Polynomial and Piecewise Functions

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1 Unit 1 Polynomial and Piecewise Functions Introduction Prior to the use of technology, polynomials were the most widely applied models because they could be calculated with paper and pencil. Given the use of graphing calculators like the FX 2.0, we no longer have this restriction. We use polynomial functions for models only when they apply. Looking at a scatter plot of data, we use linear functions when the points appear to lie in a straight line to model constant rates of change. If the scatter plot is curved but has no inflection point, a quadratic function might be used to model a consistent force of change. If the scatter plot has an inflection point, a cubic or logistic model might be appropriate. 1 In all cases, knowledge of the nature of the forces or dynamics of the data should be part of the considerations and model selection. In many situations a scatter plot may indicate a combination of two or more shapes. When this happens, a piecewise-defined function might be appropriate to model the data. Students typically have difficulty with the concept of piece-wise defined functions and for this reason, we have included a hands-on activity in this unit which might be useful as an introduction to piece-wise defined functions. It also provides a quick review of both linear and quadratic models. Problem 1 2 A herd of 100 mule deer is introduced on a small island off the coast of South Carolina. At first the herd increases rapidly, but eventually food plants are consumed and destroyed. Then the population declines to extinction. Since the dynamics suggest that the number of deer will increase rapidly, pass through an inflection point and then experience a very rapid decline to extinction, we find that the real-life situation can effectively be modeled by a quartic function. Since we know that the population will likely double before it goes into sharp decline to extinction in about 5 years we model the population p(t) of deer after t years by Copyright? 2000 by Clemson U. & Casio, Inc. Unit 1-1 Clemson Calculus Project

2 p(t) = -t t What would the domain and range of the function be? How would this compare with the domain and range of the real situation? Explain your choices considering the continuous nature of the quartic function and the discrete nature of the mule deer data. 2. What are the intercepts of the function? Use the equation solver capability of your calculator. Explain their meanings in terms of the population of mule deer on the island. 3. Is the population function even, odd or neither? Explain what your conclusion means in terms of the graph of your function. 4. Use the table capability of your calculator to investigate the end behavior of the population function. What appears to be happening to the value of the function as x approaches?? 5. Graph the population function using an appropriate domain and range in the window. One Solution 1. Answers with reasonable justifications are acceptable. t? 0 and p(t)? 0 would be an acceptable answer because negative values of the number of years and the population of deer have no meaning. 2. The x-intercepts are? 5. The population of deer on the island will be extinct in slightly less than 5 years. The y-intercept is 100, the initial population of deer. 3. Choose CAS on the menu and substitute (-t) for t in the expression. p(t) is even because p(-t) = p(t) The graph of the population function is symmetric with respect to the y-axis. Note that this is an analytic property of the mathematical model and has no special meaning in the real world of deer populations. 4. The population function approaches -? as the number of years approaches?. This has no meaning, of course, in terms of the population of deer. Copyright? 2000 by Clemson U. & Casio, Inc. Unit 1-2 Clemson Calculus Project

3 5. Problem 2 3 Piece It Together For this activity, the class should be divided into groups of 4 or 6. Each group will need scissors, grid paper, and patty paper (the waxed paper that many restaurants use to separate hamburgers). Consider the following data set: Copyright? 2000 by Clemson U. & Casio, Inc. Unit 1-3 Clemson Calculus Project

4 Enter the data into the lists and construct a scatter plot. 2. Describe the data. 3. Divide your group into 2 sub-groups, A and B. Sub-group A should work together to guess a function that would model the data from 1990 to Sub-group B should work together to guess a function that would model the data from 1996 to Graph your functions on grid paper. 5. Sub-group A: Trace your graph on to the patty paper. Sub-group B: Trace your graph onto the patty paper. 6. Pair up with a classmate from the other sub-group and place your patty papers together. You are literally piecing together a new type of function. 7. Have your calculator choose the best-fit graph for each of the two functions. Compare your graphs with the ones on the calculator. How well did your group do? Graph the piecewise-defined function on the calculator. 8. As a group, create a real-life situation that our piecewise function would model. Produce a list of questions (and solutions) like those in Problem 1 for your situation. One Solution 1. Let the independent variable be the years since 1990 (0,1,2, 9). 2. The data appears to be divided into two distinct sections. From 1990 to 1995 the data are linear in nature. From 1996 to 1999 the data appear to be quadratic. Copyright? 2000 by Clemson U. & Casio, Inc. Unit 1-4 Clemson Calculus Project

