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In the 2 scenarios below, find the change in x and the change in y. What conclusions can you draw? What are the similarities & differences? How would you model this data? X1800185019001950200020503000 Y125150172198223247272 X Y X1800181018201830184018501860 Y100150175180172149102 X Y = “change”

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So far, the data we have worked with has a linear relationship We have discussed 3 forms of linear modeling: › Best Fit line › Least Squares Regression Line › Median-Median line

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The data we have used has a linear relationship. This means that the rate of change (the slope) is constant. For linear data, every increase in the independent variable (X) has a constant increase in the dependent variable (Y). y x = constantSlope =

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Data does not always follow a linear model. Data may increase sharply, reach a maximum, then decline. Or, it may decrease sharply, reach a minimum, then increase again. The data in the graph at the right is shaped like a parabola – so it would follow a quadratic model instead of a linear model.

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Recall that parabolas are graphs of quadratic equations. They follow the model of y = ax 2 + bx +c. › If a > 0, the parabola opens up (smiling) › If a < 0, the parabola opens down (frowning) Real-life examples of data that follows a quadratic pattern include: › Stock market (Peaks and Valleys) › Disease outbreaks (Black Plague, Polio, AIDS) › Particle motion (Ball trajectory, Draining water)

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Data can also follow an exponential model. Exponential data either › Increases exponentially, where the change in y continues to increase for each change in x OR › Decreases exponentially, where the change in y continues to decrease for each change in x.

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Examples of Real-Life data that follows an exponential model include: › Population growth (increasing) › National Debt (increasing) › Radioactive Decay (decreasing) Exponential equations follow the model y = a(b) x (where a and b are constants)

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To determine if data is linear, quadratic or exponential › Create a Scatterplot of the data and look for the overall pattern › Evaluate the change in Y for each change in X y x = Constant = Increasing, 0, then Decreasing or Decreasing, 0, then Increasing = Increasing exponentially (e.g., doubles every time) or Decreases exponentially (e.g., halves every time) IF LINEAR QUADRATIC EXPONENTIAL

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The table below lists the total estimated numbers of AIDS cases, by year of diagnosis from 1999 to 2003 in the United States (Source: US Dept. of Health and Human Services, Centers for Disease Control and Prevention, HIV/AIDS Surveillance, 2003.) Notice the data peaks in 2001, then drops off. This is a good indicator that Quadratic Regression will provide the most accurate model of the data Year AIDS Cases 199941,356 200041,267 200140,833 200241,289 200343,171

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1). Plot the data, letting x = 0 correspond to the year 1998. 2). Find a quadratic function that models the data. Using your calculator, enter the Year as L1 and #of Cases as L2 Use the QuadReg function on your calculator to calculate the regression equation 3). Plot the function on the graph with the data and determine how well the graph fits the data, 4). Use the model (equation) to predict the cumulative number of AIDS cases for the year 2006.

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Copyright © Cengage Learning. All rights reserved. 4 Quadratic Functions.

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