*Functions, Data, and Models* helps undergraduates use mathematics to make sense of the enormous amounts of data coming their way in today's Information Age.

Drawing on the authors' extensive mathematical knowledge and experience, this textbook focuses on fundamental mathematical concepts and realistic problem-solving techniques that students must have to excel in a wide range of coursework, including biology, chemistry, business, finance, and economics. Thought-provoking experiments, statistical reasoning and methods, and "what if" questions help nurture the requisite mathematical skills.

*Functions, Data, and Models* can be used as a textbook in a college algebra course focusing on applications, in a quantitative literacy course, or in prerequisite courses for applied algebra or introductory statistics.

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Table of Contents

Preface

1. Data Everywhere

2. Functions Everywhere

3. Linear Functions

4. More About Linear Functions

5. Families of Nonlinear functions

6. Polynomial Functions

7. Extended Families of Functions

8. Modeling Periodic Phenomena

Appendices

Short Selected Answers

Index

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DIGMath (Dynamic Investigatory Graphical Displays for Mathematics) in Excel

*Note: Click on any of the links below to download the particular file. Most of these spreadsheets require the use of macros to operate. In order to use them, Excel must be set to accept macros. To change the security setting on macros:*

*Click on Tools, then Macros, then Security.*

Select Medium as the level of security.

Click on Trusted Publishers and then click on Trust access to Visual Basic Project.

Click on OK to close the window and the macros should load and operate smoothly.

*When you finish, you may want to undo the setting in the same manner to set it back to the original higher security level.*

**1. Linear Functions **

This DIGMath spreadsheet allows you to investigate visually three different aspects of linear functions. (1) You can enter the slope and vertical intercept and watch the effects of changing either of them via a slider on the resulting graph. (2) You can enter a point and the slope and watch the effects of changing either of them on the graph via the point-slope formula. (3) You can enter two points and change either of them to see the effects.

**2. The Sum of the Squares**

This DIGMath module allow you to investigate dynamically how the sum of the squares measures how well a line fits a set of data. You can enter a set of data and select the number of data points you want to use. You also enter the values you want for the slope and the vertical intercept of a line. The display shows the data points with the line based on those parameters and as well as the value for the sum of the squares associated with that linear fit.

**3. Simulating the Regression Line **

The user has the choice of the sample size (*n* > 2) and the number of samples. The simulation generates the random samples, calculates the equation of and plots the corresponding sample regression line, and also draws the population regression line. The students quickly see that, with small sample sizes, the likelihood of the sample regression line being close to the population regression line may be very small with widely varying slopes for many of the sample lines. As the sample size increases, the sample regression lines become ever more closely matched to the population line.

**4. Simulating the Correlation Coefficient **

This DIGMath spreadsheet lets you investigate the sample distribution for the correlation coefficient r based on repeated random samples drawn from a bivariate population. You can choose between *n* = 3 and *n* = 50 random points for each sample and between 50 and 250 such samples from the underlying population. For each sample, it then calculates the correlation coefficient and displays a histogram showing the values of **r** from the samples. It also calculates and displays the mean of the sample correlation coefficients and compares it to the correlation coefficient for the underlying bivariate population

**5. The Correlation Coefficient and The Sum of the Squares **

This DIGMath module lets you investigate the relationship between the correlation coefficient and the sum of the squares. A set of data is built-in and you can either squeeze the points together or spread them out vertically and see the effect on both the value of the correlation coefficient and the sum of the squares.

**6. Exponential Functions **

This DIGMath spreadsheet allows you to investigate visually two different aspects of exponential functions. (1) You can enter the growth/decay factor *b* in* y = b*^{x} and watch the effects on the resulting graph of changing it via a slider. (2) You can enter two points and change either of them to see the effects.

**7. Doubling Time & Half-Life **

This DIGMath spreadsheet is intended to let you investigate visually two important applications of exponential functions. First, you can explore the relationship between the growth factor *b* and the doubling time of an exponential growth process. Second, you can investigate the relationship between the decay factor *b* and the half-life of an exponential decay process

**8. Power Functions **

This DIGMath spreadsheet allows you to investigate visually the behavior of power functions. You can enter the power *p* and watch the effect on the graph of changing it via a slider.

