*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.

### 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

### DIGMath (Dynamic Investigatory Graphical Displays for Mathematics) in Excel

*Note: When you open the spreadsheet, a new bar appears near the top of the window that says "Security Warning: Macros have been disabled ".*

Click on Options.

Click on "Enable this Content" and then click OK.

**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.
**Linear Regression: Fitting a Linear Function to Data**

This DIGMath module performs a linear regression analysis on any set of up to 50 (*x, y*) data points. It shows graphically the points and the associated regression line and also displays the equation of the regression line, the value for the correlation coefficient *r*, and the value for the Sum of the Squares that measures how close the line comes to all the data points.
**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 also shows the value for the sum of the squares associated with that linear fit.
**Simulating the Regression Line**

The user has the choice of the sample size (*n* __>__ 2) and the number of samples. The simulation generates repeated 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 regression line.
**Fitting a Median-Median Line to Data**

This DIGMath module fits a median-median line to any set of up to 50 (*x, y*) data points. It shows graphically the points and the associated median-median line and also displays the equation of the median-median line and the value for the Sum of the Squares that measures how close the line comes to all the data points.
**Comparing Lines that Fit Data**

This DIGMath program lets you compare how well the least-squares line, the median-median line (that is built into many calculators), and the quartile-quartile line (based on the first and fourth quartiles of a set of data) fit sets of data. You can choose the number of random data points from an underlying population and the spreadsheet generates a random sample and displays the three lines, along with the data points, so that you can compare how well the three lines fit the data and how they compare to one another, particularly as the sample size increases.
**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.
**The Correlation Coefficient and the Sum of the Squares**

This DIGMath spreadsheet lets you investigate how the correlation coefficient and the sum of the squares capture the trend in a set of data. You use a slider to vary a parameter that represents by how much a set of data is "squeezed" or "stretched" vertically to see the effects on the correlation coefficient and on the sum of the squares.
**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.
**Exponential Regression: Fitting an Exponential Function to Data**

This DIGMath module performs an exponential regression analysis on any set of up to 50 (*x, y*) data points. It shows graphically the points and the associated exponential regression function and also displays the equation of the exponential regression function, the value for the associated correlation coefficient *r* based on the transformed (*x*, log *y*) data, and the value for the Sum of the Squares that measures how close the exponential function comes to all the data points.
**Doubling Time and 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.
**Power Functions**

ThisDIGMath 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.
**Power Regression: Fitting a Power Function to Data**

This DIGMath module performs a power regression analysis on any set of up to 50 (*x, y*) data points. It shows graphically the points and the associated power regression function and also displays the equation of the power regression function, the value for the associated correlation coefficient *r* based on the transformed (log *x*, log *y*) data, and the value for the Sum of the Squares that measures how close the power function comes to all the data points.
**Data Analysis: Fitting Functions to Data**

This DIGMath spreadsheet 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*, 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.
**Logarithmic Growth**

This DIGMath spreadsheet lets you investigate how fast (or actually how slow) logarithmic growth is. In particular, it lets you see, both graphically and numerically, what the cost is in terms of how much the variable *t *must increase in order for the logarithmic function *f(t)* = log *t* to increase by 1 unit for different values of *t*.
**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.
**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.
**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.
**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.
**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.
**Polynomial Regression: Fitting a Polynomial to Data**

This DIGMath module performs a polynomial regression analysis (linear, quadratic, ..., up to sixth degree) on any set of up to 50 (*x, y*) data points, which goes beyond what is possible with graphing calculators. It shows graphically the points and the associated polynomial regression function and also displays the equation of the regression polynomial, the value for the associated coefficient of multiple determination *R*^{2} based on multivariate linear regression of y on *x, x2, x3,* ..., *x6* (depending on the choice of degree) and its significance in terms of what percentage of the variation in the data is explained by the regression polynomial, and the value for the Sum of the Squares that measures how close the power function comes to all the data points.
**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).
**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.
**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.
**Normal Distribution Function**

This DIGMath module allows you to investigate the behavior of the normal distribution function based on its two parameters: the mean m (which produces horizontal shifts) and the standard deviation δ (which primarily produces vertical stretches and squeezes). You can change either of them using sliders to see the effect on the normal distribution curve.
**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 becomes 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 students quickly conjecture that the formula for the standard deviation of the distribution of sample means is *s*/√*n*, and that (3) as the sample size increases, the sampling distribution looks more and more like a normal distribution.
**Simulating Confidence Intervals**

his DIGMath spreadsheet 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 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. Students see that the actual (simulated) percentage is typically close to the selected value for the confidence level. They also see that typically the higher the confidence level, the longer the lines are that represent the actual confidence interval. They also see that typically those confidence intervals that do not contain the population mean are near-misses.
**Multivariate Linear Regression**

This DIGMath spreadsheet lets you perform multivariate linear regression when the dependent variable *Y* is a function of two independent variables *X*1 and *X*2 or a function of three independent variables *X*1, *X*2, and *X*3. You enter the number of data points (up to a maximum of 50) and then the values for the dependent and independent variables in the appropriate columns. The spreadsheet responds with the equation of the associated linear regression equation, the value for the sum of the squares, and the value for the coefficient of determination, *R*^{2}; note that this value tells you the percentage of the variation that is explained by the linear function.
**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.
**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*)).
**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.
**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.
**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.
**The Tangent Function**

This DIGMath module lets you investigate the definition of the tangent function as the ratio of the sine and the cosine. You can enter any desired interval in radians (or in degrees on a separate sheet of the spreadsheet). The spreadsheet draws two charts, one for the graphs of the sine and cosine functions, and the other for the graph of the tangent function. It also provides numerical results, showing the values of the sine, the cosine, and the tangent at any desired tracing point, to demonstrate that the tangent function is indeed the ratio of the sine and cosine at each point.
**DeMoivre's Theorem**

This DIGMath module lets you explore DeMoivre's Theorem. You enter the values for any desired complex number and use a slider to select the power to which *z = a + bi* is to be raised. You can investigate the results both graphically and numerically in three situations: using the trigonometric form for the complex number and its powers, using the rectangular form for both, and using non-integer powers.
**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 x 2 system, a 3 x 3 system, or a 4 x 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.
**Homogeneous Linear Systems**

This DIGMath spreadsheet lets you investigate the solutions of homogeneous systems of linear equations AX = 0 from a graphical perspective. You can use sliders to change the coefficients and watch the effects on the corresponding lines to see the conditions under which the lines have a unique point of intersection, are parallel so there is no solution, or overlap completely so that there are infinitely many solutions.
**Matrix Powers**

This DIGMath module lets you investigate the successive powers of a 2 x 2 matrix A applied to a vector X_{0}. The results are displayed both graphically and numerically.

### 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.

### 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.