Excel Graphical Simulations for Statistics and Probability

DIGMATH: Dynamic Investigatory Graphical Displays of Mathematics:

Graphical Explorations for College Algebra and Precalculus in Excel

Sheldon P. Gordon and Florence S. Gordon

Most of the following graphical explorations require the use of macros to operate. In order to use these spreadsheets, Excel must be set to accept macros. To change the security setting on macros:

When you open any of the spreadsheets, a new bar appears near the top of the window that says something like: "Some Active content has been disabled", depending on the version of Excel you are using.
Click on Options.
Click on "Enable the Content" and then click OK.

The following are the 106 DIGMath simulations that are currently (December, 2024) completed and ready for use in college algebra or precalculus. (Others are under development.) Please feel free to download (with Chrome or Edge, simply click on any of the links below; with Firefox, right click on any of the links and then select Save Link As ...) and use any or all of these files. If you want all of them, just click on the link: precalc.7z or you can send an e-mail to gordonsp@retiree.farmingdale.edu and we will try sending you a zip file with all of them. If you have any problems downloading or running any of these Excel files, please contact us at gordonsp@retiree.farmingdale.edu or flogo@optonline.net or for assistance. If you have any suggestions for improvements or for new topics, please pass them on also.

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.
Regression Simulation This DIGMath spreadsheet simulates the regression lines drawn from an underlying population to display how different randomly generated regression lines compare to one another. 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.
Median-Median Line Simulation 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 median-median line, and also draws the population median-median line. The students quickly see that, with small sample sizes, the likelihood of the sample median-median line being close to the population median-median line may be very small with widely varying slopes for many of the sample lines. As the sample size increases, the sample median-median lines slowly become more closely matched to the population median-median line.
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 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.
Moving Along Lines One of the most common mathematical concepts that arises in economics is the idea of moving along a line, typically involving the Supply and Demand lines (or curves). This DIGMath spreadsheet lets you explore this concept in which you have a pair of intersecting lines and use a slider to move one of the two lines along the other line while keeping the slope the same. The spreadsheet shows the new and the old lines and the changes in the point of intersection.
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^xand 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.
Comparing Exponential Growth Functions This DIGMath spreadsheet is designed to give you a deeper understanding of the behavior of exponential growth functions y = b^x, with b > 1. It lets you select a value for the base b between 0.15 and 0.85 using a slider and draws the graphs of the three exponential growth functions y = 1.10^x, y = b^x with your choice of b, and y = 2^x. You can also trace all three curves using a slider and see the effects on the graphs and on the printed values, all of which are color coordinated to help you see what is happening.
Comparing Exponential Decay Function s This DIGMath spreadsheet is designed to give you a deeper understanding of the behavior of exponential decay functions y = b^x, with 0 < b < 1. It lets you select a value for the base b between 0.15 and 0.85 using a slider and draws the graphs of the three exponential decay functions y = 0.10^x, y = b^x with your choice of b, and y = 0.90^x. You can also trace all three curves using a slider and see the effects on the graphs and on the printed values, all of which are color coordinated to help you see what is happening.
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.
Drug Level Models This DIGMath module lets you investigate two different models for the level of a medication in the bloodstream. The first model is for the drug level after a single dose of the medication; the second model is for the repeated (say daily) dose of a medication. In both cases, the key parameters are the amount of the medication that is taken (either once or repeatedly) and the percentage of the drug that is washed out of the blood each time period.
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.
Amortization: Financing a Home or a Car This DIGMath module is intended to let you investigate the mathematics involved in the amortization process -- taking out a loan to finance a home or a car. You have a choice of financing a home or a car on different pages of the spreadsheet. For each, you enter the amount of the loan, the length of the loan, and the annual interest rate. The spreadsheet displays the graph of the monthly balance of the loan and two different views of the breakdown of how the monthly payment is split between paying off the principal and paying for the interest on the loan. It also lets you trace across the graphs to see the amounts in any month.
