1. 1

2. Section 1.3 Linear Functions 2

3. Constant Rate of Change In the previous section, we introduced the average rate of change of a function on an interval. For many functions, the average rate of change is different on different intervals. For the remainder of this chapter, we consider functions which have the same average rate of change on every interval. Such a function has a graph which is a line and is called linear. Page 173

4. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (a) What is the average rate of change of P over every time interval? (b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. (c) Find a formula for P as a function of t. Page 18 (Example 1)4

5. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (a) What is the average rate of change of P over every time interval? This is given in the problem: 2,000 people / year Page 185

6. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. Page 186

7. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. t, yearsP, population 0 5 10 15 20 Page 187

8. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. t, yearsP, population 030,000 540,000 1050,000 1560,000 2070,000 Page 18 8

9. (b) Make a table that gives the town's population every five years over a 20-year period. Graph the population. Page 189

10. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. Page 18 10

11. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. We want: P = f(t) Page 18 11

12. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. We want: P = f(t) If we define: P = initial pop + (growth/year)(# of yrs) Page 18 12

13. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. tP 030,000 540,000 1050,000 1560,000 2070,000 If we define: P = initial pop + (growth/year)(# of yrs) Page 18 13

14. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. tP 030,000 540,000 1050,000 1560,000 2070,000 We substitute the initial value of P: P = 30,000 + (growth/year)(# of yrs) Page 18 14

15. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. tP 030,000 540,000 1050,000 1560,000 2070,000 And our rate of change: P = 30,000 + (2,000/year)(# of yrs) Page 18 15

16. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. tP 030,000 540,000 1050,000 1560,000 2070,000 And we substitute in t: P = 30,000 + (2,000/year)(t) Page 18 16

17. A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years. (c) Find a formula for P as a function of t. tP 030,000 540,000 1050,000 1560,000 2070,000 Our final answer: P = 30,000 + 2,000t Page 18 17

18. Here again is the graph and the function. Page 18 18

19. Any linear function has the same average rate of change over every interval. Thus, we talk about the rate of change of a linear function. In general: A linear function has a constant rate of change. The graph of any linear function is a straight line. Page 19 19

20. Depreciation Problem A small business spends $20,000 on new computer equipment and, for tax purposes, chooses to depreciate it to $0 at a constant rate over a five-year period. (a) Make a table and a graph showing the value of the equipment over the five-year period. (b) Give a formula for value as a function of time. Page 19 (Example 2) 20

21. Used by economists/accounts: a linear function for straight-line depreciation. Example: tax purposes-computer equipment depreciates (decreases in value) over time. Straight-line depreciation assumes: the rate of change of value with respect to time is constant. Page 19 21

22. t, yearsV, value ($) Let's fill in the table: Page 19 22

23. t, yearsV, value ($) 0 1 2 3 4 5 Let's fill in the table: Page 1923

24. t, yearsV, value ($) 0$20,000 1$16,000 2$12,000 3$8,000 4$4,000 5$0 Let's fill in the table: Page 19 24

25. And our graph: Page 19 25

26. Give a formula for value as a function of time: Page 19 26

27. Give a formula for value as a function of time: Page 19 27

28. Give a formula for value as a function of time: Page 19 28

29. Give a formula for value as a function of time: Page 19 29

30. Give a formula for value as a function of time: Page 19 30

31. Give a formula for value as a function of time: More generally, after t years? Page 19 31

32. Give a formula for value as a function of time: More generally, after t years? Page 19 32

33. Give a formula for value as a function of time: What about the initial value of the equipment? Page 19 33

34. Give a formula for value as a function of time: What about the initial value of the equipment? $20,000 Page 19 34

35. Give a formula for value as a function of time: What about the initial value of the equipment? $20,000 What is our final answer for the function? Page 19 35

36. Give a formula for value as a function of time: What about the initial value of the equipment? $20,000 What is our final answer for the function? V = 20,000 - 4,000t Page 19 36

37. Let's summarize: y x m b Page 20 37

38. Let's summarize: y x m b b = y intercept (when x=0) m = slope Page 20 38

39. Let's summarize: y x m b y = b + mx Page 20 39

40. Let's summarize: y x m b Page 20 40

41. Let's summarize: y x m b Page 20 41

42. Let's recap: example #1: P = 30,000 + 2,000t m = ? b = ? Page 20 42

43. Let's recap: example #1: P = 30,000 + 2,000t m = 2,000 b = 30,000 Page 20 43

44. Let's recap: example #2: V = 20,000 - 4,000t m = ? b = ? Page 20 44

45. Let's recap: example #2: V = 20,000 - 4,000t m = -4,000 b = 20,000 Page 20 45

46. Can a table of values represent a linear function? Page 21 46

47. Could a table of values represent a linear function? Yes, it could if: Page 21 47

48. Could a table of values represent a linear function? Yes, it could if: Page 21 48

49. xp(x)ΔxΔxΔpΔpΔp/Δx 50.10 55.11 60.12 65.13 70.14 Could p(x) be a linear function? Page 2149

