Linearizing Data. Lesson3. United States Population


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1 Lesson3 Linearizing Data You may have heard that the population of the United States is increasing exponentially. The table and plot below give the population of the United States in the census years Year Population (in millions) Source: United States Bureau of the Census. United States Population Population (in millions) Year 18 UNIT 3 LOGARITHMIC FUNCTIONS AND DATA MODELS
2 Think About This Situation Examine the pattern of U.S. population growth in the table and plot on the previous page. a b c Does it appear that the population of the United States is increasing linearly, exponentially, quadratically, or cubically? How can you decide? What method does your calculator or computer use to fit a linear function to these data? What method do you think a calculator or computer might use to fit an exponential function to these data? A power function? INVESTIGATION 1 Straightening Functions In this investigation, you will learn how transforming data can change the shape of its graph. You also will investigate how transforming data can help you decide on an appropriate model as in the case of the preceding Think About This Situation. 1. To begin, consider the data and scatterplot below that give the radii in inches of some circular tables and the surface areas in square inches of the tabletops. Surface Area Radius (in square (in inches) inches) , , ,89.56 Surface Area (in square inches) 2, 1,5 1, Radius (in inches) 25 a. Find the surface area of a table with a 15inch radius. b. Write an equation that expresses the relationship between radius x and surface area y of a circular tabletop. Describe the shape of its graph. c. Now transform each surface area from square inches to square feet. Predict what the plot of (radius in inches, surface area in square feet) will look like. Check your prediction by making a plot. LESSON 3 LINEARIZING DATA 181
3 d. Will transforming the surface area from square inches to square centimeters change the basic shape of the plot? Explain your reasoning. e. Converting square inches to square centimeters, or square inches to square feet requires multiplying or dividing by a constant. In general, will such a transformation of the y values of an (x, y) data plot change the basic shape of the plot? Why or why not? In the following activities, you will learn how to find a transformation y y* that linearizes (x, y) data; that is, when the points (x, y*) are plotted, they cluster on or about a line. 2. Examine the function y = πx 2, x, and the transformation y* = y. a. Graph the equation y = πx 2, x. b. Complete a copy of the table below. Include your own choice of x in the last row. x y = πx 2 y*= y c. Make a plot of the (x, y*) points. Are the points collinear? If so, write a linear equation expressing y* as a function of x. d. How is the slope of the graph of the equation from Part c related to the equation y = πx 2? e. What is the relationship between the original function and the linearizing (square root) function? 3. A transformation of the form y* = log y is called a logarithmic or log transformation. Consider the exponential function y = 2 x and the log transformation y* = log y. a. Graph the equation y =2 x. b. Complete a copy of the table on the next page. Include your choice of input in the last row. 182 UNIT 3 LOGARITHMIC FUNCTIONS AND DATA MODELS
4 x y =2 x y* = log y c. Make a plot of the (x, y*) points in the table above. Are the points collinear? If so, write a linear equation expressing y* as a function of x. d. How is the slope of the graph of the equation from Part c related to the equation y =2 x? e. What is the relationship between the original function y =2 x and the linearizing (log) function? 4. Suppose a set of points (x, y) falls along the graph of y = 4(5 x ). a. Graph the equation y = 4(5 x ). b. How can you transform each value of y to linearize the set of points? c. What is the relationship between the original function and the linearizing function? d. Write an equation that expresses the relationship between x and y*. What type of equation is this? e. How is the slope of the graph of the equation from Part d related to the equation y =4(5 x )? 5. Now suppose a set of points (x, y) falls along the graph of y = 6 x. a. Graph this equation. b. How can you transform each value of y to linearize the set of points? c. What is the relationship between the original function and the linearizing function? d. Imagine a plot of the transformed points. Along the graph of what equation would the transformed points fall? What type of equation is this? e. How is the slope of the graph of the equation from Part d related to the equation y = 6 x? In the next activity, you will discover how transforming data can help you determine which type of equation is a reasonable model for a set of paired data. LESSON 3 LINEARIZING DATA 183
5 6. For a science experiment, students have dropped a small weight and measured how far it falls in various lengths of time. Their data are given in the table below. Time Distance Fallen (in seconds) (in centimeters) a. Based on your work in previous courses, what type of function do you think would be a good fit for these data? Compare your conjecture to those of other groups. b. Based on your answer to Part a, what transformation y* would linearize the (time, distance fallen) data? Check your conjecture by making a scatterplot of the (time, y*) data. c. When given the (time, distance fallen) data, students in a class in California thought an exponential model would be reasonable. Compute the log of each distance fallen. Make a scatterplot of (time, log distance fallen) data. Explain why an exponential function is not a good fit. d. Find the power function that best fits the (time, distance fallen) data. Use symbolic reasoning to determine the time it would take for the weight to drop 1 cm. 7. Think back on your work with the data transformations y* = log y and y* = y. a. If a set of points (x, y) cluster on or about the graph of an exponential equation y = a(b x ), what is the equation of the graph the points (x, log y) will cluster on or about? b. If a set of points cluster on or about the graph of a quadratic power rule y = ax 2, x, what is the equation of the graph the points (x, y ) will cluster on or about? 184 UNIT 3 LOGARITHMIC FUNCTIONS AND DATA MODELS
