerequisite Skill) Getting Read for the Net Lesson Find each product. (Lesson -4) 71. 7() 7. (4)() 7. (9)(4) 74. (8)(.7) (14) PREREQUISITE SKILL Find each quotient. (To review dividing fractions, see pages 800 and 801.) Chapter 5 Analzing Linear Equations

10 Standards 1.0,.01,.0, 4.01 Mathematical Words and Everda Words You ma have noticed that man words used in mathematics are also used in everda language. You can use the everda meaning of these words to better understand their mathematical meaning. The table shows two mathematical words along with their everda and mathematical meanings. Word Everda Meaning Mathematical Meaning epression 1. something that epresses or communicates in words, art, music, or movement. the manner in which one epresses oneself, especiall in speaking, depicting, or performing one or more numbers or variables along with one or more arithmetic operations function 1. the action for which one is particularl fitted or emploed. an official ceremon or a formal social occasion. something closel related to another thing and dependent on it for its eistence, value, or significance a relationship in which the output depends upon the input Source: The American Heritage Dictionar of the English Language Notice that the mathematical meaning is more specific, but related to the everda meaning. For eample, the mathematical meaning of epression is closel related to the first everda definition. In mathematics, an epression communicates using smbols. Reading to Learn 1. How does the mathematical meaning of function compare to the everda meaning?. RESEARCH Use the Internet or other reference to find the everda meaning of each word below. How might these words appl to mathematics? Make a table like the one above and note the mathematical meanings that ou learn as ou stud Chapter 5. a. slope b. intercept c. parallel Investigating Slope-Intercept Form 6 Reading Mathematics Mathematical Words and Everda Words 6

11 Slope and Direct Variation Standards 1.0, 1.0,.01,.0, 4.01 Vocabular direct variation constant of variation famil of graphs parent graph Write and graph direct variation equations. Solve problems involving direct variation. is slope related to our shower? A standard showerhead uses about 6 gallons of water per minute. If ou graph the ordered pairs from the table, the slope of the line is 6. (minutes) (gallons) The equation is 6. The number of gallons of water depends directl on the amount of time in the shower. Gallons Gallons of Water Used in a Shower 1 4 Minutes DIRECT VARIATIN A direct variation is described b an equation of the form k, where k 0. We sa that varies directl with or varies directl as. In the equation k, k is the constant of variation. Eample 1 Slope and Constant of Variation Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. a. b. (1, ) TEACHING TIP (0, 0) (0, 0) (1, ) The constant of variation is. m 1 Slope formula 1 The constant of variation is. m 1 Slope formula m 0 ( 1, 1 ) = (0, 0) m 0 ( 1, 1 ) = (0, 0) 1 0 (, ) = (1, ) 1 0 (, ) = (1, ) m The slope is. m The slope is Chapter 5 Analzing Linear Equations Compare the constant of variation with the slope of the graph for each eample. Notice that the slope of the graph of k is k.

12 The ordered pair (0, 0) is a solution of k. Therefore, the graph of k passes through the origin. You can use this information to graph direct variation equations. Eample Direct Variation with k 0 TEACHING TIP Graph 4. Step 1 Write the slope as a ratio. ise r run Step Graph (0, 0). Step From the point (0, 0), move up 4 units and right 1 unit. Draw a dot. 4 Step 4 Draw a line containing the points. Eample Graph 1. Direct Variation with k 0 Step 1 Write the slope as a ratio. 1 1 r ise run Step Graph (0, 0). Step Step 4 From the point (0, 0), move down 1 unit and right units. Draw a dot. Draw a line containing the points. 1 A famil of graphs includes graphs and equations of graphs that have at least one characteristic in common. The parent graph is the simplest graph in a famil. Famil of Graphs The calculator screen shows the graphs of,, and 4. Think and Discuss 1. Describe an similarities among the graphs.. Describe an differences among the graphs.. Write an equation whose graph has a 4 steeper slope than 4. Check our [10, 10] scl: 1 b [10, 10] scl: 1 answer b graphing 4 and our equation. 4. Write an equation whose graph lies between the graphs of and. Check our answer b graphing the equations. 5. Write a description of this famil of graphs. What characteristics do the graphs have in common? How are the different? 6. The equations whose graphs are in this famil are all of the form m. How does the graph change as the absolute value of m increases? Lesson 5- Slope and Direct Variation 65

13 Direct variation equations are of the form k, where k 0. The graph of k alwas passes through the origin. Direct Variation Graphs The slope can be positive. k 0 The slope can be negative. k 0 k k If ou know that varies directl as, ou can write a direct variation equation that relates the two quantities. Eample 4 Write and Solve a Direct Variation Equation Suppose varies directl as, and 8 when 7. a. Write a direct variation equation that relates and. Find the value of k. k Direct variation formula 8 k(7) Replace with 8 and with 7. 8 k( 7) Divide each side b k Simplif. Therefore, 4. b. Use the direct variation equation to find when 5. 4 Direct variation equation 5 4 Replace with Divide each side b 4. 1 Simplif. Therefore, 1 when 5. More About... Biolog Snow geese migrate more than 000 miles from their winter home in the southwest United States to their summer home in the Canadian arctic. Source: Audubon Societ SLVE PRBLEMS ne of the most common uses of direct variation is the formula for distance, d rt. In the formula, distance d varies directl as time t, and the rate r is the constant of variation. Eample 5 Direct Variation Equation BILGY A flock of snow geese migrated 75 miles in 7.5 hours. a. Write a direct variation equation for the distance flown in an time. Words The distance traveled is 75 miles, and the time is 7.5 hours. Variables Equation Let r rate. Distance equals rate times time. Solve for the rate. 75 r(7.5) riginal equation 75 mi r 7.5 h ) 7. 5 Divide each side b r Simplif. Therefore, the direct variation equation is d 50t. 66 Chapter 5 Analzing Linear Equations

14 b. Graph the equation. The graph of d 50t passes through the origin with slope 50. m 5 0 r ise 1 run c. Estimate how man hours of fling time it would take the geese to migrate 000 miles. Distance (miles) Migration of Snow Geese d 50t riginal equation t Replace d with t Divide each side b 50. t 60 Simplif. At this rate, it will take 60 hours of fling time to migrate 000 miles. d d 50t (7.5, 75) t Time (hours) Concept Check Guided Practice GUIDED PRACTICE KEY 1. PEN ENDED Write a general equation for varies directl as.. Choose the equations that represent direct variations. Then find the constant of variation for each direct variation. a. 15 rs b. 4a b c. z 1 d. s 9 t. Eplain how the constant of variation and the slope are related in a direct variation equation. Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points (, 1) (0, 0) (0, 0) (, ) 1 Graph each equation Write a direct variation equation that relates and. Assume that varies directl as. Then solve. 9. If 7 when 6, find when If 10 when 9, find when If 7 when 14, find when 0. Application JBS For Eercises 1 14, use the following information. Suppose ou work at a job where our pa varies directl as the number of hours ou work. Your pa for 7.5 hours is $ Write a direct variation equation relating our pa to the hours worked. 1. Graph the equation. 14. Find our pa if ou work 0 hours. Lesson 5- Slope and Direct Variation 67

15 Practice and Appl Homework Help For Eercises See Eamples , Etra Practice See page 81. Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points (0, 0) (, 4) (1, 4) 4 (0, 0) 1 (0, 0) (, 1) (0, 0) (, ) (, ) (0, 0) (0, 0) 1 4 (4, 1) Graph each equation Write a direct variation equation that relates and. Assume that varies directl as. Then solve.. If 8 when 4, find when If 6 when 6, find when If 16 when 4, find when If 18 when 6, find when If 4 when 1, find when If 1 when 15, find when If.5 when 0.5, find when If 6.6 when 9.9, find when If when 1 4, find when If 6 when, find when Chapter 5 Analzing Linear Equations Write a direct variation equation that relates the variables. Then graph the equation. 4. GEMETRY The circumference C of a circle is about.14 times the diameter d. 44. GEMETRY The perimeter P of a square is 4 times the length of a side s. 45. SEWING The total cost is C for n ards of ribbon priced at $0.99 per ard. 46. RETAIL Kona coffee beans are $14.49 per pound. The total cost of p pounds is C.

