1. Modeling with Linear Functions Chapter 2

2. Using Lines to Model Data Section 2.1

3. Lehmann, Intermediate Algebra, 3ed Section 2.1 The number of Grand Canyon visitors is listed in the table for various years. Describe the data. Slide 3 Using Lines to Model Data Scattergrams Let v be the number (in millions) of visitors Let t be the number of years since 1960 Example Solution

4. Lehmann, Intermediate Algebra, 3ed Section 2.1 Sketch a line that comes close to (or on) the data points. Slide 4 Using Lines to Model Data Scattergrams The graph on the left does the best job of this. Example Continued

5. Lehmann, Intermediate Algebra, 3ed Section 2.1 If the points in a scattergram of data lie close to (or on) a line, then we say that the relevant variables are approximately linearly related. For the Grand Canyon situation, variables t and v are approximately linearly related. A model is a mathematical description of an authentic situation. We say that the description models the situation. Slide 5 Definitions Linear Models Definition

6. Lehmann, Intermediate Algebra, 3ed Section 2.1 A linear model is a linear function, or its graph, that describes the relationship between two quantities for an authentic situation. The Grand Canyon model is a linear model Every linear model is a linear function Functions are used to describe situations and to describe certain mathematical relationships Slide 6 Definitions Linear Models Definition Property

7. Lehmann, Intermediate Algebra, 3ed Section 2.1 Use a linear model to predict the number of visitors in 2010. Slide 7 Using a Linear Model to Make a Prediction and an Estimate Using a Linear Model to Make Estimates and Predictions Year 2010 corresponds to t = 50: 2010 – 1960 = 50 Locate point on linear model for t = 50 The v-coordinate is approximately 5.6 The model estimates 5.6 million visitors in 2010 Example Solution

8. Lehmann, Intermediate Algebra, 3ed Section 2.1 Use a linear model to estimate the year there ware 4 million visitors. Slide 8 Using a Linear Model to Make a Prediction and an Estimate Using a Linear Model to Make Estimates and Predictions 4 million visitors corresponds to v = 4 The corresponding v-coordinate is approx. t = 32 According to the linear model, there were 4 million visitors in the year 1960 + 32 = 1992 Example Solution

9. Lehmann, Intermediate Algebra, 3ed whether a linear function would model it well. Situation 1 Close to line-describes a linear function Situation 2 & 3 Points do not lie close to one line A linear model would not describe these situations Section 2.1 Consider the scattergrams. Determine Slide 9 Deciding Whether to Use a Linear Function to Model Data When to Use a Linear Function to Model Data Situation 1 Situation 2 Situation 3 Example Solution

10. Lehmann, Intermediate Algebra, 3ed Section 2.1 The wild Pacific Northwest salmon populations are listed in the table for various years. 1. Let P be the salmon Slide 10 Intercepts of a Model; Model Breakdown Intercepts of a Model and Model Breakdown population (in millions) at t years since 1950. Find a linear model that describes the situation. Data is described in terms of P and t in a table Sketch a scattergram (see the next slide) Example Solution

11. Lehmann, Intermediate Algebra, 3ed Section 2.1Slide 11 Intercepts of a Model; Model Breakdown Intercepts of a Model and Model Breakdown 2.Find the P- intercept of the model. What does it mean? 3.Use the model to predict when the salmon will become extinct. Example Continued

12. Lehmann, Intermediate Algebra, 3ed Section 2.1Slide 12 Intercepts of a Model; Model Breakdown Intercepts of a Model and Model Breakdown P- intercept is (0, 13) When P = 13, t = 0 (the year 1950) According to the model, there were 13 million salmon in 1950 T-intercept is (45, 0) When P = 0, t = 45 (the year 1950 + 45 = 1995 Salomon are still alive today Our model is a false prediction Solution

13. Lehmann, Intermediate Algebra, 3ed Section 2.1Slide 13 Definition Intercepts of a Model and Model Breakdown For situations that can be modeled by a function whose independent variable is t: when we part of the model whose t-coordinates are not between the t-coordinates of any two data points. Definition We perform interpolation

14. Lehmann, Intermediate Algebra, 3ed Section 2.1Slide 14 Definition Intercepts of a Model and Model Breakdown We perform extrapolation when we use a part of the model whose t-coordinates are not between the t- coordinates of any two data points. When a model gives a prediction that does not make sense or an estimate that is not a good approximation, we say that model breakdown has occurred. Definition

15. Lehmann, Intermediate Algebra, 3ed Section 2.1Slide 15 Modifying a Model Intercepts of a Model and Model Breakdown In 2002, there were 3 million wild Pacific Northwest salmon. For each of the following scenarios that follow, use the data for 2002 and the data in the table to sketch a model. Let P be the wild Pacific Northwest salmon population (in millions) at t years since 1950. 1.The salmon population levels off at 10 million. 2.The salmon become extinct. Example

16. Lehmann, Intermediate Algebra, 3ed Section 2.1Slide 16 Modifying a Model Intercepts of a Model and Model Breakdown Solution