Course Project ~ Curve Fitting Green Diploma Date: ____________ Name:___________________________ Course:____________ Section:_______ Instructor:________________________ Curve Fitting in the Real World Overview and Outline Mathematics plays an important part in many aspects of our world. The purpose of the following project is to help you discover how mathematics is used to model the world around us. In this project, you will be reviewing data on topics related to the environment that spans a given set of amount of time (a ten year period or a span of months). After graphing the provided data, you will determine whether the data represented can be modeled by a linear, quadratic, exponential, or logarithmic function. Once you have generated the model for the data presented, you will then be able to make predictions based upon your model. This project consists of four parts. Part I: (Complete during or at the end of Chapter One) Sketch the Graphs (20 points - 5 pts each) Determine which set of data is best represented by a Linear Model (3 pts) Calculate the Linear Model (15 pts) Part II: (Complete during or at the end of Chapter Five) Determine which set of data is best represented by an exponential model (3 pts) Determine which set of data is best represented by a logarithmic model (3 pts) Calculate the exponential and logarithmic models (30 pts - 15 pts each) Part III: (Complete during or at the end of Chapter Six) Determine which set of data is best represented by a quadratic model (3pts) Calculate the Best Fit quadratic model (15 pts) Part IV: (Complete during or at the end of Chapter Six) Write a Formal Paper Addressing Results (8 pts) Part I: You will complete the following items in Part I of the Curve Fitting Project: A. Sketch the graphs representing the given environmental data B. Determine a Linear Model Curve Fitting Project ~ Part I Green Diploma page 1 of 4 PART A: (5 pts per graph) Use a sheet of graph paper to plot the points for each of the four topics. Each topic should be on a separate sheet of paper. This work needs to be done precisely. Looking at the plotted points will help you decide which type of function might be the best fit for your data. The label at the top of the graph paper should clearly indicate River Sediment, Abandoned Island Cat Population, Invasive Plant Species, or Seasonal Lake Levels. Label your axes clearly. Let x represent the number of years or months since your study began and y represent your data values. Big Deal! Pay Attention. Choose your scale for the y - axis very carefully. Explain your decision. Plot the points on the graph for the first eight data entries of the study only. We are saving the last two entries for something later. Do NOT include them on the graph at this time. Part B: Determine a Linear Model Throughout the semester we will be studying various functions and their graphs. Each time we study a new function, we will look at our four graphs to see if this new function might be a good \"fit\" for at least one set of our data. The first function we studied is the linear function. We learned that the graph of a linear function is a straight line. The line may be increasing, decreasing or constant. Look at the four graphs from part A. Which graph best resembles a linear function? Enter your answer below. Graph (and topic) resembling a linear function: __________________________________(3 pts) Use only this topic (graph and data) for the rest of part B. For the remainder of this portion of the course project, we are going to find a linear function that best \"fits\" our data points for the graph we chose. We are going to do this in two ways: 1. 2. By Hand (8pts) We will use the concepts we learned from class to determine a linear function that best \"fits\" the data. Using a Graphing Calculator (4pts) We will use Linear Regression on the calculator to find the linear function that actually is the best fit for our data. Conclusion (3 pts) Curve Fitting Project ~ Part I Green Diploma page 2 of 4 1. By Hand: ( 8pts) Choose two points on your graph that seem to best represent where you think the line should be. List them below. Two points: ______ ________ Now calculate the linear function that contains these two points. Show your work below. y2 y1 Slope: m m = ____ x2 x1 y y1 m( x x1 ) Best-Fit Linear Function by Hand: ____________________________ Calculation for years/months 9 & 10: Use the Best-Fit Linear Function derived by hand to determine what outcome you would predict for the last 2 years/months. Show your work below. Calculations: Results: Last 2 years/months of data: Prediction (from calculations above): Actual (from collected data): _____9______ ____________ ____________ ____10_______ _____________ _____________ 2. Using a Graphing Calculator ~ Linear Regression: (4pts) Use a graphing calculator and regression to model the data. Be sure to use the same data as above. Round to two decimal places. Remember to use only the first eight years/months. Y=ax+b a= b= Best-Fit Linear Function: __________________________ Curve Fitting Project ~ Part I Green Diploma page 3 of 4 Calculation for years/months 9 & 10: Use the Best-Fit Linear Function derived by the calculator to determine what outcome you would predict for the last 2 years/months. Show your work below. Calculations: Results: Last 2 years/months of data: Prediction (from calculations above): Actual (from collected data): _____9______ ____________ ____________ ____10_______ _____________ _____________ Conclusion: Which best-fit function was the most accurate in making the predictions (by hand or calculator)? ( 3pts) Enter it here. Best-Fit Linear Function __________________________________ Note: This will be used in the formal paper you will write. Jot down notes about how reliable this will be to use for predicting into the future. Curve Fitting Project ~ Part I Green Diploma page 4 of 4