# Math Applets at SLU - Below Calculus

These following collection of applets are designed for use in mathematics courses below the level of calculus.  Another page collects applets for calculus courses.  Some of the applets were developed at SLU and some have been developed elsewhere and are included by permission.  If you would like to host the applets locally, please contact Mike May, S.J.

Applets for College Algebra
Applets for Triangles in Geometry
Applets for Vectors
Applets for Trigonometry/Pre-Calculus
Applets for Misc Other Courses
Applets for Advanced Graphing

## Applets for College Algebra

When working through the understanding of various kinds of functions it is useful to be able to graph a function with parameters a, b, and c, in the definition of the function, with the parameters controlled by sliders.  Moving the sliders lets the student explore families of functions.  The Families of Functions Applet is a GeoGebra applet for looking at the graph of such a family.  Similarly, the Family of Graphs Applet is a JCM applet designed to look at families of functions.

The Translations and Compressions Applet is a GeoGebra applet that lets you compare the graph of a function with the graph of a function transformed by translations and compressions or expansions.

The Function Composition Applet is a JCM applet that links together the graph of two functions with the graph of the function defined by the composition of these two functions.  The Visual Composition of Functions Applet is a Banchoff applet that looks at composition of functions by using three variables, with y=g(x) and z=f(y).  The projection of the curve obtained from the intersection of the two surfaces is then either the graph of f, g, or fg, depending on the plane it is projected onto.

The Linear Equations page connects the normal ways of defining a line, either by giving two points, or by giving a point and a slope, or by giving the slope and intercept.  Each window connects a set of information with a line and derives the other presentations of the line.

The Quadratic Equations applet is a GeoGebra applet that lets you connect geometric and algebraic approaches to looking at quadratic equations.

Conic Sections - The Ellipses and Hyperbolas applet lets you specify a conic section by specifying the length from the center to a vertex on the major axis and the length to a focus.  You get either an ellipse or a hyperbola, depending on which length is bigger.  The Parabolas applet looks at Parabolas defined by a focus and a directrix.

## Applets for Triangles in Geometry

### Triangle Construction

The following applets are designed to allow the user explore triangle constructions given basic data.  The applets then provide side length and angle measurements.

### Theorem Illustration

The Pythagorean Theorem applet is a GeoGebra applet that walks the user through a proof of that theorem.

The Law of Sines Applet is designed to illustrate the law of sines.  The user drags three points to define a triangle.  The applet computes the length of the three sides and the measure of the three angles.  It then computes the ratio of the length of each side to the values of the sine of the opposite angle.  It is a Geometer's Sketchpad applet.

The Triangle Middle Applet visualizes three theorems in geometry.  For each theorem, you define a method of cutting a triangle in half.  (Use angle bisector, or perpendicular bisector of a side, or line from vertex to midpoint of opposite side.)  Each theorem says the three dividing lines all intersect in a single point.  The applet lets you see this on a triangle that can be distorted and stretched.  It also shows a related construction that is useful in the proof that the three lines connect in a single point.  It is a Geometer's Sketchpad applet.

A collection of GeoGebra applets look at these theorems one at a time:

• The angle bisector applet looks at the theorem that the three angle bisectors of a triangle all meet in a single point.  The first frame lets the student explore the result with various triangles.  A second construction adds details that give a framework for a proof.
• The side bisector applet looks at the theorem that the three perpendicular bisectors of the sides of a triangle all meet in a single point.  The first frame lets the student explore the result with various triangles.  A second construction adds details that give a framework for a proof.
• The concurrent medians applet looks at the theorem that the three medians of a triangle all meet in a single point.  The first frame lets the student explore the result with various triangles.  A second construction gives a framework for establishing a lemma about tiling an area with congruent triangles.  The third construction uses the lemma to adds details that give a framework for a proof.
• The concurrent altitudes applet looks at the theorem that the three altitudes of a triangle all meet in a single point.  The first frame lets the student explore the result with various triangles.  A second construction adds details that give a framework for a proof.

