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.
The side-angle-side
applet is a GeoGebra
applet that
lets you construct a triangle by specifying the lengths of two of the
sides and the included angle.
The angle-side-angle
applet is a GeoGebra
applet that
lets you construct a triangle by specifying the lengths of two of the
sides and the included angle.
The side-side-angle
applet is a GeoGebra
applet that
lets you construct a triangle by specifying the lengths of two of the
sides and an non-included angle. If such a triangle can be
constructed, it may not be unique.
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.
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.
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.