16 47. CRITICAL THINKING Suppose varies directl as. If the value of is doubled, what happens to the value of? Eplain. BILGY Which line in the graph represents the sprinting speeds of each animal? 48. elephant, 5 mph 49. reindeer, mph 50. lion, 50 mph 51. grizzl bear, 0 mph Distance (miles) Sprinting Speeds 1 1 Time (hours) 4 SPACE For Eercises 5 and 5, use the following information. The weight of an object on the moon varies directl with its weight on Earth. With all of his equipment, astronaut Neil Armstrong weighed 60 pounds on Earth, but weighed onl 60 pounds on the moon. 5. Write an equation that relates weight on the moon m with weight on Earth e. 5. Suppose ou weigh 18 pounds on Earth. What would ou weigh on the moon? Veterinar Medicine Veterinarians compare the age of an animal to the age of a human on the basis of bone and tooth growth. nline Research For information about a career as a veterinarian, visit: careers NC Practice Standardized Test Practice ANIMALS For Eercises 54 and 55, Horse age () use the following information. Human age () Most animals age more rapidl than humans do. The chart shows equivalent ages for horses and humans. 54. Write an equation that relates human age to horse age. 55. Find the equivalent horse age for a human who is 16 ears old. 56. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How is slope related to our shower? Include the following in our answer: an equation that relates the number of gallons to the time spent in the shower for a low-flow showerhead that uses onl.5 gallons of water per minute, and a comparison of the steepness of the graph of this equation to the graph at the top of page Which equation best describes the graph at the right? A B C 1 D 1 Graphing Calculator 58. Which equation does not model a direct variation? A 4 B 1 C D 1 FAMILIES F GRAPHS For Eercises 59 6, use the graphs of 1,, and 4, which form a famil of graphs. 59. Graph 1,, and 4 on the same screen. 60. How are these graphs similar to the graphs in the Graphing Calculator Investigation on page 65? How are the different? Lesson 5- Slope and Direct Variation 69

17 61. Write an equation whose graph has a steeper slope than MAKE A CNJECTURE Eplain how ou can tell without graphing which of two direct variation equations has the graph with a steeper slope. Maintain Your Skills Mied Review Find the slope of the line that passes through each pair of points. (Lesson 5-1) (1, ) (, ) (, ) (, 0) (, ) (, 1) 66. Find the value of r so that the line that passes through (1, 7) and (r, ) has a slope of. (Lesson 5-1) Each table below represents points on a linear graph. Cop and complete each table. (Lesson 4-8) Add or subtract. (Lesson -) (1) (5) Getting Read for the Net Lesson PREREQUISITE SKILL Solve each equation for. (To review rewriting equations, see Lesson -8.) P ractice Quiz 1 Find the slope of the line that passes through each pair of points. (Lesson 5-1) 1. (4, 6), (, 8). (8, ), (11, ). (4, 8), (5, 9) 4. (0, 1), (7, 11) Find the value of r so the line that passes through each pair of points has the given slope. (Lesson 5-1) 5. (5, ), (r, 5), m 6. (6, r), (4, 9), m Graph each equation. (Lesson 5-) Lessons 5-1 and 5- Write a direct variation equation that relates and. Assume that varies directl as. Then solve. (Lesson 5-) 9. If 4 when 8, find when. 10. If 10 when 15, find when Chapter 5 Analzing Linear Equations

18 Standards 1.0,.0, 4.01 Collect the Data Cut a small hole in a top corner of a plastic sandwich bag. Loop a long rubber band through the hole. Tape the free end of the rubber band to the desktop. Use a centimeter ruler to measure the distance from the desktop to the end of the bag. Record this distance for 0 washers in the bag using a table like the one below. A Preview of Lesson 5- Investigating Slope-Intercept Form Number of Washers Distance Place one washer in the plastic bag. Then measure and record the new distance from the desktop to the end of the bag. Repeat the eperiment, adding different numbers of washers to the bag. Each time, record the number of washers and the distance from the desktop to the end of the bag. Analze the Data 1. The domain contains values represented b the independent variable, washers. The range contains values represented b the dependent variable, distance. n grid paper, graph the ordered pairs (washers, distance).. Write a sentence that describes the points on the graph.. Describe the point that represents the trial with no washers in the bag. 4. The rate of change can be found b using the formula for slope. r ise change in distance run change in number of washers Find the rate of change in the distance from the desktop to the end of the bag as more washers are added. 5. Eplain how the rate of change is shown on the graph. Make a Conjecture The graph shows sample data from a rubber band eperiment. Draw a graph for each situation. 6. A bag that hangs 10.5 centimeters from the desktop when empt and lengthens at the rate of the sample. 7. A bag that has the same length when empt as the sample and lengthens at a faster rate. 8. A bag that has the same length when empt as the sample and lengthens at a slower rate. Distance (cm) Number of Washers Investigating Slope-Intercept Form 71 Algebra Activit Investigating Slope-Intercept Form 71

19 Slope-Intercept Form Standards 1.0,.01,.0, 4.01 Vocabular slope-intercept form Write and graph linear equations in slope-intercept form. Model real-world data with an equation in slope-intercept form. is a -intercept related to a flat fee? A cellular phone service provider charges $0.10 per minute plus a flat fee of $5.00 each month. (minutes) (dollars) Dollars Total Cost of Cellular Phone Service Minutes The slope of the line is 0.1. It crosses the -ais at (0, 5). The equation of the line is charge per minute, $0.10 flat fee, $5.00 TEACHING TIP SLPE-INTERCEPT FRM An equation of the form m b is in slope-intercept form. When an equation is written in this form, ou can identif the slope and -intercept of its graph. Slope-Intercept Form Stud Tip Look Back To review intercepts, see Lesson 4-5. Words Smbols slope The linear equation m b is written in slope-intercept form, where m is the slope and b is the -intercept. m b -intercept Model (0, b) m b Eample 1 Write an Equation Given Slope and -Intercept Write an equation of the line whose slope is and whose -intercept is 5. m b Slope-intercept form 5 Replace m with and b with 5. 7 Chapter 5 Analzing Linear Equations

20 Stud Tip Vertical Lines The equation of a vertical line cannot be written in slope-intercept form. Wh? (a, 0) Horizontal Lines The equation of a horizontal line can be written in slope-intercept form as 0 b or b. b (0, b) a Eample Write an Equation Given Two Points Write an equation of the line shown in the graph. Step 1 You know the coordinates of two points on the line. Find the slope. Let ( 1, 1 ) (0, ) and (, ) (, 1). m 1 r ise run m 1 0 m 4 or The slope is. 1 Simplif. Step The line crosses the -ais at (0, ). So, the -intercept is. Step 1 0, 1, 1 Finall, write the equation. m b Slope-intercept form Replace m with and b with. The equation of the line is. Eample Graph 1. (0, ) (, 1) ne advantage of the slope-intercept form is that it allows ou to graph an equation quickl. Graph an Equation in Slope-Intercept Form Step 1 The -intercept is 1. So, graph (0, 1). Step The slope is or. r ise run From (0, 1), move down units and right units. Draw a dot. Step Draw a line connecting the points. (0, 1) 1 Eample 4 Graph an Equation in Standard Form Graph 5 6. Step 1 Solve for to find the slope-intercept form. 5 6 riginal equation Subtract 5 from each side. 6 5 Simplif (5) or Simplif. Divide each side b. Divide each term in the numerator b. (continued on the net page) Lesson 5- Slope-Intercept Form 7

21 Step The -intercept of 5 is. So, graph (0, ). Step Step 4 The slope is 5. From (0, ), move up 5 units and right units. Draw a dot. Draw a line containing the points. (0, ) 5 6 MDEL REAL-WRLD DATA If a quantit changes at a constant rate over time, it can be modeled b a linear equation. The -intercept represents a starting point, and the slope represents the rate of change. More About... Eample 5 Write an Equation in Slope-Intercept Form AGRICULTURE The natural sweeteners used in foods include sugar, corn sweeteners, srup, and hone. Use the information at the left about natural sweeteners. a. The amount of natural sweeteners consumed has increased b an average of.6 pounds per ear. Write a linear equation to find the average consumption of natural sweeteners in an ear after Words Variables The consumption increased.6 pounds per ear, so the rate of change is.6 pounds per ear. In the first ear, the average consumption was 1 pounds. Let C average consumption. Let n number of ears after Agriculture In 1989, each person in the United States consumed an average of 1 pounds of natural sweeteners. Source: USDA Agricultural utlook Equation Average rate of number of ears amount consumption equals change times after 1989 plus at start. C.6 n 1 b. Graph the equation. The graph passes through (0, 1) with slope.6. C 160 Consumption of Natural Sweeteners (10, 159) c. Find the number of pounds of natural sweeteners consumed b each person in The ear 1999 is 10 ears after So, n 10. C.6n 1 Consumption equation C.6(10) 1 Replace n with 10. C 159 Simplif. Pounds C.6n n Years Since 1989 So, the average person consumed 159 pounds of natural sweeteners in CHECK Notice that (10, 159) lies on the graph. 74 Chapter 5 Analzing Linear Equations

