Determine a Best Fit Equation for Quadratic Regression

Transcription

1 Algebra 2 Trig/Apps Lesson #44 Objective: Determine a Best Fit Equation for Quadratic Regression Warm Up A dud missile is fired straight into the air from a military instillation. The missile s height is given by the formula; h(t) = -16t t a. How high is the missile after 4.5 seconds? b. At what time will the missile reach its maximum height? c. What is the maximum height the missile will reach? d. When will the missile be 2,500 feet above the ground? e. When will the missile be 100 feel above the ground? "Regression" is a peculiar word that means you have a bunch of points, and you want to find a particular curve that most closely fits them. (For instance, you have done an experiment, and are now looking for the function that best models your data.) Example 1 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.) 1. Plot the data, letting x = 0 correspond to the year 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 Since 1998 corresponds to x = 0, the year 1999 will represent x =1, 2000 will represent x = 2, etc.

2 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 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 above, it is clear that a quadratic is not a perfect function for representing this data. We know that R 2 = , so r A graph is a perfect fit for data when r = 1. 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.

3 Using our model y = x x to predict the cumulative number of AIDS cases for the year 2006, we find that we expect that there will approximately 51,347 cumulative AIDS cases diagnosed in the year Practice/HW: 1. 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 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. Equation What would be the height of the ball if it travels 80 meters? 2. 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: Eequation: What is the concentration of medicine in the blood after 4 hours have passed? 3. At 1821 feet tall, the CN Tower in Toronto, Ontario, is the world s tallest self-supporting structure. 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. Equation Where is the penny located after falling for a total of 10.5 seconds?

4 4. The table blow lists the number of Americans (in thousands) who are expected to be over 100 years old for selected years. (Source: US Census Bureau.) Equation: How many Americans will be over 100 years in the year 2020? 5. A golf ball is hit down a straight fairway. The following table shows the height of the ball with respect to time. The ball is hit at an angle of 70 with the horizontal with a speed of 40 meters/sec. Time (sec) Height (meters) Equation: What is the height of the ball at 5.2 seconds?

5 Algebra 2 Trig/Apps Lesson #45 Objective: Determine a Best Fit Equation for Quadratic Regression Warm Up For each of the following data sets, determine whether the data can be modeled using a quadratic equation. If so, use your calculator compute the quadratic regression and write the result in the space provided.