5.3 Interpreting Rate of Change and Slope - NOTES
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1 Name Class Date 5.3 Interpreting Rate of Change and Slope NOTES Essential question: How can ou relate rate of change and slope in linear relationships? Eplore A1.3.B calculate the rate of change of a linear function represented tabularl, graphicall in contet of mathematical and realworld problems. Also A1.3.A 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 over that part of the domain. change in rate of change = _ change in The table shows the ear and the cost of sending 1ounce letter in cents. Years after 2000 () Cost (cents) change in postage Find the rate of change,, for each time period using the table. change in ear From 2003 to 200: = cent(s) per ear 3 From 200 to 2006: = = cent(s) per ear 6 From 2006 to 200: = = cent(s) per ear 6 From 200 to 2013: = = cent(s) per ear 13
2 E Plot the points represented in the table. Connect the points with line segments to make a statistical line graph. Postage Costs Cost (cents) Years after 2000 Find the rate of change for each time period using the graph. F 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?
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 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 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? 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?
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 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 (2, 3) 2 1 = 3 (1) = = 2 1 = 2 (2) = = (2, 1) m = _ = _ 1 = 1 The slope is positive. The line rises from left to right = = 2 1 = = Let (, ) be ( 1, 1 ) and (, ) be ( 2, 2 ). m = 2 1 _ 2 1 = _ The slope is and the line is.
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 realworld 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 realworld situation. 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. Maimum heart rate (beats/min) Estimated Maimum Heart Rate (20, 10) (50, 150) Age (r) slope = 15 _ rise run = = _ = Interpret the slope. The slope being means.
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) Temperature ( F) (50, 10) (77, 25) 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 realworld situation?
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