Introduction to Functions

Tutorial which is an introduction to functions including the four representations of a function, domain, and range.

Drill on finding domains of functions.
Computer program to visualize a function as "a collection of arrows from the domain to the range."

Piecewise Defined Functions

Tutorial on piecewise defined functions.

A LiveMath Notebook to explore the graphing of a piecewise defined function.
Drawing the graph of a piecewise defined function with
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator
Computer programs that will drawing the graph of a piecewise defined function.

Even and Odd Functions

Tutorial on even and odd functions.

A LiveMath Notebook to visualize when a function is symmetric with respect to the y-axis.
A LiveMath Notebook to visualize when a function is symmetric with respect to the origin.
A LiveMath Notebook to determine which function in a family of functions is odd.
A quiz on determining which functions are even, odd or neither.
Computer programs to determine graphically whether a function is even or odd or neither.

Polynomials

Tutorial on polynomial functions.

A LiveMath Notebook to compare a polynomial of degree n with a monomial of degree n.
A LiveMath Notebook to compare polynomials of different degrees.
Finding the roots of polynomials with
- TI-86 Graphing Calculator [Using Flash]

Rational Functions

Tutorial on rational functions.

Geometric Transformations of Functions

Tutorial on the translation, compression, stretching and reflection of graphs of functions.

A Java applet to explore graphically the effect of transforming a graph with translations, vertical and horizontal stretching and reflections.
A quiz to explore graphically the effect of transforming a graph with translations, vertical and horizontal stretching and reflections.
Given a function, draw the graphs of functions obtained by shifting the graph of a given function using the
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator
Computer programs that will easily draw the graphs of functions obtained by shifting the graph of a given function.
Computer programs that will easily draw the graphs of functions obtained by shifting the graph of a given function.

Algebra of Functions

A LiveMath Notebook on the graphing of the sum of two functions.
Drawing the graph of the sum of two functions with a
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator

Composition of Functions

Tutorial on the composition of functions.

A LiveMath Notebook computing the composition of functions.
A Java applet illustrating the geometrical construction of the composition of two functions
Computer programs which, given the equations of two functions, easily draws the graphs of the compositions of these functions. Also,
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator
Computer program which when given the graphs of two functions will draw the graphs of the compositions of these functions.
Drill problems on writing a function as the (nontrivial) composition of two other functions.

Trigonometric Functions

A listing of trigonometric identities.
Computer programs that draw the graphs of trigonometric functions, sine and cosine, considered as circular functions.
Computer programs that draw the graph of the tangent function considered as circular functions.

Exponential Functions

Tutorial on exponential functions.

A Javascript module to approximate a number raised to an irrational power.
A LiveMath animation of a family of exponential functions.
A LiveMath Notebook which can be used to determine which of exponential functions or polynomial functions grows more rapidly.
Computer programs which can be used to determine which of exponential functions or polynomial functions grows more rapidly. Also,
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator
LiveMath animations to explore graphically and symbolically the effect of transforming exponential functions.
A Java applet to explore graphically symbolically the effect of transforming exponential functions.
A LiveMath Notebook to approximate the exponential function y = e^x by the polynomials y = (1 + x/n)ⁿ.

Inverses of Functions

Tutorial on inverses of functions.

A Flash movie illustrating the use of the horizontal line test and the finding the equation of the inverse of a function.
Using Maple to find explicitly the inverse of a function and verify that it is the inverse.
A LiveMath Notebook to investigate which polynomials of degree 3 have inverses.
Drill on finding the inverses of functions.
Drill in finding the inverses of functions (includes inverse trigonometric functions).
Computer programs that will draw the graph of a function and its inverse. Also,
- TI-85 Graphing Calculator

Logarithms

Tutorial on logarithms.

A LiveMath animation of a family of logarithmic functions.
LiveMath animations to explore graphically and symbolically the effect of transforming logarithmic functions.
A Java applet to explore graphically and symbolically the effect of transforming logarithmic functions.
Computer programs that draw the graphs of the exponential and logarithm functions in order to visualize that they are inverses of each other.

Parameters and Functions

Tutorial on familes of functions with parameters.

A LiveMath animation to investigate a parameterized family of functions.
A LiveMath animation of a parameterized family of exponential functions.
A LiveMath animation of a family of logarithmic functions.
Computer programs which graph a family of functions with a parameter. Also,
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator
Computer programs which graph a family of functions with two parameters. Also,
- TI-86 Graphing Calculator [Using Flash]

Parametric Equations

Tutorial on parametric equations.

A LiveMath notebook to explore the graphing of a system of parametric equations in the plane.
Animation illustrating the drawing of the cycloid.
A LiveMath animation of a family of parametric equations in the plane.
Computer programs which graph a system of parametric equations in the plane. Also
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator
Computer programs which graph a system of parametric equations in the plane.
Computer programs which graph a triangle using a system of parametric equations in the plane.

Polar Coordinates

Tutorial on polar coordinates.

Computer programs which graph a function in polar coordinates. Also,
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator [Using Flash]
Computer programs that can be used to graph a family of functions with parameters in polar coordinates.
Computer programs that can be used to graph a family of functions with parameters in polar coordinates.

Graphing with Technology

Tutorial on how computer programs and graphing calculators plot the graphs of functions and some of the problems that this method of plotting has.

Problems in using a computer to draw the graph of a function. In this example, changing the number of points used in drawing the graph changes the graph significantly.
Slide show on which the page above is based.
A LiveMath notebook showing some problems in using a computer to draw the graph of a function. In this example, zooming in shows some features of the graph otherwise missed.
Computer programs and their problems in drawng the graph of a function with a discontinuity.
Computer programs and their problems in drawng the graph of the rational power of a variable.
Computer program that will plot the graph of a finite collection of ordered pairs.

Solving Equations

Tutorial on finding the intersections of the graphs of two functions.

A LiveMath notebook to graphically find the intersections of the graphs of two functions.
Computer programs to find the roots of an equation graphically by zooming in.
Find the coordinates of the points of intersections of two graphs using the
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator
Finding the roots of polynomials with
- TI-86 Graphing Calculator [Using Flash]
Drill in solving equations involving trigonometric functions.
Computer programs that can be used to solve an inequality graphically.
Computer programs that can be used to solve an inequality graphically.
Computer programs that can be used to solve an inequality graphically.
Computer programs that can be used to solve an inequality symbolically.

Curve Fitting

Tutorial on curve fitting.

A Java applet which finds and graph the least squares regression line, quadratic and cubic models which best approximates the data.
Computer programs to find and graph the least squares regression line. Also,
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator
Computer programs to find and graph the exponential model which best approximates the data. Also,
- TI-86 Graphing Calculator [Using Flash]
- TI-85 Graphing Calculator
Computer programs to find and graph the third degree polynomial model which best approximates the data.

Conic Sections

Computer programs that draw the graphs of various conic sections.
Computer programs that draw the graphs of various conic sections using polar coordinates.
A review of the concept of eccentricity through examining the effects of changing the value of the variable e in the polar equation of an ellipse.