22 Concept Check Guided Practice GUIDED PRACTICE KEY 1. PEN ENDED Write an equation for a line with a slope of 7.. Eplain wh equations of vertical lines cannot be written in slope-intercept form, but equations of horizontal lines can.. Tell which part of the slope-intercept form represents the rate of change. Write an equation of the line with the given slope and -intercept. 4. slope:, -intercept: 1 5. slope: 4, -intercept: Write an equation of the line shown in each graph (, ) (0, ) (0, 1) (, 1) Graph each equation Application MNEY For Eercises 11 1, use the following information. Suppose ou have alread saved $50 toward the cost of a new television set. You plan to save $5 more each week for the net several weeks. 11. Write an equation for the total amount T ou will have w weeks from now. 1. Graph the equation. 1. Find the total amount saved after 7 weeks. Practice and Appl Homework Help For Eercises See Eamples , Etra Practice See page 81. Write an equation of the line with the given slope and -intercept. 14. slope:, -intercept: slope:, -intercept: slope: 1, -intercept: 17. slope:, -intercept: slope: 1, -intercept: slope: 0.5; -intercept: 7.5 Write an equation of the line shown in each graph (1, 4) (0, 1) (0, 4) (, 1) (0, ) (1, ) Lesson 5- Slope-Intercept Form 75

23 Write an equation of the line shown in each graph (0, 1) (, 1) (0, 0) (, ) (0, ) (, ) 6. Write an equation of a horizontal line that crosses the -ais at (0, 5). 7. Write an equation of a line that passes through the origin with slope. Graph each equation Write a linear equation in slope-intercept form to model each situation. 40. You rent a biccle for $0 plus $ per hour. 41. An auto repair shop charges $50 plus $5 per hour. 4. A candle is 6 inches tall and burns at a rate of 1 inch per hour. 4. The temperature is 15 and is epected to fall each hour during the night. 44. CRITICAL THINKING The equations, = 4,, and 10 form a famil of graphs. What characteristic do their graphs have in common? SALES For Eercises 45 and 46, use the following information and the graph at the right. In 1991, book sales in the United States totaled $16 billion. Sales increased b about $1 billion each ear until Write an equation to find the total sales S for an ear t between 1991 and If the trend continues, what will sales be in 005? Sales (billions of dollars) S Book Sales Years Since 1991 Source: Association of American Publishers t 76 Chapter 5 Analzing Linear Equations TRAFFIC For Eercises 47 49, use the following information. In 1966, the traffic fatalit rate in the United States was 5.5 fatalities per 100 million vehicle miles traveled. Between 1966 and 1999, the rate decreased b about 0.1 each ear. 47. Write an equation to find the fatalit rate R for an ear t between 1966 and Graph the equation. 49. Find the fatalit rate in 1999.

24 50. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How is a -intercept related to a flat fee? Include the following in our answer: the point at which the graph would cross the -ais if our cellular phone service provider charges a rate of $0.07 per minute plus a flat fee of $5.99, and a description of a situation in which the -intercept of its graph is $5. NC Practice Standardized Test Practice 51. Which equation does not have a -intercept of 5? A 5 B 5 C 5 D 5 5. Which situation below is modeled b the graph? A You have $100 and plan to spend $5 each week. B You have $100 and plan to save $5 each week. C You need $100 for a new CD plaer and plan to save $5 each week. D You need $100 for a new CD plaer and plan to spend $5 each week Etending the Lesson 5. The standard form of a linear equation is A B C, where A, B, and C are integers, A 0, and A and B are not both zero. Solve A B C for. Your answer is written in slope-intercept form. 54. Use the slope-intercept equation in Eercise 5 to write epressions for the slope and -intercept in terms of A, B, and C. 55. Use the epressions in Eercise 54 to find the slope and -intercept of each equation. a. 4 b. 4 1 c. 9 Maintain Your Skills Mied Review Write a direct variation equation that relates and. Assume that varies directl as. Then solve. (Lesson 5-) 56. If 45 when 60, find when If 15 when 4, find when 10. Find the slope of the line that passes through each pair of points. (Lesson 5-1) 58. (, 0), (4, 6) 59. (, 1), (, 4) 60. (5, 5), (9, ) 61. Write the numbers.5, 4, 0.5, 7 in order from least to greatest. (Lesson -7) 8 Solve each equation. (Lesson 1-) (7) b 64. q 6 Getting Read for the Net Lesson PREREQUISITE SKILL Find the slope of the line that passes through each pair of points. (To review slope, see Lesson 5-1.) 65. (1, ), (1, ) 66. (5, 8), (, 8) 67. (1, 1), (10, 1) Lesson 5- Slope-Intercept Form 77

25 A Follow-Up of Lesson 5- Standards 4.01 Families of Linear Graphs A famil of people is a group of people related b birth, marriage, or adoption. Recall that a famil of graphs includes graphs and equations of graphs that have at least one characteristic in common. Families of linear graphs fall into two categories those with the same slope and those with the same -intercept. A graphing calculator is a useful tool for studing a group of graphs to determine whether the form a famil. Eample 1 Graph, 4, and in the standard viewing window. Describe an similarities and differences among the graphs. Write a description of the famil. Enter the equations in the Y list as Y1, Y, and Y. Then graph the equations. KEYSTRKES: Review graphing on pages 4 and 5. The graph of has a slope of 1 and a -intercept of 0. The graph of 4 has a slope of 1 and a -intercept of 4. The graph of has a slope of 1 and a -intercept of. 4 [10, 10] scl: 1 b [10, 10] scl: 1 Notice that the graph of 4 is the same as the graph of, moved 4 units up. Also, the graph of is the same as the graph of, moved units down. All graphs have the same slope and different intercepts. Because the all have the same slope, this famil of graphs can be described as linear graphs with a slope of 1. Eample Graph 1, 1, and 1 1 in the standard viewing window. Describe an similarities and differences among the graphs. Write a description of the famil. Enter the equations in the Y list and graph. The graph of 1 has a slope of 1 and a -intercept of [10, 10] scl: 1 b [10, 10] scl: Chapter 5 Analzing Linear Equations

26 The graph of 1 has a slope of and a -intercept of 1. The graph of 1 1 has a slope of 1 and a -intercept of 1. These graphs have the same intercept and different slopes. This famil of graphs can be described as linear graphs with a -intercept of 1. Sometimes a common characteristic is not enough to determine that a group of equations describes a famil of graphs. Eample Graph, 5, and 1 in the standard viewing window. Describe an similarities and differences among the graphs. The graph of has slope and -intercept 0. The graph of 5 has slope and -intercept 5. The graph of 1 has slope 1 and -intercept 0. These equations are similar in that the all have negative slope. However since the slopes are different and the -intercepts are different, these graphs are not all in the same famil. 1 5 [10, 10] scl: 1 b [10, 10] scl: 1 Eercises Graph each set of equations on the same screen. Describe an similarities or differences among the graphs. If the graphs are part of the same famil, describe the famil MAKE A CNJECTURE Write a paragraph eplaining how the values of m and b in the slope-intercept form affect the graph of the equation. 8. Families of graphs are also called classes of functions. Describe the similarities and differences in the class of functions f() c, where c is an real number. 9. Graph. Make a conjecture about the transformations of the parent graph, c and, c. Use a graphing calculator with different values of c to test our conjecture. Graphing Calculator Investigation Families of Linear Graphs 79

27 Writing Equations in Slope-Intercept Form Standards 1.0,.01,.0, 4.01 Vocabular linear etrapolation Write an equation of a line given the slope and one point on a line. Write an equation of a line given two points on the line. can slope-intercept form be used to make predictions? In 1995, the population of rlando, Florida, was about 175,000. At that time, the population was growing at a rate of about 000 per ear. (ear) (population) , , ,000 Population of rlando, Florida Population (thousands) (1995, 175,000) Year If ou could write an equation based on the slope, 000, and the point (1995, 175,000), ou could predict the population for another ear. 80 Chapter 5 Analzing Linear Equations WRITE AN EQUATIN GIVEN THE SLPE AND NE PINT You have learned how to write an equation of a line when ou know the slope and a specific point, the -intercept. The following eample shows how to write an equation when ou know the slope and an point on the line. Eample 1 Write an Equation Given Slope and ne Point Write an equation of a line that passes through (1, 5) with slope. Step 1 The line has slope. To find the -intercept, replace m with and (, ) with (1, 5) in the slope-intercept form. Then, solve for b. m b Slope-intercept form 5 (1) b Replace m with, with 5, and with 1. 5 b Multipl. 5 b Subtract from each side. b Simplif. Step Write the slope-intercept form using m and b. m b Slope-intercept form Replace m with and b with. Therefore, the equation is.