Effects of Compound Interest This DIGMath spreadsheet lets you see the effects of compounding with different frequencies -- yearly, quarterly, monthly, daily, by the hour, and by the minute -- on the balance in an account. You can enter your choice of interest rate and initial balance and see both graphically and numerically the growth in the balance over time with the different compounding periods.
Exploring the Sum Identity for Exponents This DIGMath module lets you investigate graphically and numerically the sum identity for exponents: e^a+b = e^ae^b for any a and b. You enter the values for the parameters using sliders and the spreadsheet draws the graph of the exponential function and highlights the values of a and b as horizontal distances and shows the corresponding heights along with a visualization of the identity.
Exploring the Difference Identity for Exponents This DIGMath module lets you investigate graphically and numerically the difference identity for exponents: e^a-b = e^a /e^b for any a and b. You enter the values for the parameters using sliders and the spreadsheet draws the graph of the exponential function and highlights the heights for e^a, e^b, and e^a-b. The spreadsheet focuses on the fact that e^a appears to be a multiple of e^a-b and that the multiple seems to be equal to e^b, both graphically and numerically. That realization is just a short step to the discovery of the identity.
Haldane Functions in Genetics This module draws the graph of the Haldane function relating the recombination fraction C_AB between loci A and B in a gene and the distance X_AB between the loci. A slider lets you change the value for the distance X_AB between loci and see the effect on the corresponding point on the graph of the Haldane function that involves an exponential function.
Power Functions This 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.
Comparing Power Functions This program lets you investigate the behavior of the three different kinds of power functions -- those with p> 1, 0 < p < 1, and p < 0. In each case (on a different page of the spreadsheet), the curves for two "extreme" power functions are drawn and a third graph, based on your choice of the power p between the extremes is also drawn. As you change the value of p, you can see how the graph compares to the two extreme curves.
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.
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.
DataFit: 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.
Comparing Logarithmic Functions This DIGMath spreadsheet is designed to give you a deeper understanding of the behavior of logarithmic functions y = log_bx, with b > 1. It lets you select a value for the base b between 0.15 and 0.85 using a slider and draws the graphs of the three exponential growth functions y = log₂ x, log_b x with your choice of b, and y = log₁₀ x . You can also trace along all three curves using a slider and see the effects on the graphs and on the printed values, all of which are color coordinated to help you see what is happening.
Logarithmic Functions with Fractional Bases Typically, logarithmic functions are only considered in which the base b > 1. This DIGMath program lets you investigate what happens if fractional bases with 0 < b < 1 are introduced. The resulting behavior patterns are very different from what you would expect based on the usual logarithmic functions.
The Logistic Model This DIGMath module allows you to investigate visually two different aspects of the discrete logistic model based on the logistic difference equation P_n+₁ = aP_n- bP_n². (1) You can enter, via sliders, values for the two parameters a and b, as well as the initial population value P₀and watch dynamically the effects on the resulting graph of the population, and also see the effects of changing any of these values. (2) You can also investigate visually the effects on a population of changes in the initial growth rate a and the maximum sustainable population (the limit to growth), along with the initial population value P₀, using sliders, and watching the dynamic effects on the graph of changing any of them.
Doubling Times of Various Functions This DIGMath module lets you investigate the notion of the doubling time associated with a variety of different functions, including linear, power and logarithmic. For an exponential function f(t) = Ab^t , the doubling time depends only on the growth rate b, but not on the point t or the value of the coefficient A. The module lets you see what happens with the other families of function to see which, if any, of the parameters or values of the independent variable contribute to the value for the doubling time.
The Surge Function This DIGMath module lets you investigate the properties of the surge function, which has the equation y = A t ^p b^t, where p > 0 and b < 1. It is used to model phenomena where there is an initial growth spurt in a quantity followed by a slow decay toward zero. You can experiment with different values for the parameters, using sliders, to see the effects on the behavior of the function.
Exploring the Product Identity for Logarithms This DIGMath module lets you investigate graphically and numerically the product identity for logarithms: log (a* b) = log (a) + log (b) for any a and b. You enter the values for the parameters using sliders and the spreadsheet draws the graph of the logarithmic function and highlights the values of a and b as horizontal distances and shows the corresponding heights along with a visualization of the identity.