50. xp(x)ΔxΔxΔpΔpΔp/Δx 50.10 5 55.11 5 60.12 5 65.13 5 70.14 Could p(x) be a linear function? Page 2150

51. xp(x)ΔxΔxΔpΔpΔp/Δx 50.10 5.01 55.11 5.01 60.12 5.01 65.13 5.01 70.14 Could p(x) be a linear function? Page 2151

52. xp(x)ΔxΔxΔpΔpΔp/Δx 50.10 5.01.002 55.11 5.01.002 60.12 5.01.002 65.13 5.01.002 70.14 Could p(x) be a linear function? Page 2152

53. xp(x)ΔxΔxΔpΔpΔp/Δx 50.10 5.01.002 55.11 5.01.002 60.12 5.01.002 65.13 5.01.002 70.14 Since Δp/Δx is constant, p(x) could represent a linear function. Page 2153

54. xq(x)ΔxΔxΔqΔqΔq/Δx 50.01 55.03 60.06 65.14 70.15 Could q(x) be a linear function? Page 2154

55. xq(x)ΔxΔxΔqΔqΔq/Δx 50.01 5 55.03 5 60.06 5 65.14 5 70.15 Could q(x) be a linear function? Page 2155

56. xq(x)ΔxΔxΔqΔqΔq/Δx 50.01 5.02 55.03 5 60.06 5.08 65.14 5.01 70.15 Could q(x) be a linear function? Page 2156

57. xq(x)ΔxΔxΔqΔqΔq/Δx 50.01 5.02.004 55.03 5.006 60.06 5.08.016 65.14 5.01.002 70.15 Could q(x) be a linear function? Page 2157

58. xq(x)ΔxΔxΔqΔqΔq/Δx 50.01 5.02.004 55.03 5.006 60.06 5.08.016 65.14 5.01.002 70.15 Since Δq/Δx is NOT constant, q(x) does not represent a linear function. Page 2158

59. Yearp, Price ($) Q, # sold (cars) ΔpΔpΔQΔQΔQ/Δp 19853,99049,000 19864,11043,000 19874,20038,500 19884,33032,000 What about the following example? Yugos exported from Yugoslavia to US. Page 22 59

60. Yearp, Price ($) Q, # sold (cars) ΔpΔpΔQΔQΔQ/Δp 19853,99049,000 120 19864,11043,000 90 19874,20038,500 130 19884,33032,000 What about the following example? Yugos exported from Yugoslavia to US. Page 22 60

61. Yearp, Price ($) Q, # sold (cars) ΔpΔpΔQΔQΔQ/Δp 19853,99049,000 120-6,000 19864,11043,000 90-4,500 19874,20038,500 130-6,500 19884,33032,000 What about the following example? Yugos exported from Yugoslavia to US. Page 22 61

62. Yearp, Price ($) Q, # sold (cars) ΔpΔpΔQΔQΔQ/Δp 19853,99049,000 120-6,000-50 cars/$ 19864,11043,000 90-4,500-50 cars/$ 19874,20038,500 130-6,500-50 cars/$ 19884,33032,000 What about the following example? Yugos exported from Yugoslavia to US. Page 2262

63. ΔpΔpΔQΔQΔQ/Δp 120-6,000-50 cars/$ 90-4,500-50 cars/$ 130-6,500-50 cars/$ Although Δp and ΔQ are not constant, ΔQ/Δp is. Therefore, since the rate of change (ΔQ/Δp) is constant, we could have a linear function here. Page 2263

64. Page 22 64

65. The function P = 100(1.02) t approximates the population of Mexico in the early 2000's. Here P is the population (in millions) and t is the number of years since 2000. Table 1.25 and Figure 1.21 show values of P over a 5-year period. Is P a linear function of t?1.251.21 Page 2365

66. t, yearsP (mill.)ΔtΔtΔPΔPΔP/Δt 0100 122 1102 12.04 2104.04 12.08 3106.12 12.12 4108.24 12.17 5110.41 Page 23 66

67. Page 23 67

68. t, yearsP (mill.)ΔtΔtΔPΔPΔP/Δt 0100 1021.902.190 10121.90 1026.692.669 20148.59 1032.553.255 30181.14 1039.663.966 40220.80 1048.364.836 50269.16 Page 2468

69. Page 24 69

70. The formula P = 100(1.02) t is not of the form P = b + mt, so P is not a linear function of t. Page 2470

71. This completes Section 1.3.