6 Checkpoint In this investigation, you have learned how to transform data to check whether a set of points is modeled reasonably well by an exponential equation or by a quadratic power rule. a b c How would you check if an exponential function is a reasonable fit to a set of data points (x, y)? How would you check if a quadratic power function is a reasonable fit to a set of data points (x, y)? What is the relationship between a function and its linearizing function? Be prepared to explain your methods and thinking to the entire class. On Your Own The table and plot below give the estimated population of the world for various years. Estimated Population Year (in millions) , , , , ,8 Source: World Almanac and Book of Facts, 21. World Population Estimated World Population (in millions) 6, 4, 2, Year 2 a. Make a scatterplot of (x, log y). b. Is the population of the world increasing exponentially, slower than exponentially, or faster than exponentially? Explain. LESSON 3 LINEARIZING DATA 185
7 INVESTIGATION 2 Fitting Exponential Functions Using Log Transformations In previous courses, you have used least squares regression to fit a line to data that has a linear pattern. You may recall that the least squares regression line is the one that minimizes the sum of the squared vertical distances from the data points to the line. These distances are called residuals. To find the equation of the least squares regression line, your calculator or computer uses an algorithm based on calculus. How this algorithm works for small data sets is the focus of Extending Tasks 4 and 5. If you take a calculus course, where you study general methods to find minimum and maximum values of functions, you will learn more about the mathematics of the algorithm. For now, think about the meaning of least squares regression as you complete the following two activities. 1. The graph below displays a set of points and the least squares regression line, y = 1.6x, for the points a. Find the residual for each of the points. Show each residual on a copy of the plot above. b. What is the sum of the squared residuals? 2. One of the two lines on the graph below is the least squares regression line for the given set of four points. Use the idea of least squares to determine which line it is. (Do not use your calculator to compute the regression line.) line A line B UNIT 3 LOGARITHMIC FUNCTIONS AND DATA MODELS
8 Now that you have reviewed the meaning of least squares regression, you are ready to explore how your calculator uses the method of least squares to fit an exponential function to a set of data points. 3. Examine the ordered pairs and plot below: 2 x y Consider these two candidates for a function that fits these points. y = 2 x y =.5423( x ) a. Graph these two functions on a plot of the points. Does one appear to fit the points better than the other? b. Find the sum of the squared residuals for each function by completing two copies of the table below. Record at least three decimal places in each step. x y Predicted ŷ Residual (y ŷ) Squared Residual (y ŷ) c. Which function has the smaller sum of squared residuals? d. One of these functions is the exponential function given by your calculator. Explain which one you think it will be and why. e. Check your conjecture in Part d using the exponential regression feature of your calculator. 4. In Activity 3, you found that your calculator did not give the function with the smaller sum of squared errors. In this activity, you will find out what criterion the calculator does use to fit an exponential function. a. If an exponential function is a good fit for a set of data, what transformation would you use to linearize the data? LESSON 3 LINEARIZING DATA 187
9 b. For the points in Activity 3, make a plot of (x, log y). What pattern do the points seem to follow? c. Since the transformed points are reasonably linear, you can find the least squares linear regression line for the points (x, y*). Do this using your calculator. Write the equation in the form y* = a + bx. d. Substitute log y for y* in your equation from Part c and then solve for y. Where have you seen this equation before? e. Describe how your calculator actually does use the principle of least squares to fit an exponential function. Checkpoint In this investigation, you examined how a graphing calculator fits an exponential function to data. a b What is the purpose of transforming data? Write a summary of the method your calculator appears to use to fit exponential functions to data. At what stage does the calculator minimize the sum of the squared residuals? Be prepared to share your summary and thinking with the class. There are several reasons for fitting functions to data. One is that you want a model for the situation that you can use to build a theory. For example, you might want to describe under what situations a population increases exponentially. The most important thing to consider when fitting a function to data for this purpose is whether there is any contextual, scientific, or theoretical reason why the data might follow that model. You should not just go searching for any function that comes close to the points. After all, if you look hard enough, you are bound to find some function that comes reasonably close to the points in any set of data. Another reason for fitting a function to data is that you would like to use that function for estimation purposes. For example, suppose you just want a reasonable estimate of the United States population in the year By fitting several different types of functions to the populations known from the census years and selecting the one that seems to fit the best, you would get a reasonable estimate. If you select an exponential function and the United States population isn t in fact growing exponentially, your estimate for the year 1955 would not be off by much. However, if the population isn t growing exponentially and you were to use that exponential function to predict the United States population in the year 25, you might be off by a lot. 188 UNIT 3 LOGARITHMIC FUNCTIONS AND DATA MODELS
10 On Your Own Reproduced below is the United States population data reported at the beginning of this lesson. Population Year (in millions) a. Let x = correspond to the year 19, x = 1 correspond to the year 191, and so on. Make a scatterplot of (year, log population). Does an exponential function appear to be a reasonable fit to the original data? b. Use your calculator to fit an exponential function to the original population data. What does this function say about the yearly growth rate in the United States? c. Compute this same exponential function another way by fitting a linear equation to (year, log population) and then solving that equation for population. d. What estimate would you give for the United States population in 1955? In the year 21? e. Use your exponential function to estimate the year in which the population of the United States was about 5 million; about 14 million. LESSON 3 LINEARIZING DATA 189
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