5.3 Interpreting Rate of Change and Slope

Transcription

1 Name Class Date 5.3 Interpreting Rate of Change and Slope Essential question: How can ou relate rate of change and slope in linear relationships? Resource Locker Eplore Determining Rates of Change For a function defined in terms of and, the rate of change over a part of the domain of the function is a ratio that compares the change in to the change in in that part of the domain. change in rate of change = _ change in The table shows the ear and the cost of sending 1-ounce letter in cents. Years after 2000 () Cost (cents) Houghton Mifflin Harcourt Publishing Compan Image Credits: David R. Frazier Photolibrar, Inc./Alam change in postage Find the rate of change,, for each time period using the table. change in ear A From 2003 to 200: - = cent(s) per ear B From 200 to 2006: = = cent(s) per ear C From 2006 to 200: = - 6 = cent(s) per ear - D From 200 to 2013: = 13 - = cent(s) per ear Module Lesson 3

2 E Plot the points represented in the table. Connect the points with line segments to make a statistical line graph. Postage Costs Find the rate of change for each time period using the graph. F Cost (cents) Years after 2000 Label the vertical increase (rise) and the horizontal increase (run) between points (, 37) and (6, 39). Then find the rate of change, rise run. G Label the vertical increase (rise) and the horizontal increase (run) between points (6, 39) and (, 2). Then find the rate of change, rise run. _ rise run = _ = cent(s) per ear _ rise run = = cent(s) per ear H Label the vertical increase (rise) and the horizontal increase (run) between points (, 2) and (13, 6). Then find the rate of change, rise run. _ rise run = = cent(s) per ear Reflect change in postage 1. Discussion Between which two ears is the rate of change the greatest? change in ears 2. Discussion Compare the line segment between 2006 and 200 with the line segment between 200 and Which is steeper? Which represents a greater rate of change? 3. Discuss How do ou think the steepness of the line segment between two points is related to the rate of change it represents? Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

3 Eplain 1 Determining the Slope of a Line The rate of change for a linear function can be calculated using the rise and run of the graph of the function. The rise is the difference in the -values of two points on a line. The run is the difference in the -values of two points on a line. The slope of a line is the ratio of rise to run for an two points on the line. Slope = _ rise run = difference in -values difference in -values Eample 1 A Determine the slope of each line. Use (3, ) as the first point. Subtract -values to find the change in, or rise. Then subtract -values to find the change in, or run. slope = _ = 3_ 1 = 3. Slope of the line is 3. (3, ) (2, 1) 0 B Use (-2, ) as the first point. Subtract -values to find the change in, or rise. Then subtract -values to find the change in, or run. slope = - - The slope of the line is. = _ =. (-2, 3) (1, 0) 0 Reflect. Find the rise of a horizontal line. What is the slope of a horizontal line? Houghton Mifflin Harcourt Publishing Compan 5. Find the run of a vertical line. What is the slope of a vertical line? 6. Discussion If ou have a graph of a line, how can ou determine whether the slope is positive, negative, zero, or undefined without using points on the line? Module Lesson 3

4 Your Turn Find the slope of each line. 7.. (5, 3) (-3, 2) 0 (0, -3) (-2, -3) 0 Eplain 2 Determining Slope Using the Slope Formula The slope formula for the slope of a line is the ratio of the difference in -values to the difference in -values between an two points on the line. Slope Formula If ( 1, 2 ) and ( 1, 2 ) are an two points on a line, the slope of the line is m = 2-1 _ 2-1. Eample 2 A Find the slope of each line passing through the given points using the slope formula. Describe the slope as positive, negative, zero, or undefined. The graph shows the linear relationship. 2-1 = 3 - (-1) = = 2-1 = 2 - (-2) = = m = _ = _ 1 = 1 The slope is positive. The line rises from left to right. B = - = 2-1 = - = Let (, ) be ( 1, 1 ) and (, ) be ( 2, 2 ). m = 2-1 _ 2-1 (-2, -1) = _ (2, 3) Houghton Mifflin Harcourt Publishing Compan The slope is and the line is. Module 5 22 Lesson 3

5 Your Turn Find the slope of each line passing through the given points using the slope formula. Describe the slope as positive, negative, zero, or undefined. 9. The graph shows the linear relationship. (-1, 9) 0 (2, -5) Eplain 3 Interpreting Slope Given a real-world situation, ou can find the slope and then interpret the slope in terms of the contet of the situation. Eample 3 Find and interpret the slope for each real-world situation. Houghton Mifflin Harcourt Publishing Compan A B The graph shows the relationship between a person s age and his or her estimated maimum heart rate. Use the two points that are labeled on the graph. slope = _ rise run = _ = _ = -1 Interpret the slope. The slope being -1 means that for ever ear a person s age increases, his or her maimum heart rate decreases b 1 beat per minute. The height of a plant in centimeters after das is a linear relationship. The points (30, 15) and (0, 25) are on the line. Use the two points that are given. slope = - 15 _ rise run = - = _ = Maimum heart rate (beats/min) Estimated Maimum Heart Rate (20, 10) (50, 150) Age (r) Interpret the slope. The slope being means. Module Lesson 3