5 3. Considering the linear data, the y-intercept is 23 and there is roughly one unit change in y for each unit change in x so a good first guess would be f(x) = x + 23.?? Press Def G?? Enter y 1 = x + 23?? Press Draw. y =.9x + 23 would be a better guess and would more closely model the data. Calculate the linear regression. Press DRAW. Copyright? 2000 by Clemson U. & Casio, Inc. Unit 1-5 Clemson Calculus Project

6 Considering the quadratic nature of data from 1996 to 1999, a good first guess would be the parabola with vertex at (6,19) or y? ( x? 6) 2? 19. Changing to y? 1.1( x? 6) 2? 19 would be a little closer and y? 1.3( x? 6) 2? 19 would be even closer to the data points. 4. Grid Paper. Graphs will vary, but should be close to the ones in # Paddy Paper 6. Paddy Paper 7. Separate the data into two lists; one from 1990 to 1995, the other from 1996 to Perform the linear regression on the first set of data and the quadratic regression on the second set of data. Compare to your guesses. Copyright? 2000 by Clemson U. & Casio, Inc. Unit 1-6 Clemson Calculus Project

7 8. Answers will vary. They should come up with some situations that would change drastically in 1995, and then recover quickly. Problem 3 4 My daughter and her roommate at the College of Charleston have determined that cable is too expensive for them this year. According to the Cable TV Financial Databook, the average monthly basic rate R for cable television in the United States is given for the years 1985 through 1993 in the following table: Year Basic Rate Copyright? 2000 by Clemson U. & Casio, Inc. Unit 1-7 Clemson Calculus Project

8 1. Construct a scatter plot of the data. Describe the data. 2. Find a cubic regression equation to fit the data. 3. What would constitute an appropriate domain and range for the function? 4. Interpret the meaning of the intercepts considering the nature of the data. 5. Is the function even, odd or neither. Explain. 6. Describe the behavior of the function as x???. 7. Meghan and Aven reported that the monthly basic cable rate in Charleston, SC is $65. How does this data point compare with its corresponding estimate on the model? Do you think our model should be used to predict future cable rates? Why or why not? Problem 4 After picking up a few rather lucrative mowing jobs over the summer break, you and your best friend decide to start a landscape management business. After a few months of planning you are up and running your new Mow More business. The table below shows the monthly revenue for your first year: January 93 February 115 March 144 April 177 May 223 June 274 July 335 August 425 September 531 October 550 November 569 December Enter the data. Describe it. Copyright? 2000 by Clemson U. & Casio, Inc. Unit 1-8 Clemson Calculus Project

9 2. Work with a partner to define a piecewise function that would model the data. 3. What would constitute an appropriate domain and range? 4. Interpret the intercepts in the context of the problem. 1 LaTorre, D., Kenelly, J., Fetta, I., Carpenter, L., & Harris, C. (1998), Calculus Concepts: An Informal Approach to the Mathematics of Change. Boston, New York: Houghton Mifflin Company. 2 Swokowski, E. & Cole, J. (1994), Precalculus: Functions and Graphs. Boston: PWS Publishing Company. 3 Adapted from Piecing Together Piecewise Functions, Mathematics Teacher, October, Larson, R. & Hostetler, R. (1997), Precalculus. Boston, New York: Houghton Mifflin Company. Copyright? 2000 by Clemson U. & Casio, Inc. Unit 1-9 Clemson Calculus Project

UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions. Instruction. Guided Practice Example 1

UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions. Instruction. Guided Practice Example 1 Guided Practice Example 1 A local store s monthly revenue from T-shirt sales is modeled by the function f(x) = 5x 2 + 150x 7. Use the equation and graph to answer the following questions: At what prices