28 CHECK You can check our result b graphing on a graphing calculator. Use the CALC menu to verif that it passes through (1, 5). [10, 10] scl: 1 b [10, 10] scl: 1 NC Practice Standardized Test Practice WRITE AN EQUATIN GIVEN TW PINTS Sometimes ou do not know the slope of a line, but ou know two points on the line. In this case, find the slope of the line. Then follow the steps in Eample 1. Eample Multiple-Choice Test Item Write an Equation Given Two Points The table of ordered pairs shows the coordinates of the two points on the graph of a function. Which equation describes the function? A 1 B C 1 D Read the Test Item The table represents the ordered pairs (, 1) and (6, 4). Solve the Test Item Step 1 Find the slope of the line containing the points. Let ( 1, 1 ) (,1) and (, ) (6, 4). m 1 Slope formula 1 m 4 (1) 6 () 1, 6, 1 1, 4 m or 1 9 Simplif. Step You know the slope and two points. Choose one point and find the -intercept. In this case, we chose (6, 4). m b Slope-intercept form 4 1 (6) b Replace m with 1, with 6, and with 4. Test-Taking Tip You can check our result b graphing. The line should pass through (, 1) and (6, 4). 4 b 4 b b Multipl. Add to each side. Simplif. Step Write the slope-intercept form using m 1 and b. m b Slope-intercept form 1 Replace m with 1 and b with. Therefore, the equation is 1. The answer is A. Lesson 5-4 Writing Equations in Slope-Intercept Form 81

29 Eample Write an Equation to Solve a Problem BASEBALL In the middle of the 1998 baseball season, Mark McGwire seemed to be on track to break the record for most runs batted in. After 40 games, McGwire had 45 runs batted in. After 86 games, he had 87 runs batted in. Write a linear equation to estimate the number of runs batted in for an number of games that season. Eplore Plan You know the number of runs batted in after 40 and 86 games. Let represent the number of games. Let represent the number of runs batted in. Write an equation of the line that passes through (40, 45) and (86, 87). Number Runs Batted In 90 (86, 87) (40, 45) Baseball Mark McGwire is best known for breaking Roger Maris single-season home run record of 61. In the 1998 season, McGwire hit 70 home runs. Source: USA TDAY TEACHING TIP Solve Find the slope. m 1 1 Slope formula m Let ( 1, 1 ) (40, 45) and (, ) (86, 87). m 4 or about Simplif. Choose (40, 45) and find the -intercept of the line. m b Slope-intercept form (40) b Replace m with 0.91, with 40, and with b Multipl b 6.4 Subtract 6.4 from each side. 8.6 b Simplif Games Write the slope-intercept form using m 0.91, and b m b Slope-intercept form Replace m with 0.91 and b with 8.6. Therefore, the equation is Eamine Check our result b substituting the coordinates of the point not chosen, (86, 87), into the equation riginal equation (86) 8.6 Replace with 87 and with Multipl The slope was rounded, so the answers var slightl. 8 Chapter 5 Analzing Linear Equations

30 Given the Slope and ne Point Step 1 Substitute the values of m,, and into the slope-intercept form and solve for b. Step Write the slope-intercept form using the values of m and b. Writing Equations Given Two Points Step 1 Find the slope. Step Choose one of the two points to use. Step Then, follow the steps for writing an equation given the slope and one point. When ou use a linear equation to predict values that are beond the range of the data, ou are using. linear etrapolation Eample 4 Linear Etrapolation SPRTS The record for most runs batted in during a single season is 190. Use the equation in Eample to decide whether a baseball fan following the 1998 season would have epected McGwire to break the record in the 16 games plaed that ear riginal equation 0.91(16) 8.6 Replace with Simplif. Since the record is 190 runs batted in, a fan would have predicted that Mark McGwire would not break the record. Be cautious when making a prediction using just two given points. The model ma be approimatel correct, but still give inaccurate predictions. For eample, in 1998, Mark McGwire had 147 runs batted in, which was nine less than the prediction. Concept Check Guided Practice GUIDED PRACTICE KEY NC Practice Standardized Test Practice 1. Compare and contrast the process used to write an equation given the slope and one point with the process used for two points.. PEN ENDED Write an equation in slope-intercept form of a line that has a -intercept of.. Tell whether the statement is sometimes, alwas, or never true. Eplain. You can write the equation of a line given its - and -intercepts. Write an equation of the line that passes through each point with the given slope. 4. (4, ), m 5. (, 7), m 6. (, 5), m 1 Write an equation of the line that passes through each pair of points. 7. (5, 1), (8, ) 8. (6, 0), (0, 4) 9. (5, ), (7, 4) 10. The table of ordered pairs shows the coordinates of the two points on the graph of a line. Which equation describes the line? A 7 B 7 C 5 D Lesson 5-4 Writing Equations in Slope-Intercept Form 8

31 Practice and Appl Homework Help For Eercises See Eamples , 4 Etra Practice See page 8. Write an equation of the line that passes through each point with the given slope m (1, ) (4, 1) m 1 1. (5, ), m 14. (5, 4), m (, 0), m 16. (5, ), m (, 1), m 18. (, 5), m 5 Write an equation of the line that passes through each pair of points (5, ) (0, ) (4, 1) (, 0) 1. (4, ), (, 4). (, ), (6, 4). (1, ), (, ) 4. (, ), (, ) 5. (7, ), (4, ) 6. (0, 5), (, 5) 7. (1, 1), (7, 4) 8. (5, 7), (0, 6) , 1, 1 4, 4 Write an equation of the line that has each pair of intercepts. 0. -intercept:, -intercept: intercept:, -intercept: 4. -intercept: 6, -intercept:. -intercept:, -intercept: MARRIAGE AGE For Eercises 4 7, use the information in the graphic. 4. Write a linear equation to predict the median age that men marr M for an ear t. 5. Use the equation to predict the median age of men who marr for the first time in Write a linear equation to predict the median age that women marr W for an ear t. 7. Use the equation to predict the median age of women who marr for the first time in 005. USA TDAY Snapshots Waiting on weddings Couples are marring later. The median age of men and women who tied the knot for the first time in 1970 and 1998: Men Women Men 6.7 Women 5 Source: Census Bureau, March 000 B Hilar Wasson and Sam Ward, USA TDAY 84 Chapter 5 Analzing Linear Equations

32 PPULATIN For Eercises 8 and 9, use the data at the top of page Write a linear equation to find rlando s population for an ear. 9. Predict what rlando s population will be in CANE RENTAL If ou rent a canoe for hours, ou will pa $45. Write a linear equation to find the total cost C of renting the canoe for h hours. For Eercises 41 4, consider line that passes through (14, ) and (8, 6). 41. Write an equation for line. 4. What is the slope of line? 4. Where does line intersect the -ais? the -ais? 44. CRITICAL THINKING The -intercept of a line is p, and the -intercept is q. Write an equation of the line. NC Practice Standardized Test Practice 45. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can slope-intercept form be used to make predictions? Include the following in our answer: a definition of linear etrapolation, and an eplanation of how slope-intercept form is used in linear etrapolation. 46. Which is an equation for the line with slope 1 through (, 1)? A 1 1 B 1 5 C 1 5 D 1 1 Maintain Your Skills Mied Review Getting Read for the Net Lesson 47. About 0,000 fewer babies were born in California in 1996 than in In 1995, about 560,000 babies were born. Which equation can be used to predict the number of babies (in thousands), born ears after 1995? A B C D Graph each equation. (Lesson 5-) HEALTH Each time our heart beats, it pumps.5 ounces of blood through our heart. Write a direct variation equation that relates the total volume of blood V with the number of times our heart beats b. (Lesson 5-) State the domain of each relation. (Lesson 4-) 5. {(0, 8), (9, ), (4, )} 5. {(, 1), (5, 1), (, 7), (0, )} Replace each with,, or to make a true sentence. (Lesson -7) PREREQUISITE SKILL Find each difference. (To review subtracting integers, see Lesson -.) () () Lesson 5-4 Writing Equations in Slope-Intercept Form 85

33 Writing Equations in Point-Slope Form Standards 1.0,.01, 4.01 Vocabular point-slope form Write the equation of a line in point-slope form. Write linear equations in different forms. The graph shows a line with slope that passes through (, 4). Another point on the line is (, ). m ( ) 4 Slope formula (, ) = (, ) ( 1, 1 ) = (, 4) ( ) Multipl each side b ( ). ( ) 4 Simplif. 4 ( ) Smmetric Propert of Equalit -coordinate slope -coordinate can ou use the slope formula to write an equation of a line? (, 4) (, ) PINT-SLPE FRM The equation above was generated using the coordinates of a known point and the slope of the line. It is written in point-slope form. Words The linear equation 1 m( 1 ) is written in point-slope form, where ( 1, 1 ) is a given point on a nonvertical line and m is the slope of the line. Smbols 1 m( 1 ) given point Model Point-Slope Form (, ) ( 1, 1 ) Stud Tip Point-Slope Form Remember, ( 1, 1 ) represents the given point, and (, ) represents an other point on the line. Eample 1 Write an Equation Given Slope and a Point Write the point-slope form of an equation for a line that passes through (1, 5) with slope. 1 m( 1 ) Point-slope form 5 [ (1)] ( 1, 1 ) (1, 5) 5 ( 1) Simplif. Therefore, the equation is 5 ( 1). (1, 5) 86 Chapter 5 Analzing Linear Equations

