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In order to solve problems involving quadratic regression, it is necessary to

Quadratic Regression is a process by which the equation of a parabola is found that “best fits” a given set of data. Let's look at an example of a quadratic regression problem. 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.)

 Year AIDS Cases 1999 41,356 2000 41,267 2001 40,833 2002 41,289 2003 43,171
Our approach will be to:

1. plot the data, letting x = 0 correspond to the year 1998,
2. find a quadratic function that models the data,
3. plot the function on the graph with the data and determine how well the graph fits the data,
4. use the model to predict the cumulative number of AIDS cases for the year 2006.

First we will plot the data using a TI-83 graphing calculator. Since 1998 corresponds to x = 0, the year 1999 will represent x = 1, 2000 will represent x = 2, etc. Once the data is entered, your screen should look like the following: After entering the data into the calculator, graph the data. Your screen should look like the following: Next we want to find a quadratic equation that best fits the data we have plotted. According to the calculator, the equation is the following: The graph of the function is the following: Based on the graph and the equation information listed above, it is clear that a quadratic is not a perfect function for representing this data. We know that R= 0.903486496, so . Remember that a graph is a perfect fit for data when . However, based on the graph, our function is a fair fit for the given data. It would be better to have more data so that we could determine a graph having a better fit.

Using our model to predict the cumulative number of AIDS cases for the year 2006, we find that we expect that there will be approximately 51,347 cumulative AIDS cases diagnosed in the year 2006.

Examples On Tuesday, May 10, 2005, 17 year-old Adi Alifuddin Hussin won the boys’ shot-putt gold medal for the fourth consecutive year. His winning throw was 16.43 meters. A shot-putter throws a ball at an inclination of 45° to the horizontal. The following data represent approximate heights for a ball thrown by a shot-putter as it travels a distance of x meters horizontally.

 Distance(m) Height(m) 7 8 20 15 33 24 47 26 60 24 67 21

What would be the height of the ball if it travels 80 meters?  The concentration (in milligrams per liter) of a medication in a patient’s blood as time passes is given by the data in the following table:

 Time(Hours) Concentration(mg/l) 0 0 0.5 78.1 1 99.8 1.5 84.4 2 50.1 2.5 15.6

What is the concentration of medicine in the blood after 4 hours have passed? Examples At 1821 feet tall, the CN Tower in Toronto, Ontario, is the world’s tallest self-supporting structure. [Note: This information is taken from College Algebra: A Graphing Approach by Larson, Hostetler, & Edwards (Third Edition), page 202.]

Suppose you are standing in the observation deck on top of the tower and you drop a penny from there and watch it fall to the ground. The table below shows the penny’s distance from the ground after various periods of time (in seconds) have passed. Where is the penny located after falling for a total of 10.5 seconds?

 Time(seconds) Distance(feet) 0 1821 2 1757 4 1565 6 1245 8 797 10 221  The table below lists the number of Americans (in thousands) who are expected to be over 100 years old for selected years. [Source: US Census Bureau.] How many Americans will be over 100 years old in the year 2008?

 Year Number (thousands) 1994 50 1996 56 1998 65 2000 75 2002 94 2004 110 As you can see, these types of problem require that you use a graphing calculator and a modeling approach. You must correctly enter the data into your calculator, graph the data, calculate an equation that best fits the data, graph that equation, and then make the prediction asked for in the problem. At the conclusion of the problem, you should always check for the reasonableness of your solution.

D Saye

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