Section 9B Linear Modeling

1. Section 9BLinear Modeling Pages 542-553

2. 9-B Linear Functions A Linear Functionchanges by the same absolute amount for each unit of change in the input (independent variable). A Linear Function has a constant rate of change. Examples:Straightown population as a function of time. Postage cost as a function of weight. Pineapple demand as a function of price.

3. 9-B First Class Mail – a linear function

4. 9-B First Class Mail – a linear function

5. 9-B First Class Mail – a linear function

6. 9-B First Class Mail – a linear function

7. 9-B First Class Mail – a linear function

8. 9-B First Class Mail – a linear function

9. 9-B First Class Mail – a linear function

10. 9-B First Class Mail – a linear function

11. 9-B First Class Postage

12. 9-B First class postage – a linear function

13. 9-B First class postage – a linear function

14. 9-B First class postage – a linear function

15. We define ‘rate of change’ of a linear function by: where (x1,y1) and (x2,y2) are any two ordered pairs of the function.

16. 9-B Slope = rate of change

17. 9-B Linear Functions A linear function has a constant rate of change and a straight line graph. The rate of change = slope of the graph. The greater the rate of change, the steeper the slope. positive slope negative slope

18. 9-B Example: Price-Demand Function A linear function is used to describe how the demand for pineapples varies with the price. ($2, 80 pineapples) and ($5, 50 pineapples). Find the rate of change (slope) for this function and then graph the function. independent variable = price dependent variable = demand for pineapples

19. 9-B Example: Price-Demand Function ($2, 80 pineapples) and ($5, 50 pineapples)

20. 9-B Example: Price-Demand Function • ($2, 80 pineapples) and ($5, 50 pineapples). • To graph a linear function you need 2 things: • two points or • slope and one point

21. 9-B Example: Price-Demand Function ($2, 80 pineapples) and ($5, 50 pineapples).

22. 9-B Example: Price-Demand Function ($2, 80 pineapples) and ($5, 50 pineapples).

23. 9-B General Equation for a Linear Function dependent = initial value + (slope)×independent y= initial value + (slope)×x (Initial value occurs when the independent variable = 0.) y = mx + b or y = b + mx m = slope b = y-intercept (The line goes through the point (0,b).)

24. 9-B Example: dep. variable = initial value + (slope)× indep. variable slope = -10 pineapples/$ initial value = 100 pineapples Demand = 100 - 10×(price) D = 100 – 10p

25. 9-B Example: Demand = 100 - 10×(price) D = 100 – 10p Check: $2: 100 - 10×2 = 80 pineapples $5: 100 - 10×5 = 50 pineapples

26. old example:The initial population of Straightown is 10, 000 and increases by 500 people per year. Graph Data Table

27. old example:The initial population of Straightown is 10, 000 and increases by 500 people per year. = 500 = 500 = 500 Rate of change (slope) is ALWAYS 500 (people per year). Initial population is 10,000 (people). Linear Function: Population=10,000+ 500×(year)

28. 9-B Example – First class postage Slope = $.23/ounce initial value = $0.14

29. 9-B Example: First Class Postage Slope = $.23/ounce initial value = $0.14 Postage = $0.14 + $0.23×(weight) P = $0.14+ $0.23w Check: 1 ounce: $0.14+ $0.23×1 = $0.37 6 ounces: $0.14 + $0.23×6 = $1.52

30. 9-B Example: The world record time in the 100-meter butterfly was 53.0 seconds in 1988. Assume that the record falls at a constant rate of 0.05 seconds per year. What does the model predict for the record in 2010? dependent variable = world record time (R) independent variable is time, t (years) after 1988. Slope = 0.05 seconds; initial value = 53.0 seconds; Record time = 53.0 – 0.05×(t years after 1988) R = 53 – 0.05t Record time in 2010 = 53 - .05×(22) = 51.9 seconds

31. 9-B Example: Suppose you were 20 inches long at birth and 4 ft tall on your tenth birthday. Create a linear equation that describes how your height varies with age. independent variable = age (years) dependent variable = height (inches) Two points: (0, 20) (10, 48) Initial value = 20 inches Height = 20 + 2.8t t = years

32. 9-B Example: “Fines for Certain PrePayable Violations” – Speeding other than residence zone, highway work zone and school crosswalk: $5.00 per MPH over speed limit plus processing fee ($51.00) and local fees ($5.00) independent variable = miles over speed limit dependent variable = fine ($) Initial value = $56.00 Slope = $5.00 Fine = $56 + $5(your speed-speed limit)

33. 9-B Example: Mrs. M. was given a ticket for doing 52 mph in a zone where the speed limit was 35 mph. How much was her fine? Fine = $55 + $5(her speed-35) Fine = $56 + $5(52-35) = $56 + $5(17) = $141

34. 9-B Example: “Fines for Certain PrePayable Violations” – Speeding in a residence zone: $200 plus $7.00 per MPH over speed limit (25 mph), plus processing fee ($51.00) and local fees ($5.00) independent variable = miles over speed limit dependent variable = fine ($) Initial value = $256.00 Slope = $7.00 Fine = $256 + $7(your speed-25)

35. 9-B Example: The Psychology Club plans to pay a visitor $75 to speak at a fundraiser. Tickets will be sold for $2 apiece. Find a linear equation that gives the profit/loss for the event as it varies with the number of tickets sold. independent variable = number of tickets sold dependent variable = profit/loss ($) (0, -$75) slope = +$2 (= rate of change in ticket price) Profit = -$75 +2×(number of tickets) P = -$75 +2n

36. 9-B Example: How many people must attend for the club to break even? P = -$75 +2n 0 = -$75 + 2n $75 = 2n 37.5 = n Can’t sell half a ticket -- so we’ll need to sell 38 tickets.

37. 9-B Homework Pages 553-555 # 8, 12a-b, 14a-b, 18, 26, 28, 30, 33