Quadratic Functions I

Transcription

1 Quadratic Functions I by Frank C. Wilson Activity Collection Featuring real-world contexts: Autism Awareness Autism Awareness Business Growth - USAA Changing Population Kentucky Changing Population - New York Changing Population South Carolina Falling Objects - Empire State Building Falling Objects - Sears Tower United States Population Using the Body Mass Index Formula

2 29 by Make It Real Learning Company With the purchase of this workbook, license is granted for one (1) teacher to copy the activities in this workbook for use in classes and professional development workshops. Copying pages in this workbook for any other use is prohibited without written consent from Make It Real Learning Company. For permissions, visit and complete the Contact Us form.

3 Table of Contents Introduction... 4 Activity Objectives... 5 Autism Awareness: Creating and Using Quadratic Models... 6 Solutions... 8 Autism Awareness: Solving Quadratic Equations... 1 Solutions Business Growth - USAA: Quadratic Function Modeling Solutions Changing Population Kentucky: Analyzing and Modeling Data Solutions... 2 Changing Population - New York: Modeling with Quadratics Solutions Changing Population South Carolina: Analyzing and Modeling Data Solutions Falling Objects - Empire State Building: Working with Quadratics... 3 Solutions Falling Objects - Sears Tower: Working with Quadratics Solutions United States Population: Using Quadratic Models Solutions... 4 Using the Body Mass Index Formula: Solving Equations Solutions About the Author Other Books in the Make It Real Learning Series... 46

4 Introduction When am I ever going to use this? It is a question that has plagued teachers and learners for decades. Now, with the help of the Make It Real Learning workbook series, you can answer the question. The Quadratic Functions I workbook focuses on real-world situations that may be effectively analyzed using quadratic function models. From modeling the United States population to predicting the business growth of a company, learners get to use mathematics in meaningful ways. Rest assured that each activity integrates real world information not just realistic data. These are real companies (e.g. USAA) and real world issues. The mathematical objectives of each activity are clearly specified on the Activity Objectives page following this introduction. Through the workbook series, we have consistently sought to address the content and process standards of the National Council of Teachers of Mathematics. There are multiple ways to use the activities in a teaching environment. Many teachers find that the activities are an excellent tool for stimulating mathematical discussions in a small group setting. Due to the challenging nature of each activity, group members are motivated to brainstorm problem solving strategies together. The interesting real world contexts motivate them to want to solve the problems. The activities may also be used for individual projects and class-wide discussions. As a ready-resource for teachers, the workbook also includes completely worked out solutions for each activity. To make it easier for teachers to assess student work, the solutions are included on a duplicate copy of each activity. We hope you enjoy the activities! We continue to increase the number of workbooks in the Make It Real Learning workbook series. Please visit for the most current list of activities. Thanks! Frank C. Wilson Author 4

5 Quadratic Functions I Activity Objectives Activity Title Autism Awareness: Creating and Using Quadratic Models (p. 6) Autism Awareness: Solving Quadratic Equations (p. 1) Business Growth - USAA: Quadratic Function Modeling (p. 14) Changing Population Kentucky: Doing Data Analysis and Modeling (p. 18) Changing Population - New York: Modeling with Quadratics (p. 22) Changing Population South Carolina: Doing Data Analysis and Modeling (p. 26) Falling Objects - Empire State Building: Working with Quadratics (p. 3) Falling Objects - Sears Tower: Working with Quadratics (p. 34) United States Population: Using Quadratic Models (p. 38) Using the Body Mass Index Formula: Solving Equations (p. 42) Mathematical Objectives Calculate and interpret the meaning of average rates of change Use rates of change to forecast unknown results Evaluate a quadratic function model at a given value Solve a quadratic equation with the Quadratic Formula Find the x-intercepts and vertex of a parabola from its equation Calculate and analyze trends in rates of change Draw a scatter plot of a data set Evaluate a quadratic function Determine if a linear or quadratic function best fits a data set Solve a quadratic equation with the Quadratic Formula Explain the relationship between rates of change and concavity Determine if a linear or quadratic function best fits a data set Find the vertex of a quadratic function algebraically Determine if a linear or quadratic function best fits a data set Evaluate a quadratic function at a given value Solve a quadratic equation with the Quadratic Formula Use a quadratic model for the position of a falling object Create a linear equation for the velocity of a falling object Use the quadratic formula to find x-intercepts Use a quadratic model for the position of a falling object Create a linear equation for the velocity of a falling object Use the quadratic formula to find x-intercepts Solve a quadratic equation with the Quadratic Formula Evaluate a quadratic function at a given value Estimate the point of intersection of two graphs visually Evaluate a multivariable function at a given point Solve linear and quadratic equations 5

