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2. Section 1.4 Formulas for Linear Functions 2

3. A grapefruit is thrown into the air. Its velocity, v, is a linear function of t, the time since it was thrown. (A positive velocity indicates the grapefruit is rising and a negative velocity indicates it is falling.) Check that the data in Table 1.30 corresponds to a linear function. Find a formula for v in terms of t.1.30 3 Page 27 Example 1

4. A grapefruit is thrown into the air. Its velocity, v, is a linear function of t, the time since it was thrown. (A positive velocity indicates the grapefruit is rising and a negative velocity indicates it is falling.) Check that the data in Table 1.30 corresponds to a linear function. Find a formula for v in terms of t.1.30 We will calculate: Rate of change of velocity with respect to time (or rate of change, for short). 4 Page 27

5. Finding a Formula for a Linear Function from a Table of Data t, time (secs) v, velocity (ft/sec) 148.00 216.00 3-16.00 4-48.00 5 Page 27

6. Finding a Formula for a Linear Function from a Table of Data t, time (secs) v, velocity (ft/sec) ΔtΔt 148.00 1.00 216.00 1.00 3-16.00 1.00 4-48.00 6 Page 27

7. Finding a Formula for a Linear Function from a Table of Data t, time (secs) v, velocity (ft/sec) ΔtΔtΔvΔv 148.00 1.00-32.00 216.00 1.00-32.00 3-16.00 1.00-32.00 4-48.00 7 Page 27

8. Finding a Formula for a Linear Function from a Table of Data t, time (secs) v, velocity (ft/sec) ΔtΔtΔvΔvΔv/Δt 148.00 1.00-32.00 216.00 1.00-32.00 3-16.00 1.00-32.00 4-48.00 8 Page 27

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10. Since v is a function of t, we have: v = f(t). We also remember from Section 1.3: m = ? 10 Page 28

11. Since v is a function of t, we have: v = f(t). We also remember from Section 1.3: m = slope (or the rate of change = Δv/Δt) Here, m = ? 11 Page 28

12. Since v is a function of t, we have: v = f(t). We also remember from Section 1.3: m = slope (or the rate of change = Δv/Δt) Here, m = -32. So we have: v = b + mt or v = b -32t 12 Page 28

13. v = b -32t How do we solve for b? 13 Page 28

14. What can we use from this chart? t, time (secs) v, velocity (ft/sec) ΔtΔtΔvΔvΔv/Δt 148.00 1.00-32.00 216.00 1.00-32.00 3-16.00 1.00-32.00 4-48.00 14 Page 28

15. t, time (secs)v, velocity (ft/sec) 148.00 216.00 3-16.00 4-48.00 Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-48) and ? 15 Page 27

16. t, time (secs)v, velocity (ft/sec) 148.00 216.00 3-16.00 4-48.00 Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-48) and substitute into: v = b -32t 16 Page 27

17. Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-48) and substitute into: v = b -32t (1,48): 48 = b - 32(1) → 48 + 32 = b → b = 80 17 Page 27

18. Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-32) and substitute into: v = b -32t (1,48): 48 = b - 32(1) → 48 + 32 = b → b = 80 (2,16): 16 = b - 32(2) → 16 + 64 = b → b = 80 18 Page 27

19. Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-32) and substitute into: v = b -32t (1,48): 48 = b - 32(1) → 48 + 32 = b → b = 80 (2,16): 16 = b - 32(2) → 16 + 64 = b → b = 80 (3,-16): -16 = b - 32(3) → -16 + 96 = b → b = 80 19 Page 27

20. Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-32) and substitute into: v = b -32t (1,48): 48 = b - 32(1) → 48 + 32 = b → b = 80 (2,16): 16 = b - 32(2) → 16 + 64 = b → b = 80 (3,-16): -16 = b - 32(3) → -16 + 96 = b → b = 80 (4,-48): -48 = b -32(4) → -48 + 128 = b → b = 80 20 Page 27

21. So what is our final equation? 21 Page 28

22. So what is our final equation? v = 80 - 32t 22 Page 28

23. v = 80 - 32t Note: m = -32 ft/sec per second (a.k.a. ft/sec 2 ) implies: the grapefruit’s velocity is decreasing by 32 ft/sec for every second that goes by. “The grapefruit is accelerating at -32 ft/sec per second.” Negative acceleration is also called deceleration. (Note: no shorthand way of saying “ft/sec per second”.) 23 Page 28

24. Finding a Formula for a Linear Function from a Graph We can calculate the slope, m, of a linear function using two points on its graph. Having found m, we can use either of the points to calculate b, the vertical intercept. 24 Page 28

