chapter1 modelingwith linear functions（chapter 1 modelingwith linear functions）.doc

2.1 Using Lines to Model Data Scattergram and Linear Models The graph of plotted data pairs is called a scattergram. A linear model is a straight line or an equation that describes the relationship between two quantities for a true-to-life situation. Year Years since 1960 t Number of Visitors (millions) v 1960 1970 1980 1990 2000 1.2 2.3 2.6 3.8 4.8 Example 1 Let v represent the number of visitors (in millions) to the Grand Canyon in the year that is t years since 1960. a. Fill-in the table of values for t. Identify the dependent and independent variables. b. Make a scattergram of the data. c. Use a ruler to draw (“eyeball”) a line (linear model) that fits the data well. As always, label and scale both axes. d. Use the linear model to estimate the number of visitors in 2010 (extrapolation). e. Use the linear model to estimate in what year there will be 4 million visitors to the Grand Canyon (interpolation). Interpolation, Extrapolation Model Breakdown Interpolation is making prediction within the data given. Extrapolation is making a prediction outside the data given. When a model yields a prediction that does not make sense or an estimate that is not a good approximation, we say that model breakdown has occurred. Model breakdown mostly occurs when trying to make an estimate outside the range given in the data (called extrapolation).Example 2 Year t Number of Prozac Prescriptions (millions) 1989 1991 1993 1995 1996 6.1 10.0 12.2 18.8 20.7 Prozac is an antidepressant that was approved by the FDA in 1987. Let p be the number of prescriptions of Prozac (in millions) dispensed at t years since 1980. a. Fill-in the table of values for t. b. Identify the dependent and independent variables. c. Make a scattergram of the data. d. Use a ruler to draw (“eyeball”) a line (linear model) that fits the data well. As always, label and scale both axes. e. Use the linear model to estimate the number of Prozac prescriptions in 1994.