Exploring the Logarithm of a Reciprocal This DIGMath module lets you investigate graphically and numerically the rule for the logarithm of a reciprocal that log (1/x) = - log (x), for any base b > 0 and any x > 0. You enter the values for the base b using a slider and the spreadsheet draws the graph of y = log (1/x) and shows the numerical values for both y = log (1/x) and y = log (x). One page deals with the case with b > 1 and another with 0 < b < 1.
Exploring the Quotient Identity for Logarithms This DIGMath module lets you investigate graphically and numerically the quotient identity for logarithms: log (a / b) = log (a) - log (b) for any a and b. You enter the values for the parameters using sliders and the spreadsheet draws the graph of the exponential function and highlights the values of a and b as horizontal distances and shows the corresponding heights along with a visualization of the identity.
Exploring the Logarithm of of a Power This DIGMath module lets you investigate graphically and numerically the rule that log (x^p) = p log (x), for any p and x. You enter the values for the power p using a slider and the spreadsheet draws the graphs of both y = log (x) and y = log (x^p). The display focuses on the fact that, as you trace along the two curves, the height to the second curve appears to be a fixed multiple, equal to the parameter p, of the height to y = log (x), for any x, as a visualization of the identity.
The Log of a Power Identity This DIGMath spreadsheet lets you investigate visually, numerically, and dynamically the fundamental identity log _b(b^x) = x, for any base b > 1 and any value x. It draws the graph of the exponential term y =b^x and the expression y = log _b(b^x) and allows the user to trace along both curves as it displays the algebraic steps and numerical results to show that the result is always equal to x. Simultaneously, the second graph looks linear and in fact is the line y = x.
The Power of a Log Identity This DIGMath spreadsheet lets you investigate visually, numerically, and dynamically the fundamental identity b^log _b⁽^x⁾ = x, for any base b > 1 and any value x. It draws the graph of the logarithmic term y = log _b x and the expression y = b^log _b⁽^x⁾ and allows the user to trace along both curves as it displays the algebraic steps and numerical results to show that the result is always equal to x. Simultaneously, the second graph looks linear and is, in fact, the line y = x.
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.
Polynomial Graphs 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 R² 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.
Squaring a Binomial Term This DIGMath module lets you practice squaring a binomial term (x + a)² both algebriacally and graphically. The spreadsheet randomly generates the binomial term and allows you to select the coefficients b and c in the quadratic product x² + bx + c using sliders. It displays the graphs of the squared term and the quadratic to show how you eventually get exact overlap when you get the correct result.
Product of Binomial Terms This DIGMath module lets you practice multiplying two randomly generated binomial terms (x + p) and (x + q) both algebriacally and graphically. The spreadsheet randomly generates the two binomial terms and allows you to select the coefficients b and c in the quadratic product x² + bx + c using sliders. It displays the graphs of the product of the two binomial terms and the quadratic based on b and c to show how you eventually get exact overlap when you get the correct result.
Factoring Quadratics This DIGMath module lets you practice factoring randomly generated quadratic polynomials of the form x² + bx + c by selecting the numbers p and q for two binomial terms (x + p) and (x + q). The spreadsheet displays the graphs of the target quadratic and the quadratic based on your choice of p and q to show how you eventually get exact overlap when you get the correct result.
Finding the Complex Zeros of Cubics This DIGMath spreadsheet lets you locate all three zeros, particularly any pairs of complex conjugate zeros, of cubic polynomials using a graphical exploration. The spreadsheet also displays the factorization and the values of the three zeros, either real or complex.
Finding the Complex Zeros of Quartic Polynomials This DIGMath module lets you approximate all four zeros, particularly any pairs of complex conjugate zeros, of quartic polynomials of the form y = x⁴ + ax³ + bx² + cx + d using a graphical exploration. As you vary the coefficients in an approximation to a possible quadratic factor, you see how well the resulting approximate factorization fits the original quartic both graphically and in terms of the Sum of the Squares. The spreadsheet also displays the factorization and the values of the four zeros, either real or complex.