## Applets for Vectors

### Vectors

The Adding Vectors Applet lets you visualize two vectors with their sum and difference and the vectors measured both in polar and rectangular coordinates.  It is a Geometer's Sketchpad applet.

The Projection, Dot and Cross Product Applet lets you visualize the projection of one vector onto another.  It also gives a visualization of the dot and cross product.  It is a Geometer's Sketchpad applet.

Two similar applets Banchoff applets are available to look at cross products.  They differ in the mechanism used to control the vectors.  The first  Cross Product applet gives a visual approach to cross products of vectors in 3-space with the coordinates of the two vectors controlled by sliders.  The second  Cross Product applet gives a visual approach to cross products of vectors in 3-space with the coordinates of the two vectors controlled by text boxes.

## Applets for Trigonometry/Pre-Calculus

The Trig Functions Applet connects the 6 standard trig functions with lengths of line segments from a  diagram connected to the unit circle.  It is a Geometer's Sketchpad applet.   The same idea is covered with a Trig Review Applet using GeoGebra.

The Sin Curve Fitting applet lets you fit a sinusoidal curve to a pair of specified points.

## Applets for Other Courses

The Reflection Action Applet lets the student explore the action on the standard basis obtained from reflecting across a line through the origin.  It is a Geometer's Sketchpad applet

## Applets for Advanced Graphing

The Cross Section applet looks at how the graph of a 2 variable function is built up from the cross graphs of the cross sections, which are each graphs of single variable functions.

The Level Curve applet builds up the graph from its level sets.  It connects the contours of a three dimensional surface with the contour graph obtained by looking down the z-axis.  (Currently, this one has bugs with Safari.)

The Parameterized Curve Applet is set up to look at a parameterized curve in x-y as a space curve in x-y-t space.  Rotating the axes lets you see the parameterized curve as well as the x(t) and y(t) curves in the x-t and y-t planes respectively.  The Parameterized Surface Applet lets you see the graph of a parameterized surface.  As we did with the cross section applet above, we can also see a wire frame constructed from the graphs of lines in the domain.

The 2D Grapher Applet page collects 3 applets for graphing in the plane.  One can either use the Cartesian Grapher to graph y as a function of x, or use the Parameterized Grapher to graph x and y as functions of t, or use the Polar Grapher to graph r as a function of theta.

The 3D Grapher Applet page collects 3 applets for graphing in 3-space.  One can either use the Cartesian Grapher to graph z as a function of x and y, or use the Cylindrical Grapher to graph r as a function of theta and z, or use the Spherical Grapher to graph r as a function of theta and phi.

Several of the applets above are referred to as JCM applets.  This refers to an NSF sponsored  project,  the Java Components for Mathematics project at Hobart and William Smith Colleges, that produced a series of JCM-Applets that are quite useful for calculus and pre-calculus.

Several of the applets above are referred to as Banchoff applets.  This refers to work done at Brown University under the direction of Tom Banchoff.  The project produced a demonstration applet that is quite flexible and can be configured for demonstrations and projects.  The configurations can be saved as a set of parameters used in calling the applet.  These applets are collected on the Banchoff Applet Page.

Several of the applets above are referred to as Geometer's Sketchpad® Applets.  That means that the applets were created with Java Sketchpad®, a component of Geometer's Sketchpad®.  More accurately, they were created as a set of parameter's that can be used to call the Sketchpad® Applet in a particular configuration.  These applets are collected on the Geometer's Sketchpad® Applets page.

Several of the applets above are referred to as GeoGebra Applets.  That means that the applets were created with GeoGebra, an open source program created by Markus Hohenwarter.  The program is available from http://www.geogebra.at/.  They applets are more precisely configurations of a master applet.  These applets are collected on the GeoGebra Applets page.

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Last modified November 17, 2007 by Mike May, S.J.