1. Finding a Linear Model Section 2.4

2. Lehmann, Intermediate Algebra, 4ed Section 2.4 A company’s profit was $10 million in 2005 and has increased by $3 million per year. Let p be the profit (in millions of dollars) in the year that is t years since 2005. 1.Is there a linear relationship between t and p? Since the profit is a constant $3 million per year, the variables t and p are linearly related Slide 2 Finding a Model Finding an Equation of a Linear Model Example Solution

3. Lehmann, Intermediate Algebra, 4ed Section 2.4 2.Find the p-intercept of a linear model Profit was $10 million in 2005 2005 is 0 years since 2005 This gives the ordered pair (0, 10) So, the p-intercept is (0, 10) 3.Find the slope of the linear model. Slide 3 Finding a Model Finding an Equation of a Linear Model Example Continued Solution Example Continued

4. Lehmann, Intermediate Algebra, 4ed Section 2.4 Rate of change of profit per year is $3 million per year So, the slope of the linear model is 3 4.Find an equation of the linear model. Since p-intercept is (0, 10) and slope is 3, the linear model is: Slide 4 Finding a Model Finding an Equation of a Linear Model Solution Example Continued Solution

5. Lehmann, Intermediate Algebra, 4ed Section 2.4 Verify the ordered pair (0, 10) Verify that as the input increases by 1, the output increases by 3 Slide 5 Finding a Model Finding an Equation of a Linear Model Graphing Calculator

6. Lehmann, Intermediate Algebra, 4ed Section 2.4 We perform a unit analysis of a model’s equations by determining the units of the expression on both sides of the equation. The simplified units of the expressions on both sides of the equation should be the same. Slide 6 Definition: Unit Analysis Unit Analysis of a Linear Model Definition

7. Lehmann, Intermediate Algebra, 4ed Section 2.4 A driver fills her car’s 12-gallon gasoline tank and drives as a constant speed. The car consumes 0.04 gallon per mile. Let G be the number of gallons of gasoline remaining in the tank after she has driven d miles since filling up. 1.Is there a linear relationship between d and G? Rate of change is a constant –0.04 gallons per minute, so d and G are linearly related Slide 7 Finding a Model Unit Analysis of a Linear Model Example Solution

8. Lehmann, Intermediate Algebra, 4ed Section 2.4 2.Find the G-intercept of a linear model. Tank is full at 12 gallons: ordered pair (0, 12) 3.Find the slope of the linear model. Gasoline remaining in the tank with respect to distance traveled is: Slide 8 Finding a Model Unit Analysis of a Linear Model Example Continued Solution Example Continued

9. Lehmann, Intermediate Algebra, 4ed Section 2.4 4.Find the equation of the linear model. Since p-intercept is (0, 12) and slope is –0.04, the linear model is: 5.Perform a unit analysis of the equation Here a unit analysis on the equation : Slide 9 Finding a Model Unit Analysis of a Linear Model Example Continued Solution Example Continued

10. Lehmann, Intermediate Algebra, 4ed Section 2.4 We use the fact that = 1 to simplify the units of the expression on the right-hand side of the equation: Units on both sides are gallons: Suggesting correct Slide 10 Finding a Model Unit Analysis of a Linear Model Solution Continued

11. Lehmann, Intermediate Algebra, 4ed Section 2.4 Yogurt sales (in billions of dollars) in the United States are shown in the table for various years. Let s be yogurt sales (in billions of dollars) in the year that is t years since 2000. A model of the situation is: 1.Use a graphing calculator to draw a scattergram and the model in the same viewing window. Check whether the line comes close to the data. Slide 11 Analyzing a Model Two Variables That Are Approximately Linearly Related Example

12. Lehmann, Intermediate Algebra, 4ed Section 2.4 Draw in the same screen using a graphing calculator See Sections B.8 and B.10 2.What is the slope of the model? What does it mean in this situation? Slide 12 Analyzing a Model Two Variables That Are Approximately Linearly Related Solution Example Continued

13. Lehmann, Intermediate Algebra, 4ed Section 2.4 which is of the form Since m is the slope, the slope is 0.17 Sales increase by 0.17 billion dollars per year 3.Find the rates of change in sales from one year to the next. Compare the rates of change with the results in Problem 2. Slide 13 Analyzing a Model Two Variables That Are Approximately Linearly Related Solution Example Continue

14. Lehmann, Intermediate Algebra, 4ed Section 2.4 Rates of change are shown in the table – all are close to 0.17 Slide 14 Analyzing a Model Two Variables That Are Approximately Linearly Related Solution

15. Lehmann, Intermediate Algebra, 4ed Section 2.4 4.Predict the sales in 2010. Substitute the input of 10 for t: If two quantities t and p are approximately linearly related, and if p depends on t, then the slope of a reasonable linear model is approximately equal to the average rate of change of p with respect to t. Slide 15 Analyzing a Model Two Variables That Are Approximately Linearly Related Solution Example Continue Property