Estimating the Complex Zeros of a Polynomial This DIGMath spreadsheet lets you approximate the complex zeros of any polynomial up to degree 8. You enter the desired coefficients and then enter six points that lie along the curve for the polynomial. The program draws the graph and then applies quadratic regression to find the best-fitting quadratic approximation, draws its graph, and displays its zeros, whether real or complex, as approximations to two of the zeros of the original polynomial.
Bairstow's Method for Complex Roots This DIGMath program lets you investigate Bairstow's Method for approximating the complex roots of a polynomial by creating a sequence of quadratic functions that converges to an irreducible quadratic factor of the polynomial. Once you know a quadratic factor, you can find the complex roots by applying the quadratic formula. The spreadsheet lets you use any polynomial up to 8^th degree and any choice of initial estimate for the quadratic factor. The spreadsheet draws the first five quadratic approximations to illustrate the convergence both graphically and numerically.
Components of the Lagrange Interpolating Polynomial This DIGMath spreadsheet lets you investigate the Lagrange Interpolating Polynomial and how it is formed by a set of components. You can work with a linear function based on two interpolating points, a quadratic function based on three interpolating points, or a cubic function based on four interpolating points of your choice. The spreadsheet displays the resulting Lagrange interpolating polynomial, as well as the components of that polynomial -- the two linear functions that combine to form the linear Lagrange polynomial, the three quadratic functions that combine to form the quadratic Lagrange polynomial, and the four cubic functions that combine to form the cubic Lagrange polynomial.
Difference Tables and the Newton Interpolating Polynomial This DIGMath spreadsheet lets you investigate the Newton Interpolating Polynomial and how it is based on a table of successive differences formed from a set of data. You can enter your data values with equally spaced x-values and the spreadsheet displays the corresponding table of differences, as well as the graph of the resulting Newton Interpolating Polynomial and the formula for that polynomial.
A Bouncing Ball This DIGMath module lets you investigate the mathematics behind a bouncing ball. You can work in either the English or the metric system. You input the initial height from which a ball is dropped and the percentage of the velocity that is lost on each bounce. The spreadsheet draws the graphs of the height of the ball as a function of time and the velocity of the ball as a function of time. The investigations also include the fact that the shape of each portion of the height function is a polynomial.
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 from a to b (or equivalently, from 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.
Operations on Functions This DIGMath module lets you investigate operations of adding or subtracting two functions, using a zig-zag function.
Symmetry of Inverse Functions This DIGMath program lets you investigate the symmetric nature of a function y = f(x) and its inverse function y =g(x) . It uses the exponential function y = b^x compared to its inverse, the logarithmic function y = log_b(x), where you use a slider to select the base b between 0 and 2. When b > 1, the exponential function is increasing and concave up and the logarithmic function is increasing and concave down. When b < 1, the exponential function is decreasing and concave up and the logarithmic function with fractional base is decreasing and concave up. The program demonstrates graphically and numerically how the two functions, for any b > 0, is symmetric about the diagonal line y = x, which is true for any function and its inverse.
The Bisection Method This DIGMath module illustrates the convergence of the Bisection Method for finding the real zeros of any function y = f(x). To use it, you enter the desired function and an initial interval from a to b over which the function has at least one zero and the program performs successive iterations, or repetitions, of the method, displaying the results both graphically and in a table to illustrate how the successive approximations converge to a real zero.
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.
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 s (which primarily produces vertical stretches and squeezes). You can change either of them using sliders to see the effect on the normal distribution curve.
Multivariate Linear Regression This DIGMath spreadsheet lets you perform multivariate linear regression when the dependent variable Y is a function of two independent variables X1 and X2 or a function of three independent variables X1, X2, and X3. 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, R²; note that this value tells you the percentage of the variation that is explained by the linear function.
Comparing Moving Averages This DIGMath program lets you compare different moving averages to one another as well as to the underlying set of data. Moving averages are used widely in many different fields to identify patterns and trends in data where there are wild fluctuations on a day-to-day basis. This spreadsheet uses some actual data on the spread of the COVID pandemic to illustrate the concept by plotting the underlying data, the 3-day moving average, the 20-day moving average, and a third moving average that you can choose based on either 4-days, 5-days, up through 19-days.