**9. Fitting Functions to Data **

This DIGMath module is provided as a visual and computational tool for investigating the issue of fitting linear, exponential, and power functions to data. You can enter a set of data and the spreadsheet displays six graphs:

(1) For a linear fit: the regression line superimposed over the original (*x, y*) data;

(2) For an exponential fit: the regression line superimposed over the transformed (*x*(*x,* log *y*) data values;

(3) The exponential function superimposed over the original (*x, y*) data;

(4) For a power fit: the regression line superimposed over the transformed (log *x*, log *y*) data values;

(5) The power function superimposed over the original (*x, y*) data;

(6) All three functions superimposed over the original (*x, y*) data.

The spreadsheet also shows the values for the correlation coefficients associated with all three linear fits and the values for the sums of the squares associated with each of the three fits to the original data.

**10. Systems of Linear Equations **

This DIGMath modules provides a tool for solving systems of linear equations using matrix methods. You have the choice of a 2 2 system, a 3 3 system, or a 4 4 system. In each case, you enter the components of the matrix of coefficients **A** and the vector (matrix) of constants **B** in **AX** = **B** and the program responds with the corresponding solution vector **X**.

**11. Matrix Powers ** This DIGMath module lets you investigate the successive powers of a 2 2 matrix **A** applied to a vector **X**_{0}. The results are displayed both graphically and numerically.

**12. Quadratic Functions**

This DIGMath spreadsheet allows you to investigate visually two different aspects of quadratic functions. (1) You can enter, via sliders, values for the three coefficients in a quadratic function and watch dynamically the effects on the resulting graph of changing any of them via sliders. (2) You can also investigate visually the fact that a quadratic is determined by three points by entering the coordinates of three points, using sliders, and watching the dynamic effects on the graph of changing any of them.

**13. Cubic Functions**

This DIGMath spreadsheet allows you to investigate visually two different aspects of cubic functions. (1) You can enter, via sliders, values for the four coefficients in a cubic function and watch dynamically the effects on the resulting graph of changing the values of any of them via sliders. (2) You can also investigate visually the fact that a cubic is determined by four points by entering the coordinates of four points, using sliders, and watching the dynamic effects on the graph of changing any of them.

**14. Quartic Functions**

This DIGMath spreadsheet allows you to investigate visually two different aspects of quartic functions. (1) You can enter, via sliders, values for the five coefficients in a cubic function and watch dynamically the effects on the resulting graph of changing the values of any of them via sliders. (2) You can also investigate visually the fact that a quartic is determined by five points by entering the coordinates of five points, using sliders, and watching the dynamic effects on the graph of changing any of them.

**15. Polynomials**

This DIGMath spreadsheet lets you investigate the graph of any polynomial up to eighth degree by entering the values for the coefficients and the interval over which you want to see the graph. You can also control a point on the graph by means of a slider to see the coordinates of that point and so locate real roots, turning points, and inflection points.

**16. End Behavior: A Polynomial vs. Its Power Function**

This DIGMath module lets you investigate the end behavior of any polynomial up to eighth degree. You must enter the values for the coefficients. The spreadsheet then displays the graph of the polynomial as well as the graph of the power function corresponding to the leading term of the polynomial. A slider lets you expand the interval for the display, so that you can see how different the two graphs are when the interval is small and the turning points and inflection points of the polynomial are clearly in view. As the interval expands, the polynomial looks more and more like the power function.

**17. Graph of a Function**

This DIGMath spreadsheet allows you to investigate the graph of any desired function of the form *y = f(x)* on any desired interval *a* to *b* (or equivalently, xMin to xMax).

