Words Algebra Graph. m 5 y 2 2 y 1. slope. Find slope in real life

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1 . 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. 6. Ke Vocabular slope parallel perpendicular rate of change reciprocal, p. 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 5 rise run slope run rise (, ) (, ) E XAMPLE Find slope in real life SKATEBOARDING A skateboard ramp has a rise of 5 inches and a run of 5 inches. What is its slope? slope 5 rise run c The slope of the ramp is 5 8. run 5 5 in. rise 5 5 in. E XAMPLE Standardized Test Practice What is the slope of the line passing through the points (, ) and (, )? A B C D AVOID ERRORS When calculating slope, be sure to subtract the - and -coordinates in a consistent order. Let (, ) 5 (, ) and (, ) 5 (, ). m 5 5 () 5 c The correct answer is A. A B C D (, ) (, ) 8 Chapter Linear Equations and Functions

2 GUIDED PRACTICE for Eamples and. WHAT IF? In Eample, suppose that the rise of the ramp is changed to inches without changing the run. What is the slope of the ramp?. What is the slope of the line passing through the points (, 9) and (8, )? A B C Find the slope of the line passing through the given points.. (0, ), (, 8). (5, ), (5, ) 5. (, ), (6, ) 6. (7, ), (, 7) D 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 Classif lines using slope Without graphing, tell whether the line through the given points rises, falls, is horizontal, or is vertical. a. (5, ), (, ) b. (6, 0), (, ) c. (, ), (5, 8) d. (, 6), (, ) a. m 5 (5) Because m 5 0, the line is horizontal. 8 b. m 5 0 (6) c. m () d. m Because m < 0, the line falls. Because m > 0, the line rises. Because m is undefined, the line is vertical. GUIDED PRACTICE for Eample Without graphing, tell whether the line through the given points rises, falls, is horizontal, or is vertical. 7. (, ), (, 6) 8. (7, ), (7, ) 9. (, ), (5, ) 0. (5, 6), (, ). Find Slope and Rate of Change 8

3 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 and l with slopes m and m. Parallel Lines The lines are parallel if and onl if the have the same slope. l l m 5 m Perpendicular Lines The lines are perpendicular if and onl if their slopes are negative reciprocals of each other. m 5 m, or m m 5 l l E XAMPLE Classif parallel and perpendicular lines Tell whether the lines are parallel, perpendicular, or neither. a. Line : through (, ) and (0, ) b. Line : through (, ) and (, ) Line : through (, ) and (, ) Line : through (, ) and (, ) a. Find the slopes of the two lines. m 5 0 () 5 5 m 5 () () c Because m m 5 p 5, m and m are negative reciprocals of each other. So, the lines are perpendicular. (, ) (, ) (, ) Line (0, ) Line b. Find the slopes of the two lines. m m 5 () (, ) Line (, ) Line c Because m 5 m (and the lines are different), ou can conclude that the lines are parallel. (, ) (, ) 8 Chapter Linear Equations and Functions

4 GUIDED PRACTICE for Eample Tell whether the lines are parallel, perpendicular, or neither.. Line : through (, 8) and (, ). Line : through (, ) and (, 7) Line : through (5, ) and (, ) Line : through (, ) and (, 5) 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. REVIEW RATES Remember that a rate is a ratio of two quantities that have different units. EXAMPLE 5 Solve a 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 065. IN IN STEP Find the average rate of change. Change in diameter Change in time Average rate of change 5 in. 7 in in. 5 0 ears 5 0. inch per ear STEP Predict the diameter of the sequoia in 065. Find the number of ears from 005 to 065. Multipl this number b the average rate of change to find the total increase in diameter during the period Number of ears Increase in diameter 5 (60 ears)(0. inch/ear) 5 6 inches c In 065, the diameter of the sequoia will be about inches. GUIDED PRACTICE for Eample 5. WHAT IF? In Eample 5, suppose that the diameter of the sequoia is 8 inches in 965 and 5 inches in 005. Find the average rate of change in the diameter, and use it to predict the diameter in 05.. Find Slope and Rate of Change npe-00.indd /0/05 9:0: AM