Fitting Functions to Moving Averages This DIGMath spreadsheet lets you investigate how moving averages can be used to identify the pattern in a set of data that has extreme daily fluctuations. The program creates charts for the 5-day moving average, the 10-day moving average, the 15-day moving average, and the 20-day moving average. It also allows you to select the type of function you would like to use -- linear, exponential, power, quadratic, or cubic -- using a slider and displays the resulting functions that fit both the underlying data and the corresponding moving average. It also displays the resulting equations of the functions and the values of the corresponding correlation coefficients, r, (for the linear, exponential, and power fits) and the coefficient of multiple correlation, R, (for the polynomial fits) to assess how well each function fits the data.
Visualizing the 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)).
Parameters of the Sine Function This DIGMath module lets you explore the effects of each of the four parameters A, B, C and D (each on a different page) of the sine function f(x) = A + B sin (C(x - D)). In each case, three curves are shown corresponding to specific values of the parameter and the fourth curve corresponds to the value of the parameter you select via a slider.
Parameters of the Cosine Function This DIGMath module lets you explore the effects of each of the four parameters A, B, C and D (each on a different page) of the cosine function f(x) = A + B cos (C(x - D)). In each case, three curves are shown corresponding to specific values of the parameter and the fourth curve corresponds to the value of the parameter you select via a slider.
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.
Level of the Tides Model This DIGMath spreadsheet lets you explore a simple sinusoidal model for the level of the tides. You enter the time (hour and minute) of the projected high tide, the following low tide, and the heights of those two tides. The program generates the graph of the sinusoidal model over a 24 hour day and you can trace along the curve to see the corresponding level of the tide at each time and the associated rate at this the tide is rising or falling at that time. The graph of the rate of rise/fall of the tide is also shown.
Visualizing Trig Identities This DIGMath module lets you investigate dynamically, graphically, and numerically the notion of a trig identity. You have the choice of sin 2x, cos 2x, sin 3x, ..., cos 5x. The spreadsheet draws the graphs of the function and the identity function to show that they agree exactly at every point on the interval [0, 2p]. As you trace along the curves, the spreadsheet also displays the values of y along both curves associated with the tracing point to illustrate that the agreement is identical numerically at each point. Finally, in most of the cases, the spreadsheet also draws the graphs of the components of the identity function and shows the difference in heights between the two as well as the graph of the function to demonstrate how the identity function is created.
Verifying Trig Identities This DIGMath spreadsheet lets you verify graphically whether a supposed trig identity is actually true for all values of x on any desired interval. You enter the expressions for the function on the left-hand side and the right-hand side of the supposed identity. The spreadsheet then draws the two graphs and displays the coordinates of a tracing point. If it is truly an identity, the two graphs should completely overlap across the entire interval and the values associated with the tracing point should be the same; otherwise, it is not an identity.
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)x²+ (1/d)x³ + (1/e)x⁴ + (1/f)x⁵+ (1/g)x⁶, 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^xwith 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) x²+ (1/d) x³+ (1/e) x⁴+ (1/f) x⁵, 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.
Approximating the Natural Logarithmic Function This DIGMath spreadsheet lets you investigate dynamically the idea of approximating the natural logarithmic function f(x) = ln 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) + (1/c) (x-1)²+ (1/d) (x-1)³+ (1/e) (x-1)⁴+ (1/f) (x-1)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.
Approximating the Sine with Lagrange Interpolation This DIGMath module lets you investigate dynamically the way that the Lagrange Interpolating Polynomial is used to approximate values of the sine function on the interval [0, p/4] using either a quadratic polynomial (based on three interpolating points) or a cubic polynomial (based on four interpolating points).
Approximating the Cosine with Lagrange Interpolation This DIGMath module lets you investigate dynamically the way that the Lagrange Interpolating Polynomial is used to approximate values of the cosine function on the interval [0, p/4] using either a quadratic polynomial (based on three interpolating points) or a cubic polynomial (based on four interpolating points).
