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Modeling with Linear Functions

Chapter 2. Modeling with Linear Functions. Section 2.1. Using Lines to Model Data. Section 2.1. Slide 3. Using Lines to Model Data. Scattergrams. Example. The number of Grand Canyon visitors is listed in the table for various years. Describe the data. Solution.

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Modeling with Linear Functions

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  1. Chapter 2 Modeling with Linear Functions

  2. Section 2.1 Using Lines to Model Data

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

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

  5. Section 2.1 Slide 5 Definitions Linear Models 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. Definition

  6. Section 2.1 Slide 6 Definitions Linear Models Definition 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 Property

  7. Section 2.1 Slide 7 Using a Linear Model to Make a Prediction and an Estimate Using a Linear Model to Make Estimates and Predictions Example Use a linear model to predict the number of visitors in 2010. Solution • 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

  8. Section 2.1 Slide 8 Using a Linear Model to Make a Prediction and an Estimate Using a Linear Model to Make Estimates and Predictions Example Use a linear model to estimate the year there ware 4 million visitors. Solution • 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

  9. Section 2.1 Slide 9 Deciding Whether to Use a Linear Function to Model Data When to Use a Linear Function to Model Data Example Consider the scattergrams. Determine Situation 1 Situation 2 Situation 3 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 Solution

  10. Section 2.1 Slide 10 Intercepts of a Model; Model Breakdown Intercepts of a Model and Model Breakdown Example The wild Pacific Northwest salmon populations are listed in the table for various years. 1. Let P be the salmon 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) Solution

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

  12. Section 2.1 Slide 12 Intercepts of a Model; Model Breakdown Intercepts of a Model and Model Breakdown Solution • 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

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

  14. Section 2.1 Slide 14 Definition Intercepts of a Model and Model Breakdown Definition 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. Definition 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.

  15. Section 2.1 Slide 15 Modifying a Model Intercepts of a Model and Model Breakdown Example 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.

  16. Section 2.1 Slide 16 Modifying a Model Intercepts of a Model and Model Breakdown Solution

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