6 Your Turn Find and interpret the slope. 11. The graph shows the relationship between the temperature epressed in F and the temperature epressed in C. Temperature ( C) 25 (77, 25) (50, 10) Temperature ( F) 12. The number of cubic feet of water in a reservoir hours after the water starts flowing into the reservoir is a linear function. The points (0, 3000) and (60, 000) are on the line of the function. v Elaborate 13. How can ou relate the rate of change and slope in the linear relationships? 1. How is the slope formula related to the definition of slope? 15. How can ou interpret slope in a real-world situation? Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

7 Evaluate: Homework and Practice Determine the slope of each line. 1. (5, 7) 2. (-6, 5) Online Homework Hints and Help Etra Practice 0 (2, -2) (, -3) 0 3. (5, 9). 0 (5, -2) 0 (-3, -7) (6, -7) Houghton Mifflin Harcourt Publishing Compan 5. (, 7) (2, ) 6. (-7, -9) (-1, 5) 0 Module Lesson 3

8 Find the slope of each line passing through the given points using the slope formula. Describe the slope as positive, negative, zero, or undefined. 7. (5, 3) and (10, ). (-5, 1) and (-1, 2) 9. (-5, 6) and (, 6) 10. (, -17) and (, -3) 11. (12, -7) and (2, -2) 12. (-3, -10) and (-1, -1) Find and interpret the slope for each real-world situation Mone earned ($) Jars of peanut butter (, 170) (, 110) Time worked (hr) (960, 2) Peanuts (3360, 7) Houghton Mifflin Harcourt Publishing Compan Image Credits: Jiri Hera/ Shutterstock Module 5 22 Lesson 3

9 15. Cost ($) (3500, 60) (1000, 310) Pages printed 16. Balance ($) (3, 60) (, 55) Time (months) 17. a. The table shows the distance that a group of hikers has traveled from the start of the trail. Time (hr) Distance (km) Use the table to plot the points on the graph and join the points using line segments. Distance (km) Time (hr) Houghton Mifflin Harcourt Publishing Compan b. Find the slope for each of the three line segments. c. Which line segment has the greatest slope? Does this line segment appear to be the steepest on the graph? Module Lesson 3

10 1. Determine whether each set of points is on a line that has a positive slope, negative slope, zero slope, or undefined slope. Select the correct answer for each part. a. (5, 0) and (, ) positive negative zero undefined b. (-6, 1) and (-6, 9) positive negative zero undefined c. (2, 6) and (11,-3) positive negative zero undefined d. (3, ) and (-2, 12) positive negative zero undefined e. (-3, 5) and (7, 5) positive negative zero undefined 19. What is the slope of the segment shown for a staircase with 10-inch treads and 7.75-inch risers? As ou walk up (or down) the stairs, our vertical distance from the floor is a linear function of our horizontal distance from the point on the floor where ou started. Is the function discrete or continuous? Eplain. Riser Tread 20. The Mount Washington Cog Railwa in New Hampshire is one of the steepest cog railwas in the world. A section of the railwa has a slope of approimatel In this section, a vertical change of 1 unit corresponds to a horizontal change of what length? Round our answer to the nearest hundredth. 21. a. Biolog The table shows how the number of cricket chirps per minute changes with the air temperature. Temperature ( F) Chirps per minute Find the rates of change. Houghton Mifflin Harcourt Publishing Compan Image Credits: alesvirid/ Shutterstock b. Is the graph of the data a line? If so, what is the slope? If not, eplain wh not. Module Lesson 3

11 H.O.T. Focus on Higher Order Thinking 22. Eplain the Error A student is asked to find the slope of a line containing the points (, 3) and (-2, 15) and finds the slope as shown. Eplain the error. slope = _ rise run = _ - (-2) 3-15 = 6_ -12 = - 1_ Critical Thinking In this lesson, ou learned that the slope of a line is constant. Does this mean that all lines with the same slope are the same line? Eplain. 2. a. Represent Real-World Problems A ladder is leaned against a building. The bottom of the ladder is 11 feet from the building. The top of the ladder is 19 feet above the ground. What is the slope of the ladder? b. What does the slope of the ladder mean in the real world? c. If the ladder were set closer to the building, would it be harder or easier to climb? Eplain in terms of the slope of the ladder. 25. a. The table shows the cost, in dollars, charged b an electric compan for various amounts of energ in kilowatt-hours. Graph the data and show the rates of change. Energ (kwh) Cost ($) Cost ($) Houghton Mifflin Harcourt Publishing Compan b. Compares the rates of change for each interval. Are the all the same? Eplain. c. What do the rates of change represent? d. Describe in words the electric compan s billing plan Energ (kwh) Module Lesson 3

12 Lesson Performance Task A cit has three Internet service providers (ISP), each of which charges a usage fee when a subscriber goes over 100 megabtes (MB) per billing ccle. The table below relates the amount of data a subscriber uses with the cost for each ISP. ISP 100 MB 200 MB 00 MB A $5 $7 $9 B $2 $57 $7 C $60 $72 $96 Use the table to find the rate of change for each interval of each ISP, and use the rates of change to determine whether the usage fee is constant for each ISP. Interpret the meaning of the rates of change for each ISP. Then determine and eplain which ISP would be the least epensive and which ISP would be the most epensive for a subscriber that uses a high amount of data. Houghton Mifflin Harcourt Publishing Compan Module Lesson 3