Approximating the Exponential Function with Lagrange Interpolation This DIGMath spreadsheet lets you investigate dynamically the way that the Lagrange Interpolating Polynomial is used to approximate values of the exponential function. One page lets you work with a quadratic polynomial based on three points on the exponential curve and another page lets you use a cubic polynomial based on four points.
Approximating the Sine with Newton Interpolation This DIGMath module lets you investigate dynamically the way that the Newton Interpolating Polynomial is used to approximate values of the sine function on the interval [0, p/4] using either a quadratic polynomial (based on three uniformly spaced interpolating points) or a cubic polynomial (based on four uniformly spaced interpolating points).
Approximating the Cosine with Newton Interpolation This DIGMath module lets you investigate dynamically the way that the Newton Interpolating Polynomial is used to approximate values of the cosine function on the interval [0, p/4] using either a quadratic polynomial (based on three uniformly spaced interpolating points) or a cubic polynomial (based on four uniformly spaced interpolating points).
Approximating the Exponential Function with Newton Interpolation This DIGMath spreadsheet lets you investigate dynamically the way that the Newton Interpolating Polynomial is used to approximate values of the exponential function. One page lets you work with a quadratic polynomial based on three uniformly spaced points on the exponential curve and another page lets you use a cubic polynomial based on four uniformly spaced points. The spreadsheet calculates and draws the graph of the interpolating function as it passes through the interpolating points and seeks to match the exponential curve. It also draws the graph of the error function -- the difference between the polynomial and the exponential function.
Approximating the Natural Logarithm with Newton Interpolation This DIGMath spreadsheet lets you investigate dynamically the way that the Newton Interpolating Polynomial is used to approximate values of the natural logarithm function. The first page lets you compare the effectiveness of the quadratic, the cubic, the quartic, and the quintic Newton polynomials based on uniformly spaced points on the curve. The spreadsheet calculates and draws the graphs of the four interpolating polynomials as they pass through the interpolating points on the natural logarithm curve. It also lets you select a particular degree for the polynomial and draws the graph of the associated error function -- the difference between the polynomial and the exponential function. A second page shows all four interpolating polynomials separately and all four error functions separately.
Chebyshev Nodes This DIGMath module lets you investigate the so-called Chebyshev nodes that are often the best points to use to construct an interpolating polynomial based on a set of points. The Chebyshev nodes have the property that they provide the minimum value of the maximum error in the approximation at the interpolating points. The spreadsheet lets you enter your choice of either three or four interpolating points on the interval [-1, 1] and compares, both numerically and graphically, the maximum error to the corresponding maximum error using the Chebyshev nodes.
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.
Tangentoidal Functions This DIGMath module lets you investigate the behavior of the so-called tangentoidal functions (a generalization of the tangent function) that are defined as f(x) = sin x /[a + cos x]. You can enter any desired value for the parameter a and any desired interval in radians. The spreadsheet draws the graph of the tangentoidal function. It also raises questions about the behavior of these functions in terms of the location and existence of vertical asymptotes and how that is related to the value of a.
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 X₀. The results are displayed both graphically and numerically.
Rotation Matrices This DIGMath module lets you investigate the effects of successively applying a rotation matrix A to any desired initial vector X₀ using any desired angle. The results are displayed both graphically and numerically.
Markov Chains This DIGMath module lets you investigate the effects of successive terms in a Markov chain by applying a transition matrix A to any desired initial vector X₀. The results are displayed both graphically and numerically.
Matrix Transformations of Vectors This DIGMath spreadsheet lets you visualize how the process of multiplying a column vector X by a 2 x 2 matrix A transforms the vector X into a new vector Y = AX. It simultaneously provides a way of visualizing how an eigenvector may occur in the process.
Finding Eigenvectors Graphically This DIGMath program lets you investigate graphically the notion of the eigenvectors and eigenvalues of the 2 x 2 matrix A. You can enter the four entries in the matrix and the components of an initial vector X using sliders. The spreadsheet draws the graph of the vector X and the resultant vector AX. You can estimate the eigenvectors by adjusting the components of X until X and AX are essentially pointing in the same direction. The program also displays the coordinates and shows the actual eigenvalues, either real or complex.