25. Figure 1.25 shows oxygen consumption as a function of heart rate for two people.1.25 (a) Assuming linearity, find formulas for these two functions. (b) Interpret the slope of each graph in terms of oxygen consumption. 25 Page 28 Example 2

26. 26 Page 28

27. Let's calculate m: 27 Page 29

28. Let's calculate m: 28 Page 29

29. Let's calculate m: 29 Page 29

30. What are our 2 linear equations so far? 30 Page 29

31. What are our 2 linear equations so far? For person A: 31 Page 29

32. What are our 2 linear equations so far? For person A: For person B: 32 Page 29

33. Now let's calculate b: 33 Page 29

34. Now let's calculate b: 34 Page 29

35. Now let's calculate b: 35 Page 29

36. What are our 2 linear equations? 36 Page 29

37. What are our 2 linear equations? For person A: 37 Page 29

38. What are our 2 linear equations? For person A: For person B: 38 Page 29

39. Figure 1.25 shows oxygen consumption as a function of heart rate for two people.1.25 (b) Interpret the slope of each graph in terms of oxygen consumption. What about (b)? 39 Page 29

40. Here are two reminders- this slide and the next: 40 Page 29

41. m=.01 m=.0067 41 Page 28

42. Since the slope for person B is smaller than for person A, person B consumes less additional oxygen than person A. 42 Page 29

43. We have $24 to spend on soda and chips for a party. A six-pack of soda costs $3 and a bag of chips costs $2. The number of six-packs we can afford, y, is a function of the number of bags of chips we decide to buy, x. (a) Find an equation relating x and y. (b) Graph the equation. Interpret the intercepts and the slope in the context of the party. 43 Page 30 Example 3

44. Let: x = # of bags of chips y = # of six-packs of soda 44 Page 30

45. Let: x = # of bags of chips $2x = amount spent on chips y = # of six-packs of soda 45 Page 30

46. Let: x = # of bags of chips $2x = amount spent on chips y = # of six-packs of soda $3y = amount spent on soda 46 Page 30

47. Let: x = # of bags of chips $2x = amount spent on chips y = # of six-packs of soda $3y = amount spent on soda & 2x + 3y = 24 47 Page 30

48. 2x + 3y = 24 Let's solve for y: 48 Page 30

49. 49 Page 30

50. What is the slope and what is the y intercept? 50 Page 30

51. What is the slope and what is the y intercept? 51 Page 30

52. (b) Graph the equation. Interpret the intercepts and the slope in the context of the party. 52 Page 30

53. x (chips)y (soda) Let's plot some points: 53 Page N/A

54. x (chips)y (soda) 0 1 2 3 4 5 6 7 8 9 10 11 12 Let's plot some points: 54 Page N/A

55. x (chips)y (soda) 08.0000 17.3333 26.6667 36.0000 45.3333 54.6667 64.0000 73.3333 82.6667 92.0000 101.3333 110.6667 120.0000 Let's plot some points: 55 Page N/A

56. 56 Page 30

57. x (chips)y (soda) 08.0000 17.3333 26.6667 36.0000 45.3333 54.6667 64.0000 73.3333 82.6667 92.0000 101.3333 110.6667 120.0000 What conclusions can we draw from this table and from the equation below? 57 Page 31

58. x (chips)y (soda) 08.0000 17.3333 26.6667 36.0000 45.3333 54.6667 64.0000 73.3333 82.6667 92.0000 101.3333 110.6667 120.0000 What conclusions can we draw from this table and from the equation below? 58 Page 31

59. x (chips)y (soda) 08.0000 17.3333 26.6667 36.0000 45.3333 54.6667 64.0000 73.3333 82.6667 92.0000 101.3333 110.6667 120.0000 What conclusions can we draw from this table and from the equation below? 59 Page 31

60. x (chips)y (soda) 08.0000 17.3333 26.6667 36.0000 45.3333 54.6667 64.0000 73.3333 82.6667 92.0000 101.3333 110.6667 120.0000 What conclusions can we draw from this table and from the equation below? +3 -2 60 Page 31

61. x (chips)y (soda) 08.0000 17.3333 26.6667 36.0000 45.3333 54.6667 64.0000 73.3333 82.6667 92.0000 101.3333 110.6667 120.0000 What conclusions can we draw from this table and from the equation below? -3 +2 61 Page 31

62. Alternative Forms for the Equation of a Line: Slope-Intercept Form: y = b + mx [m=slope, y=y-int] Point-Slope Form: y – y o = m(x – x o ) [m=slope, (x o,y o ) point on the line] Standard Form: Ax + By + C = 0 [A, B, C constants] 62 Page 31

63. End of Section 1.4 63