34 Vertical lines cannot be written in point-slope form because the slope is undefined. However, since the slope of a horizontal line is 0, horizontal lines can be written in point-slope form. Eample Write an Equation of a Horizontal Line Write the point-slope form of an equation for a horizontal line that passes through (6, ). 1 m( 1 ) () 0( 6) 0 Point-slope form ( 1, 1 ) (6, ) Simplif. Therefore, the equation is 0. (6, ) Stud Tip Look Back To review standard form, see Lesson 4-5. FRMS F LINEAR EQUATINS common forms of linear equations. You have learned about three of the most Form Equation Description Forms of Linear Equations Slope-Intercept m b m is the slope, and b is the -intercept. Point-Slope 1 m( 1 ) m is the slope and ( 1, 1 ) is a given point. Standard A B C A and B are not both zero. Usuall A is nonnegative and A, B, and C are integers whose greatest common factor is 1. Linear equations in point-slope form can be written in slope-intercept or standard form. Eample Write an Equation in Standard Form Write 5 5 ( ) in standard form. 4 In standard form, the variables are on the left side of the equation. A, B, and C are all integers. 5 5 ( ) 4 riginal equation 4( 5) ( ) Multipl each side b 4 to eliminate the fraction ( ) Distributive Propert Distributive Propert Subtract 0 from each side Simplif Add 5 to each side Simplif. The standard form of the equation is Lesson 5-5 Writing Equations in Point-Slope Form 87

35 Eample 4 Write an Equation in Slope-Intercept Form Write 1 ( 5) in slope-intercept form. In slope-intercept form, is on the left side of the equation. The constant and are on the right side. 1 ( 5) riginal equation Distributive Propert Add to each side and The slope-intercept form of the equation is 1 9. You can draw geometric figures on a coordinate plane and use the point-slope form to write equations of the lines. Stud Tip Geometr The hpotenuse is the side of a right triangle opposite the right angle. Eample 5 Write an Equation in Point-Slope Form GEMETRY The figure shows right triangle ABC. a. Write the point-slope form of the line containing the hpotenuse AB. Step 1 First, find the slope of AB. m 1 Slope formula or 6 4 ( 1, 1 ) (, 1), (, ) (6, 4) Step You can use either point for ( 1, 1 ) in the point-slope form. Method 1 Use (6, 4). Method Use (, 1). 1 m( 1 ) 1 m( 1 ) 4 4 ( 6) 1 ( ) 4 b. Write each equation in standard form. 4 4 ( 6) riginal equation 1 ( ) 4 A (, 1) (6, 4) B (6, 1) C 4( 4) 4 4 ( 6) Multipl each side b 4. 4( 1) 4 4 ( ) 4 16 ( 6) Multipl. 4 4 ( ) Distributive Propert Add to each side. 4 4 Subtract from each side. 4 4 Multipl each side b 1. 4 Regardless of which point was used to find the point-slope form, the standard form results in the same equation. 88 Chapter 5 Analzing Linear Equations

36 Concept Check 1. Eplain what 1 and 1 in the point-slope form of an equation represent.. FIND THE ERRR Tana and Akira wrote the point-slope form of an equation for a line that passes through (, 6) and (1, 6). Tana sas that Akira s equation is wrong. Akira sas the are both correct. Tana +6=4(+) Akira 6=4( 1) Who is correct? Eplain our reasoning.. PEN ENDED Write an equation in point-slope form. Then write an equation for the same line in slope-intercept form. Guided Practice GUIDED PRACTICE KEY Write the point-slope form of an equation for a line that passes through each point with the given slope (1, ) m (1, ) m 0 m (, ) Write each equation in standard form ( ) 8. ( 1) 9..5( 1) 4 Write each equation in slope-intercept form ( ) 11. ( 6) ( 4) Application GEMETRY For Eercises 1 and 14, use parallelogram ABCD. A parallelogram has opposite sides parallel. 1. Write the point-slope form of the line containing AD. 14. Write the standard form of the line containing AD. (1, ) A D (, 1) (6, ) B (4, 1) C Practice and Appl Homework Help For Eercises See Eamples Etra Practice See page 8. Write the point-slope form of an equation for a line that passes through each point with the given slope. 15. (, 8), m 16. (4, ), m (, 4), m 18. (6, 1), m (, 6), m 0 0. (9, 1), m 1. (8, ), m 4. (6, ), m. (1, ), m (9, 5), m 0 5. (4, 8), m 7 6. (1, 4), m 8 Lesson 5-5 Writing Equations in Point-Slope Form 89

37 7. Write the point-slope form of an equation for a horizontal line that passes through (5, 9). 8. A horizontal line passes through (0, 7). Write the point-slope form of its equation. Write each equation in standard form ( ) 0. ( 5) 1. 5 ( 6). 5( 1). 7 1 ( ) ( 4) ( 8) ( 1) 7. 5 ( 6) 8. 6 ( 4) ( 7) 40..5( 1) Write each equation in slope-intercept form. 41. ( 1) ( 1) 4. ( 5) ( ) ( 4) ( 9) ( ) ( 15) Write the point-slope form, slope-intercept form, and standard form of an equation for a line that passes through (5, ) with slope Line passes through (1, 6) with slope. Write the point-slope form, slope-intercept form, and standard form of an equation for line. BUSINESS For Eercises 55 57, use the following information. Ahome securit compan provides securit sstems for $5 per week, plus an installation fee. The total fee for 1 weeks of service is $ Write the point-slope form of an equation to find the total fee for an number of weeks. 56. Write the equation in slope-intercept form. 57. What is the flat fee for installation? Movies In 1907, movie theaters were called nickelodeons. There were about 5000 movie screens, and the average movie ticket cost 5 cents. Source: National Association of Theatre wners MVIES For Eercises 58 60, use the following information. Between 1990 and 1999, the number of movie screens in the United States increased b about 1500 each ear. In 1996, there were 9,690 movie screens. 58. Write the point-slope form of an equation to find the total number of screens for an ear. 59. Write the equation in slope-intercept form. 60. Predict the number of movie screens in the United States in 005. Number (thousands) 40 U.S. Movie Screens nline Research Data Update What has happened to the number of movie screens since 1999? Visit to learn more (1996, 9,690) Year Source: Motion Picture Association of America 90 Chapter 5 Analzing Linear Equations

38 GEMETRY For Eercises 61 6, use square PQRS. 61. Write a point-slope equation of the line containing each side. 6. Write the slope-intercept form of each equation. 6. Write the standard form of each equation. S P Q 64. CRITICAL THINKING A line contains the points (9, 1) and (5, 5). Write a convincing argument that the same line intersects the -ais at (10, 0). R NC Practice Standardized Test Practice 65. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can ou use the slope formula to write an equation of a line? Include the following in our answer: an eplanation of how ou can use the slope formula to write the point-slope form. 66. Which equation represents a line that neither passes through (0, 1) nor has a slope of? A 1 B 1 ( 6) C ( 6) D PEN ENDED Write the slope-intercept form of an equation of a line that passes through (, 5). Etending the Lesson For Eercises 68 71, use the graph at the right. 68. Choose three different pairs of points from the (1, ) graph. Write the slope-intercept form of the line (0, 1) using each pair. 69. Describe how the equations are related. 70. Choose a different pair of points from the graph and predict the equation of the line determined b these points. Check our conjecture b finding the equation. 71. MAKE A CNJECTURE What conclusion can ou draw from this activit? (1, 1) (, ) Maintain Your Skills Mied Review Write the slope-intercept form of an equation of the line that satisfies each condition. (Lessons 5- and 5-4) 7. slope and -intercept 5 7. passes through (, 4) with slope 74. passes through (, 4) and (0, 6) 75. a horizontal line through (1, 1) Solve each equation. (Lesson -4) 76. 4a c v Evaluate (5 4) ( 1 ). (Lesson 1-) Getting Read for the Net Lesson PREREQUISITE SKILL Write the multiplicative inverse of each number. (For review of multiplicative inverses, see pages 800 and 801.) Lesson 5-5 Writing Equations in Point-Slope Form 91

39 Geometr: Parallel and Perpendicular Lines Standards 1.0,.01, 4.01 Vocabular parallel lines perpendicular lines Write an equation of the line that passes through a given point, parallel to a given line. Write an equation of the line that passes through a given point, perpendicular to a given line. can ou determine whether two lines are parallel? The graphing calculator screen shows a famil of linear graphs whose slope is 1. Notice that the lines do not appear to intersect. PARALLEL LINES Lines in the same plane that do not intersect are called parallel lines. Parallel lines have the same slope. Words Two nonvertical lines are parallel if the have the same slope. All vertical lines are parallel. Parallel Lines in a Coordinate Plane Model same slope vertical lines 9 Chapter 5 Analzing Linear Equations You can write the equation of a line parallel to a given line if ou know a point on the line and an equation of the given line. Eample 1 Parallel Line Through a Given Point Write the slope-intercept form of an equation for the line that passes through (1, ) and is parallel to the graph of. The line parallel to has the same slope,. Replace m with, and ( 1, 1 ) with (1, ) in the point-slope form. 1 m( 1 ) () [ (1)] ( 1) 5 Point-slope form Replace m with, with, and with 1. Simplif. Distributive Propert Subtract from each side. Therefore, the equation is 5. Write the equation in slope-intercept form.