Eigenvalues and Eigenvectors This DIGMath module finds the eigenvalues and eigenvectors of any two by two matrix that you enter, whether there are two distinct real eigenvalues, a double real eigenvalue, or a pair of complex conjugate eigenvalues. It also displays the characteristic equation, the value of the discriminant, and graphs of the real eigenvectors.
Eigenvalues of Matrix Powers ThisDIGMath spreadsheet lets you discover, graphically and numerically, the relationship between the eigenvalues of a 2 x 2 matrix A and the associated eigenvalues of the matrix powers A² and A³.
Eigenvalues of the Inverse Matrix ThisDIGMath module lets you discover, graphically and numerically, the relationship between the eigenvalues of a 2 x 2 non-singular matrix A and the associated eigenvalues of the inverse matrix A^-1.
The Ellipse This DIGMath spreadsheet lets you investigate the properties of the ellipse. One sheet is based on the equation of the ellipse and you can select and vary the values for the center x = c of the ellipse and the parameters a and b in the ellipse and see the effects on the resulting graphs instantly. The second sheet is based on the definition of the ellipse as the set of points such that the sum of the distances to two fixed points, the foci, is a constant. You can enter and vary the locations of the two foci that lie on the x-axis and the value for the sum of the two distances and see the effects on the ellipse so determined, as well as its equation.
The Hyperbola This DIGMath spreadsheet lets you investigate the properties of the hyperbola. One sheet is based on the equation of the hyperbola where the foci are located on the horizontal axis and you can select and vary the values for the center x = x₀ of the hyperbola and the parameter a in its equation and see the effects on the resulting graph instantly. A second sheet is based on equation of the hyperbola where the foci are located on the vertical axis and you can select and vary the values for the center y = y₀ of the hyperbola and the parameter a in its equation and see the effects on the resulting graph instantly. The third sheet is based on the definition of the hyperbola as the set of points such that the difference in the distances to two fixed points, the foci, is a constant. You can enter and vary the location of the foci that lie on the x-axis symmetrically and the value for the difference of the two distances and see the effects on the hyperbola so determined, as well as its equation.
The Parabola This DIGMath spreadsheet lets you investigate the geometric definition of the parabola. One sheet is based on the equation of the parabola where the focus is located on the vertical axis and you can select and vary the value for the focus c and the width of the display window and see the effects on the resulting graph instantly. It also shows the directrix and displays the distances from a tracing point both to the focus and to the directrix. A second sheet is based on equation of the parabola where the focus lies on the horizontal axis and you can select and vary the values for the value for the focus c and the height of the display window and see the effects on the resulting graph instantly. It also shows the distances from a moving point to the focus and the directrix.
Rational Functions This DIGMath spreadsheet lets you investigate the properties of rational functions. On different sheets, you have the choice of a variety of different forms for a rational function, including (1) the product of two linear terms divided by a quadratic term, (2) the product of three linear terms divided by a quadratic term, and (3) the product of one linear and one quadratic term divided by a quadratic term. In each case, you enter the values of the parameters in each factor and the spreadsheet displays the corresponding graph instantly, as well as the equation of that rational function.
Exploring Vertical Asymptotes of Rational Functions This DIGMath module lets you explore the vertical asymptotes of rational functions of the form f(x) = (x - a)(x - b)/(x - c)(x - d). You can enter the values for the four parameters and the spreadsheet produces the graph of the function, highlighting both the real roots and the vertical asymptotes. As you vary the parameters c and d, you can see how the vertical asymptotes change accordingly to give a dynamic feel to the location of the asymptotes and their correspondence to the roots of the denominator.
The Fibonacci Sequence and the Golden Ratio This DIGMath module lets you explore the Fibonacci Sequence {1, 1, 2, 3, 5, 8, 13, ...} in which each term is the sum of the previous two terms. It draws the graph of the points in the sequence along with a visual depiction of how each term is the sum of the preceding two. It also draws the values of the ratios of the successive terms to illustrate how they converge to the Golden Ratio and the graph of the differences between each pair of successive ratios and the value of the Golden Ratio to highlight how the convergence is oscillatory in manner.
