1. Section 9B Linear Modeling Pages 542-553

2. Linear Functions 9-B A linear function describes a relation between independent (input) and dependent (output) variables with a constant rate of change (and a straight-line graph). Examples: Straightown population as a function of time. Postage cost as a function of weight. Pineapple demand as a function of price.

3. 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.

4. We define slope of a straight line by: where (x 1,y 1 ) and (x 2,y 2 ) are any two points on the graph of the straight line.

5. 9-B General Equation for a Linear Function dependent = initial value + (rate of change x independent) or y = m x + b where m is slope and b is y intercept.

6. 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.

7. 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 is ALWAYS 500 (people per year). Initial population is 10000 (people). Linear Function: P = 10000 + 500 x t

10. Slope of the straight line graph is 500. Y-intercept of the straight line graph is 10000. Linear Function: P = 10000 + 500 x t

11. What is the population after 25 years? When will the population be 28,000?

12. Example: The table below gives the cost of US mail based on weight. What is the rate of change? Graph the cost as a function of weight and determine the slope. Weight (independent) Postage cost (dependent) 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 Rate of change is ALWAYS 0.23 (dollars per ounce). Initial cost is ??? (dollars). Linear Function: C = ?? +.23 x w

13. 9-B Example: The table below gives the cost of US mail based on weight. What is the rate of change? Graph the cost as a function of weight and determine the slope.

14. 9-B Example: The table below gives the cost of US mail based on weight. What is the rate of change? Graph the cost as a function of weight and determine the slope.

15. 9-B Example: The table below gives the cost of US mail based on weight. What is the rate of change? Graph the cost as a function of weight and determine the slope.

16. 9-B Example: The table below gives the cost of US mail based on weight. What is the rate of change? Graph the cost as a function of weight and determine the slope.

17. Slope of the straight line graph is 0.23. Y-intercept of the straight line graph is.37-.23 =.14. Linear Function: C = 0.14 + 0.23 x w

18. How much will it cost to mail 6.5 ounces? How many ounces can be mailed for $1.10?

19. For linear functions: Slope = Rate of Change Use any two ordered pairs (points on the graph) to calculate rate of change (slope).

20. How does rate of change (slope) affect steepness… 9-B …the greater the rate of change (slope), the steeper the graph.

21. 9-B ex2/545 A linear function is used to describe how the demand for pineapples varies with the price. We know at a price of $2, the demand is 80 pineapples and at a price of $5, the demand is 50 pineapples. Find the rate of change (slope) for this function and then graph the function. Express as a linear function. Independent variable: price Dependent variable: demand (of pineapples) Demand is a function of price. ($2,80) and ($5,50)

22. 9-B ($2, 80 pineapples) and ($5, 50 pineapples) ex2/545 A linear function is used to describe how the demand for pineapples varies with the price. We know at a price of $2, the demand is 80 pineapples and at a price of $5, the demand is 50 pineapples. Find the rate of change (slope) for this function and then graph the function. Express as a linear function. For every dollar increase in price, the demand for pineapples decreases by 10.

23. 9-B ($2, 80 pineapples) and ($5, 50 pineapples). ex2/545 A linear function is used to describe how the demand for pineapples varies with the price. We know at a price of $2, the demand is 80 pineapples and at a price of $5, the demand is 50 pineapples. Find the rate of change (slope) for this function and then graph the function For every dollar increase in price, the demand for pineapples decreases by 10.

24. 9-B ($2, 80 pineapples) and ($5, 50 pineapples). For every dollar increase in price, the demand for pineapples decreases by 10. Slope of the straight line graph is -10. Y-intercept of the straight line graph is 100. Linear Function: D = 100 -10 x p

25. 9-B Linear Function: D = 100 -10 x p What is the demand for pineapples if the price is $8.50? If the demand is 75, what is the corresponding price?

26. For linear functions: Slope = Rate of Change Use any two ordered pairs (points on the graph) to calculate rate of change (slope). Postive Slope Negative Slope

27. More Practice 23/555 The price of a particular model car is $12,000 today and rises with time at a constant rate of $1200 per year. A) Clearly identify independent and dependent variable. B) Find a linear equation to describe the situation. C) How much will a new car cost in 2.5 years. 25/555 A snowplow has a maximum speed of 30 miles per hour on a dry highway. Its maximum speed decreases by 0.5 miles per hour for every inch of snow on the highway. A) Clearly identify independent and dependent variable. B) Find a linear equation to describe the situation. C) How much will a new car cost in 2.5 years? 27/555 You can rent time on computers at the local copy center for $5 setup charge and an additional $3 for every 5 minutes. A) Clearly identify independent and dependent variable. B) Find a linear equation to describe the situation. C) How much time can you rent for $15?

28. More Practice 29/555 Suppose that you were 20 inches long at birth and 4 feet tall on your tenth birthday. A) Clearly identify independent and dependent variable. B) Find a linear equation to describe the situation. C) Use the equation to predict your height at ages 2,6,20,50. D) Comment on the validity of the model. 31/555 A YMCA fundraiser offers raffle tickets for $5 each. The prize for the raffle is a $350 television set, which must be purchased with proceeds from the ticket sales. Find an equation that gives the profit/loss for the raffle as it varies with the number of tickets sold. How many tickets must be sold for the raffle sales to equal the cost of the prize? 33/555 A $1000 washing machine is depreciated for tax purposes at a rate of $50 per year. Find an equation for the depreciated value of the washing machine as it varies with time. When does the depreciated value reach $0?

29. Linear Functions Constant Rate of Change Straight Line Graph Dependent = Initial + Rate x Independent Y = mX + b

30. Homework: Pages 553-555 # 12a-b, 14a-b, 24, 26, 28, 32 9-B