6 Autism Awareness Creating and Using Quadratic Models Autism is a developmental disorder of the brain that is often characterized by impaired social interaction, limited communication, and repetitive behavior. With early intervention, some children with milder forms of autism are able to grow into self-sufficient adults. However, individuals with more severe forms of the disorder may need assistance for the rest of their lives. The table below shows the number of reported cases of autism between 1995 and 25. Years Children with Autism (since 1995) (Thousands) Source: Statistical Abstract of the United States: 28, Table Calculate the average rate of change in number of children with autism for each consecutive two-year period : = 6.7 thousand cases per year : thousand cases per year : = 16.3 thousand cases per year 21 3: thousand cases per year 23 5: thousand cases per year 2. Explain what the results from (1) tell us about the number of reported cases of autism. The number of reported cases of autism is increasing at an increasing rate. 6

7 3. Calculate the change in each consecutive pair of rates of change from (1). Based on the result, what type of function will fit the data well? Explain. We look at the changes in the rates of change: 6.7 to 11.8 is a change of 5.1; 11.8 to 16.3 is a change of 4.5; 16.3 to 21.3 is a change of 5; 21.3 to 26.3 is a change of 5. Since the change in the rates of change is close to being constant, a quadratic function will fit the data well. The average of the changes in the rates of change is = Based on the pattern of the rates of change in (1) and the observations from (3), predict the number of reported cases of autism for 27. In (3) we observed that the rates of change are increasing by about 4.9 thousand cases per year each year. Since the average rate of change for 23 5 is 26.3 thousand cases per year, we predict the average rate of change for 25 7 to be 4.9 thousand cases per year more than this. That is, 31.2 thousand cases per year. Since the number of cases in 25 is thousand and the rate of change for 25 7 is 31.2 thousand cases per year, 27 cases = 25 cases + 2( 25 to 27 rate of change) = ( 31.2) = = 256. We predict 256. thousand cases of childhood autism for Use quadratic regression to find a model for the data. Then use the model to predict the number of autism cases in 27. Let a be the number of autism cases (in thousands) and t be the number of years since The 2 quadratic regression model is at ( ) = 1.21t t Since t = 12 corresponds with 27, we must find a ( 12). 2 a (12) = 1.21( 12) ( 12) Using the model, we predict thousand autism cases in How can the mathematical model in (5) be used to benefit society? Explain. People with autism may need special support services such as physical therapy, occupational therapy, speech therapy, special education, and so on. The mathematical model can be used to help government planners budget appropriately to meet the needs of children with autism. 7

8 Autism Awareness Creating and Using Quadratic Models Autism is a developmental disorder of the brain that is often characterized by impaired social interaction, limited communication, and repetitive behavior. With early intervention, some children with milder forms of autism are able to grow into self-sufficient adults. However, individuals with more severe forms of the disorder may need assistance for the rest of their lives. The table below shows the number of reported cases of autism between 1995 and 25. Years Children with Autism (since 1995) (Thousands) Source: Statistical Abstract of the United States: 28, Table Calculate the average rate of change in number of children with autism for each consecutive two-year period : = 6.7 thousand cases per year : thousand cases per year : = 16.3 thousand cases per year 21 3: thousand cases per year 23 5: thousand cases per year 2. Explain what the results from (1) tell us about the number of reported cases of autism. The number of reported cases of autism is increasing at an increasing rate. 8

9 3. Calculate the change in each consecutive pair of rates of change from (1). Based on the result, what type of function will fit the data well? Explain. We look at the changes in the rates of change: 6.7 to 11.8 is a change of 5.1; 11.8 to 16.3 is a change of 4.5; 16.3 to 21.3 is a change of 5; 21.3 to 26.3 is a change of 5. Since the change in the rates of change is close to being constant, a quadratic function will fit the data well. The average of the changes in the rates of change is = Based on the pattern of the rates of change in (1) and the observations from (3), predict the number of reported cases of autism for 27. In (3) we observed that the rates of change are increasing by about 4.9 thousand cases per year each year. Since the average rate of change for 23 5 is 26.3 thousand cases per year, we predict the average rate of change for 25 7 to be 4.9 thousand cases per year more than this. That is, 31.2 thousand cases per year. Since the number of cases in 25 is thousand and the rate of change for 25 7 is 31.2 thousand cases per year, 27 cases = 25 cases + 2( 25 to 27 rate of change) = ( 31.2) = = 256. We predict 256. thousand cases of childhood autism for Use quadratic regression to find a model for the data. Then use the model to predict the number of autism cases in 27. Let a be the number of autism cases (in thousands) and t be the number of years since The 2 quadratic regression model is at ( ) = 1.21t t Since t = 12 corresponds with 27, we must find a ( 12). 2 a(12) = 1.21( 12) ( 12) Using the model, we predict thousand autism cases in How can the mathematical model in (5) be used to benefit society? Explain. People with autism may need special support services such as physical therapy, occupational therapy, speech therapy, special education, and so on. The mathematical model can be used to help government planners budget appropriately to meet the needs of children with autism. 9