40 CHECK You can check our result b graphing both equations. The lines appear to be parallel. The graph of 5 passes through (1, ). (1, ) 5 PERPENDICULAR LINES Lines that intersect at right angles are called perpendicular lines. There is a relationship between the slopes of perpendicular lines. Perpendicular Lines Model A scalene triangle is one in which no two sides are equal. Cut out a scalene right triangle ABC so that C is a right angle. Label the vertices and the B sides as shown. c a Draw a coordinate plane on grid paper. Place ABC on the coordinate plane so that A is at the origin and side b lies along the positive -ais. A C b Analze 1. Name the coordinates of B.. What is the slope of side c?. Rotate the triangle 90 counterclockwise so that A is still at the origin and side b is along the positive -ais. Name the coordinates of B. 4. What is the slope of side c? 5. Repeat the activit for two other different scalene triangles. 6. For each triangle and its rotation, what is the relationship between the first position of side c and the second? 7. For each triangle and its rotation, describe the relationship between the coordinates of B in the first and second positions. 8. Describe the relationship between the slopes of c in each position. Make a Conjecture 9. Describe the relationship between the slopes of an two perpendicular lines. Words Two nonvertical lines are perpendicular if the product of their slopes is 1. That is, the slopes are opposite reciprocals of each other. Vertical lines and horizontal lines are also perpendicular. Perpendicular Lines in a Coordinate Plane Model m m 1 horizontal line vertical line Lesson 5-6 Geometr: Parallel and Perpendicular Lines 9

41 Eample Determine Whether Lines are Perpendicular KITES The outline of a kite is shown on a coordinate plane. Determine whether AC is perpendicular to BD. Find the slope of each segment. Slope of AC: m 5 1 or 5 7 Slope of BD: m 4 0 or D (0, 0) A (5, 5) B (8, 4) C (7, 1) The line segments are perpendicular because 1 () 1. You can write the equation of a line perpendicular to a given line if ou know a point on the line and the equation of the given line. Kites In India, kite festivals mark Makar Sankranti, when the Sun moves into the northern hemisphere. Source: Eample Perpendicular Line Through a Given Point Write the slope-intercept form for an equation of a line that passes through (, ) and is perpendicular to the graph of 4 1. Step 1 Find the slope of the given line riginal equation Subtract 1 from each side Simplif Divide each side b Simplif. Step Step The slope of the given line is 1. So, the slope of the line perpendicular 4 to this line is the opposite reciprocal of 1, or 4. 4 Use the point-slope form to find the equation. 1 m( 1 ) Point-slope form () 4[ ()] ( 1, 1 ) (, ) and m 4 Stud Tip Graphing Calculator The lines will not appear to be perpendicular on a graphing calculator if the scales on the aes are not set correctl. After graphing, press ZM 5 to set the aes for a correct representation. 4( ) Simplif. Distributive Propert Subtract from each side. Simplif. Therefore, the equation of the line is CHECK You can check our result b graphing both equations on a graphing calculator. Use the CALC menu to verif that 4 10 passes through (, ). [ , ] scl: 1 b [10, 10] scl: 1 94 Chapter 5 Analzing Linear Equations

42 Eample 4 Perpendicular Line Through a Given Point Write the slope-intercept form for an equation of a line perpendicular to the graph of 1 and passes through the -intercept of that line. Step 1 Step Find the slope of the perpendicular line. The slope of the given line is 1, therefore a perpendicular line has slope because 1 1. Find the -intercept of the given line. 1 riginal equation 0 1 Replace with 0. 1 Subtract from each side. 6 Multipl each side b. The -intercept is at (6, 0). Step Substitute the slope and the given point into the point-slope form of a linear equation. Then write the equation in slope-intercept form. 1 m( 1 ) Point-slope form 0 ( 6) Replace with 6, with 0, and m with. 18 Distributive Propert Concept Check 1. Eplain how to find the slope of a line that is perpendicular to the line shown in the graph.. PEN ENDED Give an eample of two numbers that are negative reciprocals.. Define parallel lines and perpendicular lines. 1 Guided Practice GUIDED PRACTICE KEY Write the slope-intercept form of an equation of the line that passes through the given point and is parallel to the graph of each equation (0, 1) (, ) 6. (1, ), 1 7. (, ), 4 8. GEMETRY Quadrilateral ABCD has vertices A(, 1), B(, ), C(5, 7), and D(0, 5). Determine whether AC is perpendicular to BD. Write the slope-intercept form of an equation that passes through the given point and is perpendicular to the graph of each equation. 9. (, 1), (6, ), (, ), 5 5 Lesson 5-6 Geometr: Parallel and Perpendicular Lines 95

43 Application 1. GEMETRY The line with equation 4 contains side AC of right triangle ABC. If the verte of the right angle C is at (, 5), what is an equation of the line that contains side BC? C B A Practice and Appl Homework Help For Eercises See Eamples , 4 Etra Practice See page 8. Write the slope-intercept form of an equation of the line that passes through the given point and is parallel to the graph of each equation. 1. (, 7), 14. (, 1), 15. (, ), (4, 1), (5, 4), (, ), (4, ), 1 0. (1, ), (, 0), 1. (, ), 6. (, ), (, ), GEMETRY A parallelogram is a quadrilateral in which opposite sides are parallel. Is ABCD a parallelogram? Eplain. 6. Write an equation of the line parallel to the graph of 5 and through the origin. 7. Write an equation of the line that has -intercept 6 and is parallel to the graph of = 8. 1 A D B C Write the slope-intercept form of an equation that passes through the given point and is perpendicular to the graph of each equation. 8. (, 0), 6 9. (1, 1), (, 1), 7 1. (0, 5), 8 4. (1, ), 1 4. (4, 7), 1 4. (0, 4), (, 7), 5 6. (6, 1), 7. (0, 1), 5 8. (8, ), (, ), Find an equation of the line that has a -intercept of and is perpendicular to the graph of Write an equation of the line that is perpendicular to the line through (9, 10) and (, ) and passes through the -intercept of that line. Determine whether the graphs of each pair of equations are parallel, perpendicular, or neither GEMETRY The diagonals of a square are segments that connect the opposite vertices. Determine the relationship between the diagonals AC and BD of square ABCD. D A 46. CRITICAL THINKING What is a if the lines with equations a 5 and (a 4) 1 are parallel? C B 96 Chapter 5 Analzing Linear Equations

44 NC Practice Standardized Test Practice 47. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can ou determine whether two lines are parallel? Include the following in our answer: an equation whose graph is parallel to the graph of 5, with an eplanation of our reasoning, and an equation whose graph is perpendicular to the graph of 5, with an eplanation of our reasoning. 48. What is the slope of a line perpendicular to the graph of 4 4? A 4 B 4 C 4 D How can the graph of 4 be used to graph? A Move the graph of the line right units. B Change the slope of the graph from 4 to. C Change the -intercept from 4 to. D Move the graph of the line left units. Maintain Your Skills Mied Review Write the point-slope form of an equation for a line that passes through each point with the given slope. (Lesson 5-5) 50. (, 5), m 51. (4, 7), m 5 5. (1, ), m 1 TELEPHNE For Eercises 5 and 54, use the following information. An international calling plan charges a rate per minute plus a flat fee. A 10-minute call to the Czech Republic costs $.19. A 15-minute call costs $4.9. (Lesson 5-4) 5. Write a linear equation in slope-intercept form to find the total cost C of an m-minute call. 54. Find the cost of a 1-minute call. Getting Read for the Net Lesson PREREQUISITE SKILL Write the slope-intercept form of an equation of the line that passes through each pair of points. (To review slope-intercept form, see Lesson 5-4.) 55. (5, 1), (, ) 56. (0, ), (8, 0) 57. (, 1), (, 4) 58. (5, 5), (8, 1) 59. (6, 9), (4, 9) 60. (6, 4), (, ) P ractice Quiz Write the slope-intercept form for an equation of the line that satisfies each condition. 1. slope 4 and -intercept (Lesson 5-). passes through (1, ) with slope (Lesson 5-4). passes through (1, ) and (1, ) (Lesson 5-4) 4. parallel to the graph of and passes through (, ) (Lesson 5-6) 5. Write 4 1 ( ) in standard form and in slope-intercept form. (Lesson 5-5) Lessons 5- through Lesson 5-6 Geometr: Parallel and Perpendicular Lines 97