The Lucas Sequence and the Golden Ratio This DIGMath spreadsheet lets you explore the Lucas Sequence, which is an extension of the Fibonacci Sequence in which each entry is likewise the sum of the previous two, but the first two terms can be selected rather than being 1 and 1. The spreadsheet draws the graph of the points in the sequence along with a visual depiction of how each term is the sum of the preceding two. It also draws the ratios of the successive terms in the sequence to illustrate how they also converge to the Golden Ratio and the graph of the differences between each pair of successive ratios and the value of the Golden Ratio to highlight how the convergence is oscillatory in manner.
Variations on the Fibonacci Sequence This DIGMath module lets you investigate some variations on the Fibonacci Sequence in which the underlying rule is G_n+₂ = aG_n+₁ + bG_n, where a and b are two constant multiples and the two starting values can also be selected. The spreadsheet displays a chart showing the points in the resulting sequence and a second chart showing the successive ratios of the terms in the sequence. For many values of a and b, the terms of the sequence increase rapidly in an exponential-type pattern and the successive ratios approach a limiting value. For other values of a and b, the terms of the sequence oscillate in a periodic pattern, as do the successive ratios. Associated with the relationship among the G's is a quadratic polynomial r²- ar - b = 0, known as the characteristic polynomial. When this quadratic has two real roots (depending on the sign of the discriminant), the sequence of the successive ratios approach a limiting value and when there are a pair of complex conjugate roots, the sequence of values oscillate periodically and the successive ratios also oscillate periodically. The spreadsheet also has a chart of the graph of this polynomial and it is reasonably evident that the larger of the two real roots is equal to the limiting value for the successive ratios.
The Tribonacci Sequence This DIGMath module lets you explore the so-called Tribonacci Sequence as an extention of the Fibonacci Sequence. Instead of each term being the sum of the previous two terms, each term is now the sum of the previous three terms. For instance, if the first three terms are 1, 1, and 3, then the following term is 5 and the one after that is 9. The spreadsheet draws the graph of the points in the sequence along with a visual depiction of how each term is the sum of the preceding three. It also draws the values of the ratios of the successive terms to illustrate how they converge to a fixed limit (not the Golden Ratio) and the graph of the differences between each pair of successive ratios and the value of the limiting value to highlight how the convergence is oscillatory in manner.
The Fibonacci-like Sequence with Differences instead of Sums This DIGMath spreadsheet lets you explore a variation on the Fibonacci Sequence in which each entry is the difference of the previous two terms. You can select the first two terms using sliders to determine the specific sequence based on them. The spreadsheet draws the graph of the points in the sequence along with a visual depiction of how each term is the difference of the preceding two. Surprisingly, the terms in this sequence are periodic with period 6.
The Fibonacci-like Sequence with Products instead of Sums This DIGMath spreadsheet lets you investigate a variation on the Fibonacci Sequence in which each entry is the product of the previous two terms. You can select the first two terms using sliders to determine the specific sequence based on them. The spreadsheet draws the graph of the points in the sequence. The exploration of this sequence suggests that you write out the terms in the solution as powers of the two initial values and look for a very surprisingly pattern in those powers.
Characteristic Polynomials and Metallic Ratios Every recurrence relationship, such as the Fibonacci Sequence F_n+₂ = F_n+₁ + F_n, that is linear with constant coefficients, has an associated characteristic polynomial such as r²- r - 1 = 0 one of whose real roots is the Golden Ratio. Similarly, the generalized Tribonacci Sequence G_n+₃ = G_n+₂ + G_n+₁ + G_n has as its characteristic polynomial r³ - r²- r - 1 = 0 and the limit of the successive terms is equal to its largest real root, known as the Silver Ratio. This spreadsheet lets you investigate graphically and numerically the characteristic polynomials for similar recurrance relationships up to 7th degree. The ratio of the succesive terms, which is equal to the largest real root, is designated as a different metallic ratio.

All of these files were developed under the support from a variety of grants from the National Science Foundation, to whom the authors are very appreciative.