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Section 1.3 Linear Functions

Section 1.3 Linear Functions. 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.

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Section 1.3 Linear Functions

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  1. Section 1.3 Linear Functions

  2. 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 17

  3. 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. 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 18

  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. (b)  Make a table that gives the town's population every five years over a 20-year period. Graph the population. Page 18

  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 18

  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. Page 18

  8. (b)  Make a table that gives the town's population every five years over a 20-year period. Graph the population. Page 18

  9. 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. 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. 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. 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. If we define: P = initial pop + (growth/year)(# of yrs) Page 18

  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. We substitute the initial value of P: P = 30,000 + (growth/year)(# of yrs) Page 18

  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. And our rate of change: P = 30,000 + (2,000/year)(# of yrs) Page 18

  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. And we substitute in t: P = 30,000 + (2,000/year)(t) Page 18

  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. Our final answer: P = 30,000 + 2,000t Page 18

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

  18. 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. 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. 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. Let's fill in the table: Page 19

  22. Let's fill in the table: Page 19

  23. Let's fill in the table: Page 19

  24. And our graph: Page 19

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  48. Could p(x) be a linear function? Page 21

  49. Could p(x) be a linear function? Page 21

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