1. Section 9B Linear Modeling Pages 542-553

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

3. First Class Mail – a linear function WeightPostage cost 1 oz $0.37 2 oz $0.60 3 oz $0.83 4 oz $1.06 5 oz $1.29 6 oz $1.52 7 oz $1.75 9-B

4. First Class Mail – a linear function WeightPostage costDifference 1 oz $0.37 2 oz $0.60 3 oz $0.83 4 oz $1.06 5 oz $1.29 6 oz $1.52 7 oz $1.75 9-B

5. First Class Mail – a linear function WeightPostage costDifference 1 oz $0.37 2 oz $0.60$0.23 3 oz $0.83 4 oz $1.06 5 oz $1.29 6 oz $1.52 7 oz $1.75 9-B

6. First Class Mail – a linear function WeightPostage costDifference 1 oz $0.37 2 oz $0.60$0.23 3 oz $0.83$0.23 4 oz $1.06 5 oz $1.29 6 oz $1.52 7 oz $1.75 9-B

7. First Class Mail – a linear function WeightPostage costDifference 1 oz $0.37 2 oz $0.60$0.23 3 oz $0.83$0.23 4 oz $1.06$0.23 5 oz $1.29 6 oz $1.52 7 oz $1.75 9-B

8. First Class Mail – a linear function WeightPostage costDifference 1 oz $0.37 2 oz $0.60$0.23 3 oz $0.83$0.23 4 oz $1.06$0.23 5 oz $1.29$0.23 6 oz $1.52 7 oz $1.75 9-B

9. First Class Mail – a linear function WeightPostage costDifference 1 oz $0.37 2 oz $0.60$0.23 3 oz $0.83$0.23 4 oz $1.06$0.23 5 oz $1.29$0.23 6 oz $1.52$0.23 7 oz $1.75 9-B

10. First Class Mail – a linear function WeightPostage costDifference 1 oz $0.37 2 oz $0.60$0.23 3 oz $0.83$0.23 4 oz $1.06$0.23 5 oz $1.29$0.23 6 oz $1.52$0.23 7 oz $1.75$0.23 9-B

11. First Class Postage 9-B

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

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

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

15. We define ‘rate of change’ of a linear function by: where ( x 1,y 1 ) and ( x 2,y 2 ) are any two ordered pairs of the function.

16. Slope = rate of change 9-B

17. 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 positive slope negative slope 9-B

18. 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 9-B

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

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

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

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

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

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

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

26. tP=f(t) 0f(0)=10,000 5f(5)=12,500 10f(10)=15,000 15f(15)=17,500 20f(20)=20,000 40f(40)=30,000 Data Table Graph old example: The initial population of Straightown is 10, 000 and increases by 500 people per year.

27. tP=f(t) 010,000 512,500 1015,000 1517,500 2020,000 4030,000 old example: The initial population of Straightown is 10, 000 and increases by 500 people per year. = 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. Example – First class postage WeightPostage cost 1 oz $0.37 2 oz $0.60 3 oz $0.83 4 oz $1.06 5 oz $1.29 6 oz $1.52 7 oz $1.75 9-B Slope = $.23/ounce initial value = $0.14

29. 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 6 ounces: $0.14 + $0.23×6 = $1.52 9-B

30. 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 9-B

31. 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 + 2.8tt = years

32. Example: 9-B “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 = $56.00Slope = $5.00 Fine = $56 + $5(your speed-speed limit)

33. Example: 9-B 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. Example: 9-B “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 = $256.00Slope = $7.00 Fine = $256 + $7(your speed-25)

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