**18. Shifting and Stretching**

This DIGMath spreadsheet allows you to investigate visually the four different aspects of shifting and stretching/squeezing a function. The function used in the dynamic presentation is a zig-zag function (basically a saw-tooth wave that serves as a precursor to the sine function). (1) The first investigation involves experimenting with the effects of changing the parameters *a* and *c* in the zig-zag function *y* - *c* = zig (*x* - *a*). You can enter, via sliders, values for these parameters and watch dynamically the effects on the resulting graph of changing their values via sliders to see the horizontal and vertical shifts. (2) The second investigation involves experimenting with the effects of changing the parameters *k* and *m* in the zig-zag function *k* ** y* = zig (*m* * *x*). You can enter, via sliders, values for these parameters and watch dynamically the effects on the resulting graph of changing their values via sliders to see the horizontal and vertical stretches and squeezes that occur.

**19. Newton's Laws of Heating and Cooling**

This DIGMath module lets you explore both Newton's Law of Heating and Newton's Law of Cooling. Using sliders, you can enter the temperature of the medium, the heating or cooling constant (essentially, the rate at which the object heats up or cools), and the initial temperature of the object. The program draws the graph of the temperature function and allows the user to trace along the curve to see the temperature value at different times.

**20. Normal Distribution Function**

This DIGMath spreadsheet allows you to investigate the behavior of the normal distribution function based on its two parameters: the mean Î¼ (which produces horizontal shifts) and the standard deviation Î´ (which primarily produces vertical stretches and squeezes). You can change either of them using sliders to see the effect on the normal distribution curve.

**21. Normal Probabilities**

This DIGMath module lets you find, both graphically and geometrically, the probability that a randomly chosen value from a normal distribution is between two given values of *x*. You enter any desired values of the mean Î¼ and standard deviation Î´ of a normal distribution, as well as the interval of *x* values from *x*_{L} to *x*_{R}. The spreadsheet displays the normal distribution and highlights the corresponding area under the curve and calculates and displays the area of that region, which is equivalent to the probability the *x* is between *x*_{L} and *x*_{R}.

**22. The Central Limit Theorem and the Distribution of Sample Means**

This DIGMath spreadsheet lets you investigate the distribution of sample means. You can choose any of four underlying populations (normal, uniformly distributed, skewed, and bimodal), the sample size, and the number of random samples. The simulation randomly generates the samples and plots the means of each sample. From the graphical display and the associated numerical displays, it should become apparent that (1) the distribution of sample means is centered very close to the mean of the underlying population, that (2) the spread in the sample means is a fraction of the standard deviation of the underlying population (about one-half as large when *n* = 4, about one-third as large when *n* = 9, about one-quarter as large when *n* = 16, etc.), so that you might conjecture a relatively simple formula for the standard deviation of the distribution of sample means in terms of the sample standard deviation s and the sample size *n*, and that (3) as the sample size increases, the sampling distribution looks more and more like a normal distribution.

**23. Simulating Confidence Intervals**

This DIGMath module lets you investigate the notion of creating confidence intervals to estimate the mean of a population. You have the choice of the same four underlying populations as in the Central Limit Theorem simulation (to see that the population chosen does not affect the results) and the confidence level (90%, 95%, 98%, 99%). The simulation generates a fixed number of samples from the selected population, calculates and plots the corresponding confidence interval, and summarizes the number and percentage of confidence intervals that actually contain the mean of the underlying population. You can see that the actual (simulated) percentage is typically close to the selected value for the confidence level. You should also see that typically the higher the confidence level, the longer the lines are that represent the actual confidence interval. You can also see that typically those confidence intervals that do not contain the population mean are near-misses.

**24. Visualizing Cosine and Sine**

This DIGMath spreadsheet is intended to introduce, visually and dynamically, the graphs of the cosine and sine functions based on the movement of the minute hand of a clock over a 60 minute period. (1) The cosine function is introduced as the vertical distance, at time *t*, of the end of the minute hand above/below the horizontal axis. Two graphs are shown, one being that of the clock as time passes and the other being that of the associated vertical distances as time passes. You control the time *t* via a slider to see how the curve generated is related to the time on the clock. (2) Similarly, the sine function is introduced as the horizontal distance, at time *t*, from the vertical axis to the end of the minute hand.