5 . EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 9, 9, and 5 5 STANDARDIZED TEST PRACTICE Es., 7, 5, 6,, 5, and 8. VOCABULARY Cop and complete: The? of a nonvertical line is the ratio of vertical change to horizontal change.. WRITING How can ou use slope to decide whether two nonvertical lines are parallel? whether two nonvertical lines are perpendicular? EXAMPLES and on pp. 8 8 for Es. 7 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.. (, ), (, ). (8, 9), (, ) 5. (5, ), (8, ) 6. (, ), (, ) 7. (, ), (, ) 8. (6, 5), (6, 5) 9. (5, ), (, ) 0. (, 6), (7, ). (, ), (, 9). (5, 5), (7, ). (0, ), (, ). (, ), (, ) at classzone.com ERROR ANALYSIS Describe and correct the error in finding the slope of the line passing through the given points. 5. (, ), (, ) 6. (, ), (5, ) m 5 5 () 5 () m MULTIPLE CHOICE What is true about the line through (, ) and (5, )? A It rises from left to right. C It is horizontal. B It falls from left to right. D It is vertical. EXAMPLE on p. 8 for Es. 8 EXAMPLE 5 on p. 85 for Es. 7 CLASSIFYING LINES Tell whether the lines are parallel, perpendicular, or neither. 8. Line : through (, ) and (6, ) 9. Line : through (, 5) and (, ) Line : through (, 5) and (, 7) Line : through (, ) and (, 0) 0. Line : through (, ) and (, 5). Line : through (5, 8) and (7, ) Line : through (6, ) and (0, ) Line : through (7, ) and (, ). Line : through (, ) and (5, 0). Line : through (, ) and (, ) Line : through (, ) and (, ) Line : through (8, ) and (, 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.. (, ), (5, 0) is measured in hours and is measured in dollars 5. (0, ), (, 50) is measured in gallons and is measured in miles 6. (, 0), (5, 8) is measured in seconds and is measured in feet 7. (, 8), (7, 0) is measured in seconds and is measured in meters 86 Chapter Linear Equations and Functions

6 8. REASONING The Ke Concept bo on page 8 states that lines l and l must be nonvertical. Eplain wh this condition is necessar. FINDING SLOPE Find the slope of the line passing through the given points. 9.,, 0, 7 0.,, 5,., 5, 5,. (., 0.), (., 0.). (0.,.), (.7, 0.8). (.5, ), (.5, 0.5) 5. SHORT RESPONSE 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 (, ) and which point is (, ) in the formula for slope? Support our answers using three different pairs of points on the line shown. P ΠR S T 6. OPEN-ENDED MATH Find two additional points on the line that passes through (0, ) and has a slope of. CHALLENGE Find the value of k so that the line through the given points has the given slope. Check our solution. 7. (, ) and (k, 7); m 5 8. (0, k) and (, ); m 5 9. (, k) and (k, 5); m 5 0. (, k) and (k, ); m PROBLEM SOLVING EXAMPLE on p. 8 for Es.. ESCALATORS An escalator in an airport rises 8 feet over a horizontal distance of 8 feet. What is the slope of the escalator?. INCLINE RAILWAY The Duquesne Incline, a cable car railwa, rises 00 feet over a horizontal distance of 685 feet on its ascent to an overlook of Pittsburgh, Pennslvania. What is the slope of the incline?. ROAD GRADE A road s grade is its slope epressed as a percent. A road rises 95 feet over a horizontal distance of 000 feet. What is the grade of the road?. SHORT RESPONSE 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 MULTIPLE CHOICE Over a 0 da period, the amount of propane in a tank that stores propane for heating a home decreases from 00 gallons to gallons. What is the average rate of change in the amount of propane? A 6. gallons per da B 6 gallons per da C 0.6 gallon per da D 6 gallons per da. Find Slope and Rate of Change 87

7 6. BIOLOGY A red sea urchin grows its entire life, which can last 00 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 0 Age 0 d =.9 cm d = 5.5 cm 7. MULTI-STEP PROBLEM A building code requires the minimum slope, or pitch, of an asphalt-shingle roof to be a rise of feet for each 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? 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? 8. EXTENDED RESPONSE 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 foot for ever 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. 9. CHALLENGE A car travels 6 miles per gallon of gasoline in highwa driving and 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.) 80 ft 5 ft MIXED REVIEW Identif the propert that the statement illustrates. (p. ) 50. 5(8 ) 5 5(8) 5() 5. (7 9) 5 7 (9 ) 5. () p p p p 5 PREVIEW Prepare for Lesson. in Es Solve the equation for. (p. 6) Solve the equation or inequalit. (p. 5) < > Chapter EXTRA Linear Equations PRACTICE and Functions for Lesson., p. 0 ONLINE QUIZ at classzone.com