45 Statistics: Scatter Plots and Lines of Fit Standards.0, 4.01 Vocabular scatter plot positive correlation negative correlation line of fit best-fit line linear interpolation Interpret points on a scatter plot. Write equations for lines of fit. do scatter plots help identif trends in data? The points of a set of real-world data do not alwas lie on one line. But, ou ma be able to draw a line that seems to be close to all the points. The line in the graph shows a linear relationship between the ear and the number of bushels of apples. As the ears increase, the number of bushels of apples also increases. Number (millions of bushels) Apples in Storage in U.S Year 00 Source: U.S. Apple Association INTERPRET PINTS N A SCATTER PLT A scatter plot is a graph in which two sets of data are plotted as ordered pairs in a coordinate plane. Scatter plots are used to investigate a relationship between two quantities. In the first graph below, there is a positive correlation between and. That is, as increases, increases. In the second graph below, there is a negative correlation between and. That is, as increases, decreases. In the third graph below, there is no correlation between and. That is, and are not related. If the pattern in a scatter plot is linear, ou can draw a line to summarize the data. This can help identif trends in the data. Positive Correlation Negative Correlation No Correlation negative slope Scatter Plots positive slope 98 Chapter 5 Analzing Linear Equations

46 Eample 1 Analze Scatter Plots Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. a. NUTRITIN The graph shows fat grams and Calories for selected choices at a fast-food restaurant. The graph shows a positive correlation. As the number of fat grams increases, the number of Calories increases. Calories Fast-Food Choices Fat Grams 40 Source: len Publishing Co. TEACHING TIP b. CARS The graph shows the weight and the highwa gas mileage of selected cars. The graph shows a negative correlation. As the weight of the automobile increases, the gas mileage decreases. Gas Mileage (mpg) Automobiles Source: Yahoo! Weight (pounds) Is there a relationship between the length of a person s foot and his or her height? Make a scatter plot and then look for a pattern. Making Predictions Collect the Data Measure our partner s foot and height in centimeters. Then trade places. Add the points (foot length, height) to a class scatter plot. Analze the Data 1. Is there a correlation between foot length and height for the members of our class? If so, describe it.. Draw a line that summarizes the data and shows how the height changes as the foot length changes. Make a Conjecture. Use the line to predict the height of a person whose foot length is 5 centimeters. Eplain our method. Lesson 5-7 Statistics: Scatter Plots and Lines of Fit 99

47 LINES F FIT If the data points do not all lie on a line, but are close to a line, ou can draw a line of fit. This line describes the trend of the data. nce ou have a line of fit, ou can find an equation of the line. In this lesson, ou will use a graphical method to find a line of fit. In the follow-up to Lesson 5-7, ou will use a graphing calculator to find a line of fit. The calculator uses a statistical method to find the line that most closel approimates the data. This line is called the best-fit line. Eample Find a Line of Fit BIRDS The table shows an estimate for the number of bald eagle pairs in the United States for certain ears since Years since 1985 Bald Eagle Pairs Source: U.S. Fish and Wildlife Service Birds The bald eagle was listed as an endangered species in 196, when the number of breeding pairs had dropped below 500. Source: U.S. Fish and Wildlife Service Stud Tip Lines of Fit When ou use the graphical method, the line of fit is an approimation. So, ou ma draw another line of fit using other points that is equall valid. Some valid lines of fit ma not contain an of the data points. a. Draw a scatter plot and determine what relationship eists, if an, in the data. Let the independent variable be the number of ears since 1985, and let the Bald Eagle Pairs dependent variable be the number of bald eagle pairs The scatter plot seems to indicate that as the number of ears increases, the 4500 number of bald eagle pairs increases. There is a positive correlation between the two variables. 500 b. Draw a line of fit for the scatter plot. 500 No one line will pass through all of the data points. Draw a line that passes 0 close to the points. A line of fit is shown in the scatter plot at the right. c. Write the slope-intercept form of an equation for the line of fit. The line of fit shown above passes through the data points (, 500) and (11, 5000). Step 1 Step Find the slope. m 1 Slope formula 1 m Let ( 11 1, 1 ) (, 500) and (, ) (11, 5000). m 5 00 or 1.5 Simplif. 8 Use m 1.5 and either the point-slope form or the slope-intercept form to write the equation. You can use either data point. We chose (, 500). Number (pairs) Years Since 1985 Point-slope form 1 m( 1 ) ( 5) Using either method, Slope-intercept form m b () b b b Chapter 5 Analzing Linear Equations

48 CHECK Check our result b substituting (11, 5000) into Line of fit equation (11) Replace with 11 and with Multipl Add. The solution checks. In Lesson 5-4, ou learned about linear etrapolation, which is predicting values that are outside the range of the data. You can also use a linear equation to predict values that are inside the range of the data. This is called linear interpolation. Eample Linear Interpolation BIRDS Use the equation for the line of fit in Eample to estimate the number of bald eagle pairs in Use the equation , where is the number of ears since 1985 and is the number of bald eagle pairs riginal equation 1.5(1) Replace with or Simplif. There were about 565 bald eagle pairs in Concept Check Guided Practice GUIDED PRACTICE KEY 1. Eplain how to determine whether a scatter plot has a positive or negative correlation.. PEN ENDED Sketch scatter plots that have each tpe of correlation. a. positive b. negative c. no correlation. Compare and contrast linear interpolation and linear etrapolation. Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation Test Scores Weekl Activities Test Score Eercise (hours) Stud Time (min) TV (hours) Lesson 5-7 Statistics: Scatter Plots and Lines of Fit 01

49 Application BILGY For Eercises 6 9, use the table that shows the average bod temperature in degrees Celsius of 9 insects at a given air temperature. Air Bod Temperature ( C) Draw a scatter plot and determine what relationship eists, if an, in the data. 7. Draw a line of fit for the scatter plot. 8. Write the slope-intercept form of an equation for the line of fit. 9. Predict the bod temperature of an insect if the air temperature is 40. F. Practice and Appl Homework Help For Eercises See Eamples , Etra Practice See page 8. Determine whether each graph shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation Census Forms Returned Hurricanes Percent Year Number Year Source: U.S. Census Bureau Source: USA TDAY Electronic Ta Returns Cereal Bars Number (millions) Calories Source: IRS Year Sugar (grams) Source: Vitalit 0 Chapter 5 Analzing Linear Equations FARMING For Eercises 14 and 15, refer to the graph at the top of page 98 about apple storage. 14. Use the points (1997, 8.1) and (1999, 1.4) to write the slope-intercept form of an equation for the line of fit. 15. Predict the number of bushels of apples in storage in 00.

50 USED CARS For Eercises 16 and 17, use the scatter plot that shows the ages and prices of used cars from classified ads. 16. Use the points (, 9600) and (5, 6000) to write the slope-intercept form of an equation for the line of fit shown in the scatter plot. 17. Predict the price of a car that is 7 ears old. Price (thousands of dollars) Used Cars (5, 6000) (, 9600) Age (ears) Source: Columbus Dispatch Aerospace Engineer Aerospace engineers design, develop, and test aircraft and spacecraft. Man specialize in a particular tpe of aerospace product, such as commercial airplanes, militar fighter jets, helicopters, or spacecraft. nline Research For information about a career as an aerospace engineer, visit: careers PHYSICAL SCIENCE For Eercises Hdrocarbons 18, use the following information. Hdrocarbons like methane, ethane, Number Boiling propane, and butane are composed Name Formula of Carbon Point of onl carbon and hdrogen atoms. Atoms ( C) The table gives the number of carbon Ethane C H 6 89 atoms and the boiling points for Propane C H 8 4 several hdrocarbons. Butane C 4 H Draw a scatter plot comparing Heane C 6 H the numbers of carbon atoms to the boiling points. ctane C 8 H Draw a line of fit for the data. 0. Write the slope-intercept form of an equation for the line of fit. 1. Predict the boiling point for methane (CH 4 ), which has 1 carbon atom.. Predict the boiling point for pentane (C 5 H 1 ), which has 5 carbon atoms.. The boiling point of heptane is 98.4 C. Use the equation of the line of fit to predict the number of carbon atoms in heptane. SPACE For Eercises 4 8, use the table that shows the amount the United States government has spent on space and other technologies in selected ears. Year Spending (billions of dollars) Federal Spending on Space and ther Technologies Source: U.S. ffice of Management and Budget 4. Draw a scatter plot and determine what relationship, if an, eists in the data. 5. Draw a line of fit for the scatter plot Let represent the number of ears since Let represent the spending in billions of dollars. Write the slope-intercept form of the equation for the line of fit. 7. Predict the amount that will be spent on space and other technologies in The government projects spending of $14. billion in space and other technologies in 005. How does this compare to our prediction? Lesson 5-7 Statistics: Scatter Plots and Lines of Fit 0