**25. Sinusoidal Functions**

This DIGMath spreadsheet allows you to investigate dynamically the effects of the four parameters *A, B, C,* and *D* on a sinusoidal function. (1) For the sine curve *f*(*x*) = *A* + *B* sin (*C*(*x* - *D*)), you can vary the values of the midline, the amplitude, the frequency, and the phase shift via sliders and see the effects on the corresponding graph. (2) You can conduct the same kind of experiments on the cosine function *f*(*x*) = *A* + *B* cos (*C*(*x* - *D*)).

**26. Fitting Sinusoidal Functions to Data**

This DIGMath spreadsheet lets you investigate dynamically the problem of fitting a sinusoidal function to a set of data. You can opt to use either a sine or a cosine function. You enter the desired set of periodic data values and then enter values for the four parameters - the midline, the amplitude, the period, and the phase shift. The spreadsheet displays the corresponding sinusoidal function superimposed over the data, so you can visually assess how well the function fits the data. It also gives the value for the sum of the squares associated with the function, so you can assess numerically how well the function fits the data. You can then adjust any of the four parameters that are reasonable to see if you can improve on the fit.

**27. Approximating Sinusoidal Functions**

This DIGMath spreadsheet lets you investigate dynamically the idea of approximating a sinusoidal function with a polynomial. You can choose to work with either the sine or the cosine function and can use an approximating polynomial up to sixth degree in the form y = *a* + (1/*b*)*x* + (1/*c*)*x*^{2}+ (1/*d*)*x*^{3} + (1/*e*)*x*^{4} + (1/*f*)*x*^{5} + (1/*g*)*x*^{6}, where the seven parameters are all integers. You use sliders to change each of the parameter values and the associated graph shows how well the corresponding polynomial fits the sinusoidal curve. The spreadsheet also shows the value for the sum of the squares to provide a numerical measure for the goodness of the fit. This investigation is intended as a precursor to the notion of Taylor polynomial approximations while reinforcing the behavioral characteristics of polynomials and sinusoidal functions.

**28. Approximating the Exponential Function**

This DIGMath spreadsheet lets you investigate dynamically the idea of approximating the exponential function *f*(*x*) = *b*^{x} with base *e* = 2.71828 with a polynomial. You can use an approximating polynomial up to fifth degree in the form *y* = *a* + (1/*b*)*x* +(1/*c*) *x*^{2}+ (1/*d*) *x*^{3} + (1/*e*) *x*^{4 }+ (1/*f*) *x*^{5}, where the six parameters *a, b, c, d, e,* and *f* are all integers. You use sliders to change each of the parameter values and the associated graph shows how well the corresponding polynomial fits the exponential curve. The spreadsheet also shows the value for the sum of the squares to provide a numerical measure for the goodness of the fit. This investigation is intended as a precursor to the notion of Taylor polynomial approximations while reinforcing the behavioral characteristics of polynomials and exponential functions.

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Excerpt: Interpreting the Graphs (p. 266)

Often, the single best approach to deciding which of several functions is the best fit to a set of data is to plot all possible functions superimposed over the scatterplot and examine the resulting image carefully. Our eyes and minds have developed over the ages to process information visually and to identify patterns and trends in those visual images. This ability also applies to mathematical images. Use either your graphing calculator or a spreadsheet to create graphs with an appropriate viewing window showing all the important details. When you examine the graphs carefully, it is usually fairly evident which function is the best overall fit to the data and which functions are poor choices.

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About the Authors

**Sheldon Gordon** (Farmington State College), who earned his Ph.D. in mathematics from McGill University, has written almost 200 articles on mathematics and math education. He has served on numerous Association committees, including Curriculum Renewal Across the First Two Years, and co-edited MAA Notes volumes.

**Florence Gordon** (New York Institute of Technology), who earned her Ph. D. in mathematics from McGill University, has written over 75 articles on statistics, statistical education, and mathematics education. She has served on several MAA committees, including the Committee on Contributed Paper Sessions, and co-edited MAA Notes volumes.