Unit Circle Symmetry: a GeoGebraBook Exploration

I have completed a GeoGebraBook of nine Unit Circle Symmetry applets, which you can use by clicking here. There are three applets per type of symmetry on the unit circle, one focusing on the unit circle only, and the other two linking unit circle properties to patterns in the graphs of the sine and cosine functions.

When two angle expressions, such as \theta and (\pi -\theta ), exhibit symmetry on the unit circle, mastery of unit circle symmetries and reference angles often allow function arguments to be simplified. Mastery of symmetries and reference angles will also be very handy when expanding inverse trigonometric function results to describe all possible answers to a problem.

Suggestions for improvements to these applets, or additional applets, are always welcome via comments on this post.

Angle Measures

Suppose nobody had ever thought of measuring the size of an angle, and someone asked you “How can I describe the size of an angle?” What approach might you take in answering this question?

You might start by arbitrarily picking some angle, any angle, such as angle ABC in the image below, and call its measure “1”. All other angles could be
Continue reading Angle Measures

Function Translation: a Transformation

A function has been “translated” when it has been moved in a way that does not change its shape or rotate it in any way. Such changes are a subset of the possible “transformations” of a function, and can be accomplished through vertical translation, horizontal translation, or both.

Imagine a graph that has been drawn on tracing paper that was loosely lain over a printed set of axes. If you move the tracing paper left, right, up, or down some distance, without rotating it in any way, you are “translating” the graph.

It is very useful to be able to interpret a function as a translation of a “parent function”. Understanding the behavior of

g(x)=(x-3)^2+1

is much easier if you think of it as the graph of a simpler looking function that you should already be very familiar with

f(x)=x^2

shifted right by 3, and up by 1.

This approach helps us understand the behavior of Continue reading Function Translation: a Transformation

GeoGebra is Great!

I finally figured out how to post GeoGebra applets on Google Sites so that I can link to them from WordPress.com (which does not support them).

The greatest value of GeoGebra, or Geometer’s Sketchpad, or other such packages, in my eyes is their ability to help students dynamically visualize the effect each constant has on the graph of an equation. I find them an invaluable “thinking aid” as I ponder a new equation form, and they help me formulate my own answers to questions such as “why does it do that?”

Check out ones for Linear functions:

Quadratic functions:

Exponential and Logarithmic functions:

Rational functions:

Trigonometric functions: