TEKS 2.2 a.1, a.4, a.5 Find Slope and Rate of Change Before You graphed linear functions. Now You will find slopes of lines and rates of change. Wh? So ou can model growth rates, as in E. 46. Ke Vocabular slope parallel perpendicular rate of change reciprocal, p. 4 KEY CONCEPT Slope of a Line For Your Notebook Words Algebra Graph The slope m of a nonvertical line is the ratio of vertical change (the rise) to horizontal change (the run). m 5 2 2 1 } 2 2 1 5 rise } run slope run 2 2 1 rise 2 2 1 (1, 1) ( 2, 2 ) E XAMPLE 1 Find slope in real life SKATEBOARDING A skateboard ramp has a rise of 15 inches and a run of 54 inches. What is its slope? slope 5 } rise 5 } 15 5 } 5 run 54 18 run 5 54 in. rise 5 15 in. c The slope of the ramp is 5 } 18. E XAMPLE 2 TAKS PRACTICE: Multiple Choice What is the slope of the line passing through the points (22, 1) and (3, 5)? A 2 5 } 4 B 2 4 } 5 C 4 } 5 D 5 } 4 AVOID ERRORS When calculating slope, be sure to subtract the - and -coordinates in a consistent order. Let ( 1, 1 ) 5 (22, 1) and ( 2, 2 ) 5 (3, 5). m 5 2 2 1 } 2 2 1 5 5 2 1 } 3 2 (22) 5 4 } 5 c The correct answer is C. A B C D 4 5 4 (22, 1) 1 (3, 5) 82 Chapter 2 Linear Equations and Functions
GUIDED PRACTICE for Eamples 1 and 2 1. WHAT IF? In Eample 1, suppose that the rise of the ramp is changed to 12 inches without changing the run. What is the slope of the ramp? 2. What is the slope of the line passing through the points (24, 9) and (28, 3)? A 2 2 } 3 B 2 1 } 2 C 2 } 3 D 3 } 2 Find the slope of the line passing through the given points. 3. (0, 3), (4, 8) 4. (25, 1), (5, 24) 5. (23, 22), (6, 1) 6. (7, 3), (21, 7) KEY CONCEPT For Your Notebook Classification of Lines b Slope The slope of a line indicates whether the line rises from left to right, falls from left to right, is horizontal, or is vertical. READING A vertical line has undefined slope because for an two points, the slope formula s denominator becomes 0, and division b 0 is undefined. Positive slope Rises from left to right Negative slope Falls from left to right Zero slope Horizontal Undefined slope Vertical E XAMPLE 3 Classif lines using slope Without graphing, tell whether the line through the given points rises, falls, is horizontal, or is vertical. a. (25, 1), (3, 1) b. (26, 0), (2, 24) c. (21, 3), (5, 8) d. (4, 6), (4, 21) a. m 5} 1 2 1 5 } 0 5 0 Because m 5 0, the line is horizontal. 3 2 (25) 8 b. m 5 24 2 0 } 2 2 (26) 5 24 } 8 5 2 1 } 2 Because m < 0, the line falls. c. m 5 8 2 3 } 5 2 (21) 5 5 } 6 Because m > 0, the line rises. d. m 5 21 2 6 } 4 2 4 5 27 } 0 Because m is undefined, the line is vertical. GUIDED PRACTICE for Eample 3 Without graphing, tell whether the line through the given points rises, falls, is horizontal, or is vertical. 7. (24, 3), (2, 26) 8. (7, 1), (7, 21) 9. (3, 22), (5, 22) 10. (5, 6), (1, 24) 2.2 Find Slope and Rate of Change 83
PARALLEL AND PERPENDICULAR LINES Recall that 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. KEY CONCEPT For Your Notebook Slopes of Parallel and Perpendicular Lines Consider two different nonvertical lines l 1 and l 2 with slopes and. Parallel Lines The lines are parallel if and onl if the have the same slope. l 1 l 2 5 Perpendicular Lines The lines are perpendicular if and onl if their slopes are negative reciprocals of each other. 52 1 } m2, or 521 l 1 l 2 E XAMPLE 4 Classif parallel and perpendicular lines Tell whether the lines are parallel, perpendicular, or neither. a. Line 1: through (22, 2) and (0, 21) b. Line 1: through (1, 2) and (4, 23) Line 2: through (24, 21) and (2, 3) Line 2: through (24, 3) and (21, 22) a. Find the slopes of the two lines. 5 21 2 2 } 0 2 (22) 5 23 } 2 52 3 } 2 5 3 2 (21) } 2 2 (24) 5 4 } 6 5 2 } 3 c Because 52 3 } 2 p 2 } 3 521, and are negative reciprocals of each other. So, the lines are perpendicular. 2 (22, 2) 21 (24, 21) (2, 3) Line 2 (0, 21) Line 1 b. Find the slopes of the two lines. 5} 23 2 2 5 } 25 52} 5 4 2 1 3 3 (24, 3) Line 2 (1, 2) 5 22 2 3 } 21 2 (24) 5 25 } 3 52 5 } 3 1 1 Line 1 c Because 5 (and the lines are different), ou can conclude that the lines are parallel. (21, 22) (4, 23) 84 Chapter 2 Linear Equations and Functions
GUIDED PRACTICE for Eample 4 Tell whether the lines are parallel, perpendicular, or neither. 11. Line 1: through (22, 8) and (2, 24) 12. Line 1: through (24, 22) and (1, 7) Line 2: through (25, 1) and (22, 2) Line 2: through (21, 24) and (3, 5) REVIEW RATES Remember that a rate is a ratio of two quantities that have different units. RATE OF CHANGE Slope can be used to represent an average rate of change, or how much one quantit changes, on average, relative to the change in another quantit. A slope that is a real-life rate of change involves units of measure such as miles per hour or degrees per da. E XAMPLE 5 TAKS REASONING: Multi-Step Problem FORESTRY Use the diagram, which illustrates the growth of a giant sequoia, to find the average rate of change in the diameter of the sequoia over time. Then predict the sequoia s diameter in 2065. STEP 1 Find the average rate of change. Average rate of change 5 5 Change in diameter } Change in time 141 in. 2 137 in. } 2005 2 1965 5 4 in. } 40 ears 5 0.1 inch per ear STEP 2 Predict the diameter of the sequoia in 2065. Find the number of ears fro005 to 2065. Multipl this number b the average rate of change to find the total increase in diameter during the period 2005 2065. Number of ears 5 2065 2 2005 5 60 Increase in diameter 5 (60 ears)(0.1 inch/ear) 5 6 inches c In 2065, the diameter of the sequoia will be about 141 1 6 5 147 inches. GUIDED PRACTICE for Eample 5 13. WHAT IF? In Eample 5, suppose that the diameter of the sequoia is 248 inches in 1965 and 251 inches in 2005. Find the average rate of change in the diameter, and use it to predict the diameter in 2105. 2.2 Find Slope and Rate of Change 85
2.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 9, 19, and 45 5 TAKS PRACTICE AND REASONING Es. 17, 35, 36, 44, 45, 48, 50, and 51 1. VOCABULARY Cop and complete: The? of a nonvertical line is the ratio of vertical change to horizontal change. 2. WRITING How can ou use slope to decide whether two nonvertical lines are parallel? whether two nonvertical lines are perpendicular? EXAMPLES 2 and 3 on pp. 82 83 for Es. 3 17 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. 3. (2, 24), (4, 21) 4. (8, 9), (24, 3) 5. (5, 1), (8, 24) 6. (23, 22), (3, 22) 7. (21, 4), (1, 24) 8. (26, 5), (26, 25) 9. (25, 24), (21, 3) 10. (23, 6), (27, 3) 11. (4, 4), (4, 9) 12. (5, 5), (7, 3) 13. (0, 23), (4, 23) 14. (1, 21), (21, 24) at classzone.com ERROR ANALYSIS Describe and correct the error in finding the slope of the line passing through the given points. 15. 16. (21, 4), (5, 1) (24, 23), (2, 21) 21 2 (23) m 5} 52} 1 24 2 2 3 m 5 5 2 (21) } 1 2 4 522 17. TAKS REASONING What is true about the line through (2, 24) and (5, 1)? A It rises from left to right. B It falls from left to right. C It is horizontal. D It is vertical. EXAMPLE 4 on p. 84 for Es. 18 23 EXAMPLE 5 on p. 85 for Es. 24 27 CLASSIFYING LINES Tell whether the lines are parallel, perpendicular, or neither. 18. Line 1: through (3, 21) and (6, 24) 19. Line 1: through (1, 5) and (3, 22) Line 2: through (24, 5) and (22, 7) Line 2: through (23, 2) and (4, 0) 20. Line 1: through (21, 4) and (2, 5) 21. Line 1: through (5, 8) and (7, 2) Line 2: through (26, 2) and (0, 4) Line 2: through (27, 22) and (24, 21) 22. Line 1: through (23, 2) and (5, 0) 23. Line 1: through (1, 24) and (4, 22) Line 2: through (21, 24) and (3, 23) Line 2: through (8, 1) and (14, 5) AVERAGE RATE OF CHANGE Find the average rate of change in relative to for the ordered pairs. Include units of measure in our answer. 24. (2, 12), (5, 30) is measured in hours and is measured in dollars 25. (0, 11), (3, 50) is measured in gallons and is measured in miles 26. (3, 10), (5, 18) is measured in seconds and is measured in feet 27. (1, 8), (7, 20) is measured in seconds and is measured in meters 86 Chapter 2 Linear Equations and Functions
28. REASONING The Ke Concept bo on page 84 states that lines l 1 and l 2 must be nonvertical. Eplain wh this condition is necessar. FINDING SLOPE Find the slope of the line passing through the given points. 29. 1 21, 3 } 2 2, 1 0, 7 } 2 2 30. 1 23 } 4, 222, 1 5 } 4, 232 31. 1 21 } 2, 5 } 2 2, 1 5 } 2, 32 32. (24.2, 0.1), (23.2, 0.1) 33. (20.3, 2.2), (1.7, 20.8) 34. (3.5, 22), (4.5, 0.5) 35. TAKS REASONING Does it make a difference which two points on a line ou choose when finding the slope? Does it make a difference which point is ( 1, 1 ) and which point is ( 2, 2 ) in the formula for slope? Support our answers using three different pairs of points on the line shown. P Π2 R S 4 T 36. TAKS REASONING Find two additional points on the line that passes through (0, 3) and has a slope of 24. CHALLENGE Find the value of k so that the line through the given points has the given slope. Check our solution. 37. (2, 23) and (k, 7); m 522 38. (0, k) and (3, 4); m 5 1 39. (24, 2k) and (k, 25); m 521 40. (22, k) and (2k, 2); m 520.25 PROBLEM SOLVING EXAMPLE 1 on p. 82 for Es. 41 44 41. ESCALATORS An escalator in an airport rises 28 feet over a horizontal distance of 48 feet. What is the slope of the escalator? 42. INCLINE RAILWAY The Duquesne Incline, a cable car railwa, rises 400 feet over a horizontal distance of 685 feet on its ascent to an overlook of Pittsburgh, Pennslvania. What is the slope of the incline? 43. ROAD GRADE A road s grade is its slope epressed as a percent. A road rises 195 feet over a horizontal distance of 3000 feet. What is the grade of the road? 44. TAKS REASONING The diagram shows a three-section ramp to a bridge. Each section has the same slope. Compare this slope with the slope that a single-section ramp would have if it rose directl to the bridge from the same starting point. Eplain the benefits of a three-section ramp in this situation. EXAMPLE 5 on p. 85 for Es. 45 46 45. TAKS RE ASONING Over a 30 da period, the amount of propane in a tank that stores propane for heating a home decreases from 400 gallons to 214 gallons. What is the average rate of change in the amount of propane? A 26.2 gallons per da B 26 gallons per da C 20.16 gallon per da D 6 gallons per da 2.2 Find Slope and Rate of Change 87
46. BIOLOGY A red sea urchin grows its entire life, which can last 200 ears. The diagram gives information about the growth in the diameter d of one red sea urchin. What is the average growth rate of this urchin over the given period? Growth of Red Sea Urchin Age 30 Age 110 d = 11.9 cm d = 15.5 cm 47. MULTI-STEP PROBLEM A building code requires the minimum slope, or pitch, of an asphalt-shingle roof to be a rise of 3 feet for each 12 feet of run. The asphalt-shingle roof of an apartment building has the dimensions shown. a. Calculate What is the slope of the roof? b. Interpret Does the roof satisf the building code? 15 ft c. Reasoning If ou answered no to part (b), b how much must the rise be increased to satisf the code? If ou answered es, b how much does the rise eceed the code minimum? 80 ft 48. TAKS REASONING Plans for a new water slide in an amusement park call for the slide to descend from a platform 80 feet tall. The slide will drop 1 foot for ever 3 feet of horizontal distance. a. What horizontal distance do ou cover when descending the slide? b. Use the Pthagorean theorem to find the length of the slide. c. Engineers decide to shorten the slide horizontall b 5 feet to allow for a wider walkwa at the slide s base. The plans for the platform remain unchanged. How will this affect the slope of the slide? Eplain. 49. CHALLENGE A car travels 36 miles per gallon of gasoline in highwa driving and 24 miles per gallon in cit driving. If ou drive the car equal distances on the highwa and in the cit, how man miles per gallon can ou epect to average? (Hint: The average fuel efficienc for all the driving is the total distance traveled divided b the total amount of gasoline used.) MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 1.5; TAKS Workbook REVIEW Lesson 1.4; TAKS Workbook 50. TAKS PRACTICE A cit is building a rectangular plaground in a communit park. The cit has 560 feet of fencing to enclose the plaground. The length of the plaground should be 40 feet longer than the width. What is the length of the plaground if all of the fencing is used? TAKS Obj. 10 A 120 ft C 200 ft B 160 ft D 300 ft 51. TAKS PRACTICE A computer technician charges $185 for parts needed to fi a computer and $45 for each hour that he works on the computer. Which equation best represents the relationship between the number of hours, h, the technician works on the computer and the total charges, c? TAKS Obj. 1 F c 5 45 2 185h G c 5 45 1 185h H c 5 185 2 45h J c 5 185 1 45h 88 EXTRA PRACTICE for Lesson 2.2, p. 1011 ONLINE QUIZ at classzone.com