1. Section 9B Linear Modeling Pages 571-585

2. Linear Modeling 9-B LINEAR constant rate of change

3. Understanding Rate of Change 9-B Example: The population of Straightown increases at a rate of 500 people per year. How much will the population grow in 2 years? 10 years? The population of Straightown varies with respect to time (year) with a rate of change of 500 people per year. P = f(y) In 2 years, the population will change by: (500 people/year ) x 2 years = 1000 people

4. Understanding Rate of Change 9-B Example/571: During a rainstorm, the rain depth reading in a rain gauge increases by 1 inch each hour. How much will the depth change in 30 minutes? The rain depth varies with respect to time (hour) with a rate of change of 1 inch per hour. D = f(h) In 30 minutes, the rain depth will change by: (1 inch/hour ) x (1/2 hour) = (1/2) inch

5. Understanding Rate of Change 9-B Example 27/583: The water depth in a lake decreases at a rate of 1.5 inches per day because of evaporation. How much does the water depth change in 6.5 days? in 12.5 days? The water depth varies with respect to time (days) with a rate of change of -1.5 inches per day. W = f(d) In 6.5 days, the water depth will change by: (-1.5 inches/day ) x (6.5 days) = -9.75 inches

6. Understanding Linear Equations 9-B Example: The population of Straightown is 10,000 and increasing at a rate of 500 people per year. What will the population be in 2 years? The population of Straightown varies with respect to time (years) with an initial value of 10,000 and a rate of change of 500 people per year. P = f(y) P = 10000 + 500y P = 10000 + (500)(2) = 11000 people

7. Understanding Linear Equations 9-B Example: The rain depth at the beginning of a storm is ½ inch and is increasing at a rate of 1 inch per hour? What is the depth in the gauge after 3 hours? The rain depth varies with respect to time (hours) with an initial value of ½ inch and a rate of change of 1 inch per hour. D = f(h) P = 1/2+ (1)(h) P = 1/2 + (1)(3) = 7/2 inches or 3.5 inches

8. Understanding Linear Equations 9-B Example 27*/583: The water depth in a lake is 100 feet and decreases at a rate of 1.5 inches per day because of evaporation? What is the water depth after 6.5 days? The water depth varies with respect to time (days) with an initial value of 100 feet (1200 inches) and a rate of change of 1.5 inches per day. W = f(d) P = 1200-(1.5)(d) P = 1200-(1.5)(6.5) = 1200 – 9.75 = 1190.25 inches

9. Understanding Linear Equations 9-B General Equation for a Linear Function (p576): dependent var. = initial value + (rate of change x independent var.) NOTE: rate of change = dependent variable per independent variable

10. yP 010,000 110,500 211,000 311,500 512,500 1015,000 Graphing Linear Equations Example - Straightown: P = 10000 + 500y

11. hD 01/2 13/2 25/2 37/2 511/2 1021/2 Graphing Linear Equations Example – Rain Depth: D = 1/2 + (1)(h)

12. hD 01200 11190.25 21180.5 31170.75 51151.25 101102.5 Graphing Linear Equations Example – Lake Water Depth: W = 1200 - (9.75)(d)

13. Linear Modeling 9-B LINEAR constant rate of change (slope) straight line graph

14. 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. Understanding Slope

15. Example: Calculate the slope of the Straightown graph.

16. Understanding Slope Example: Calculate the slope of the Water Lake Depth graph.

17. More Practice 33/583 The price of a particular model car is $15,000 today and rises with time at a constant rate of $1200 per year. A) Find a linear equation to describe the situation. B) How much will a new car cost in 2.5 years. 35/583 A snowplow has a maximum speed of 40 miles per hour on a dry highway. Its maximum speed decreases by 1.1 miles per hour for every inch of snow on the highway. A) Find a linear equation to describe the situation. B) At what snow depth will the plow be unable to move? 37/583 You can rent time on computers at the local copy center for $8 setup charge and an additional $1.50 for every 5 minutes. B) Find a linear equation to describe the situation. C) How much time can you rent for $25? 53, 55, 57/584

18. Homework: Pages 582-583 34, 36, 38, 54, 56, 58 formulas, answers and graphs for each problem. 9-B

19. Algebraic Linear Equations Slope Intercept Form y = b + mx b is the y intercept or initial value m is the slope or rate of change. More Practice/584: 45, 47, 49, 51

20. Homework: Pages 584 # 40, 44, 48, 50, 54, 56, 58 9-B