51 FRESTRY For Eercises 9, use the table that shows the number of acres burned b wildfires in Florida each ear and the corresponding number of inches of spring rainfall. Year Florida s Burned Acreage and Spring Rainfall Rainfall Acres Rainfall Acres Year (inches) (thousands) (inches) (thousands) Source: Florida Division of Forestr You can use a line of fit to describe the trend in winning lmpic times. Visit webquest to continue work on our WebQuest project. 9. Draw a scatter plot with rainfall on the -ais and acres on the -ais. 0. Draw a line of fit for the data. 1. Write the slope-intercept form of an equation for the line of fit.. In 000, there was onl 8.5 inches of spring rainfall. Estimate the number of acres burned b wildfires in In 1998, there was. inches of rainfall, et 507,000 acres were burned. Where was this data graphed in the scatter plot? How did this affect the line of fit? nline Research Data Update What has happened to the number of acres burned b wildfires in Florida since 1999? Visit to learn more. 4. CRITICAL THINKING A test contains 0 true-false questions. Draw a scatter plot that shows the relationship between the number of correct answers and the number of incorrect answers. RESEARCH For Eercises 5 and 6, choose a topic to research that ou believe ma be correlated, such as arm span and height. Find eisting data or collect our own. 5. Draw a line of fit for the data. 6. Use the line to make a prediction about the data. NC Practice Standardized Test Practice 7. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How do scatter plots help identif trends in data? Include the following in our answer: a scatter plot that shows a person s height and his or her age, with a description of an trends, and an eplanation of how ou could use the scatter plot to predict a person s age given his or her height. 8. Which graph is the best eample of data that show a negative linear relationship between the variables and? A B C D 04 Chapter 5 Analzing Linear Equations

52 9. Choose the equation for the line that best fits the data in the table at the right A B C D Etending the Lesson GEGRAPHY For Eercises 40 44, use the following information. The latitude of a place on Earth is the measure of its distance from the equator. 40. MAKE A CNJECTURE What do ou think is the relationship between a cit s latitude and its Januar temperature? latitude 40 N latitude 0 N 41. RESEARCH Use the Internet or other reference to find the latitude of 15 cities in the northern hemisphere and the corresponding Januar mean temperatures. latitude 0 S 4. Make a scatter plot and draw a line of fit for the data. 4. Write an equation for the line of fit. 44. MAKE A CNJECTURE Find the latitude of our cit and use the equation to predict its mean Januar temperature. Check our prediction b using another source such as the newspaper. Maintain Your Skills Mied Review Write the slope-intercept form of an equation for the line that satisfies each condition. (Lesson 5-6) 45. parallel to the graph of 4 5 and passes through (, 5) 46. perpendicular to the graph of and passes through (0, 0) Write the point-slope form of an equation for a line that passes through each point with the given slope. (Lesson 5-5) m 1 (, ) m m (, ) (1, ) Find the - and -intercepts of the graph of each equation. (Lesson 4-5) Solve each equation. Then check our solution. (Lesson -4) r 7 r n ( 4) Lesson 5-7 Statistics: Scatter Plots and Lines of Fit 05

53 Standards.0, 4.01 Regression and Median-Fit Lines ne tpe of equation of best-fit ou can find is a linear regression equation. Linear regression is sometimes called the method of least squares. A Follow-Up of Lesson 5-7 EARNINGS The table shows the average hourl earnings of U.S. production workers for selected ears. Year Earnings $ Source: Bureau of Labor Statistics Find and graph a linear regression equation. Then predict the average hourl earnings in 010. Find a regression equation. Enter the ears in L1 and the earnings in L. KEYSTRKES: Review entering a list on page 04. Graph the regression equation. Use STAT PLT to graph the scatter plot. KEYSTRKES: Review statistical plots on page 04. Find the regression equation b selecting LinReg(a+b) on the STAT CALC menu. KEYSTRKES: STAT 4 ENTER Cop the equation to the Y= list and graph. KEYSTRKES: VARS 5 1 GRAPH The equation is in the form a b. The equation is about r is the linear correlation coefficient. The closer the absolute value of r is to 1, the better the equation models the data. Because the r value is close to 1, the model fits the data well. [1950, 00] scl: 10 b [0, 0] scl: 5 The residual is the difference between actual and predicted data. The predicted earnings in 1970 using this model were $.94. (To calculate, press nd [CALC] ENTER.) So the residual for 1970 was $.94 $. or $0.71. Predict using the regression equation. Find when 010 using value on the CALC menu. KEYSTRKES: nd [CALC] ENTER According to the regression equation, the average hourl earnings in 010 will be about $ The graph and the coordinates of the point are shown Chapter 5 Analzing Linear Equations

54 A second tpe of best-fit line that can be found using a graphing calculator is a median-fit line. The equation of a median-fit line is calculated using the medians of the coordinates of the data points. Find and graph a median-fit equation for the data on hourl earnings. Then predict the average hourl earnings in 010. Compare this prediction to the one made using the regression equation. Find a median-fit equation. The data are alread in Lists 1 and. Find the median-fit equation b using Med-Med on the STAT CALC menu. KEYSTRKES: STAT ENTER Graph the median-fit equation. Cop the equation to the Y= list and graph. KEYSTRKES: CLEAR VARS 5 1 GRAPH The median-fit equation is [1950, 010] scl: 10 b [0, 0] scl: 5 Predict using the median-fit equation. KEYSTRKES: nd [CALC] ENTER According to the median-fit equation, the average hourl earnings in 010 will be about $15.8. This is slightl less than the predicted value found using the regression equation. Eercises Refer to the data on bald eagles in Eample on pages 00 and Find regression and median-fit equations for the data.. What is the correlation coefficient of the regression equation? What does it tell ou about the data?. Use the regression and median-fit equations to predict the number of bald eagle pairs in Compare these to the number found in Eample on page 01. For Eercises 4 and 5, use the table that shows the number of votes cast for the Democratic presidential candidate in selected North Carolina counties in the 1996 and 000 elections. 4. Find regression and median-fit equations for the data. 5. In 1996, New Hanover Count had,89 votes for the Democratic candidate. Use the regression and median-fit equations to estimate the number of votes for the Democratic candidate in that count in 000. How do the predictions compare to the actual number of 9,9? ,447 16,84 19,458 19,81 8,674 0,91 1,658 8,545,79 8,66 46,54 5,457 49,186 5,907 69,08 80,787 10,49 16,911 10,574 1,466 Source: NC State Board of Elections Graphing Calculator Investigation Regression and Median-Fit Lines 07

55 Vocabular and Concept Check best-fit line (p. 00) constant of variation (p. 64) direct variation (p. 64) famil of graphs (p. 65) linear etrapolation (p. 8) linear interpolation (p. 01) line of fit (p. 00) negative correlation (p. 98) parallel lines (p. 9) parent graph (p. 65) perpendicular lines (p. 9) point-slope form (p. 86) positive correlation (p. 98) rate of change (p. 58) scatter plot (p. 98) slope (p. 56) slope-intercept form (p. 7) Eercises Choose the correct term to complete each sentence. 1. An equation of the form k, where k 0, describes a ( direct variation, linear etrapolation).. The ratio of ( rise, run), or vertical change, to the (rise, run ), or horizontal change, as ou move from one point on a line to another, is the slope of a nonvertical line.. The lines with equations 7 and 6 are ( parallel, perpendicular) lines. 4. The equation ( 1) is written in ( point-slope, slope-intercept) form. 5. The equation 1 6 is written in ( slope-intercept, standard) form. 6. The (-intercept, -intercept ) of the equation 4 is See pages Slope Concept Summar The slope of a nonvertical line is the ratio of the rise to the run. m ( 1, 1 ) (, ) Eample Determine the slope of the line that passes through (0, 4) and (, ). Let (0, 4) ( 1, 1 ) and (, ) (, ). m 1 Slope formula 1 m (4) 0 1 0,, 1 4, (0, 4) (, ) m 6 or Simplif. Eercises Find the slope of the line that passes through each pair of points. See Eamples 1 4 on page (1, ), (, 6) 8. (0, 5), (6, ) 9. (6, 4), (6, ) 10. (8, ), (, ) 11. (.9, 4.7), (0.5, 1.1) 1. 1, 1, 1, 08 Chapter 5 Analzing Linear Equations